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mathematics Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...
, the differential geometry of surfaces deals with the
differential geometry Differential geometry is a mathematical discipline that studies the geometry of smooth shapes and smooth spaces, otherwise known as smooth manifolds. It uses the techniques of differential calculus, integral calculus, linear algebra and mult ...
of
smooth Smooth may refer to: Mathematics * Smooth function, a function that is infinitely differentiable; used in calculus and topology * Smooth manifold, a differentiable manifold for which all the transition maps are smooth functions * Smooth algebrai ...
surface A surface, as the term is most generally used, is the outermost or uppermost layer of a physical object or space. It is the portion or region of the object that can first be perceived by an observer using the senses of sight and touch, and is ...
s with various additional structures, most often, a
Riemannian metric In differential geometry, a Riemannian manifold or Riemannian space , so called after the German mathematician Bernhard Riemann, is a real, smooth manifold ''M'' equipped with a positive-definite inner product ''g'p'' on the tangent space '' ...
. Surfaces have been extensively studied from various perspectives: ''extrinsically'', relating to their
embedding In mathematics, an embedding (or imbedding) is one instance of some mathematical structure contained within another instance, such as a group that is a subgroup. When some object X is said to be embedded in another object Y, the embedding is g ...
in
Euclidean space Euclidean space is the fundamental space of geometry, intended to represent physical space. Originally, that is, in Euclid's ''Elements'', it was the three-dimensional space of Euclidean geometry, but in modern mathematics there are Euclidea ...
and ''intrinsically'', reflecting their properties determined solely by the distance within the surface as measured along curves on the surface. One of the fundamental concepts investigated is the
Gaussian curvature In differential geometry, the Gaussian curvature or Gauss curvature of a surface at a point is the product of the principal curvatures, and , at the given point: K = \kappa_1 \kappa_2. The Gaussian radius of curvature is the reciprocal of . ...
, first studied in depth by
Carl Friedrich Gauss Johann Carl Friedrich Gauss (; german: Gauß ; la, Carolus Fridericus Gauss; 30 April 177723 February 1855) was a German mathematician and physicist who made significant contributions to many fields in mathematics and science. Sometimes refer ...
, who showed that curvature was an intrinsic property of a surface, independent of its isometric embedding in Euclidean space. Surfaces naturally arise as graphs of functions of a pair of variables, and sometimes appear in parametric form or as loci associated to space curves. An important role in their study has been played by
Lie group In mathematics, a Lie group (pronounced ) is a group that is also a differentiable manifold. A manifold is a space that locally resembles Euclidean space, whereas groups define the abstract concept of a binary operation along with the addi ...
s (in the spirit of the
Erlangen program In mathematics, the Erlangen program is a method of characterizing geometries based on group theory and projective geometry. It was published by Felix Klein in 1872 as ''Vergleichende Betrachtungen über neuere geometrische Forschungen.'' It is nam ...
), namely the
symmetry group In group theory, the symmetry group of a geometric object is the group of all transformations under which the object is invariant, endowed with the group operation of composition. Such a transformation is an invertible mapping of the amb ...
s of the
Euclidean plane In mathematics, the Euclidean plane is a Euclidean space of dimension two. That is, a geometric setting in which two real quantities are required to determine the position of each point ( element of the plane), which includes affine notions ...
, the
sphere A sphere () is a geometrical object that is a three-dimensional analogue to a two-dimensional circle. A sphere is the set of points that are all at the same distance from a given point in three-dimensional space.. That given point is the c ...
and the
hyperbolic plane In mathematics, hyperbolic geometry (also called Lobachevskian geometry or Bolyai– Lobachevskian geometry) is a non-Euclidean geometry. The parallel postulate of Euclidean geometry is replaced with: :For any given line ''R'' and point ' ...
. These Lie groups can be used to describe surfaces of constant Gaussian curvature; they also provide an essential ingredient in the modern approach to intrinsic differential geometry through connections. On the other hand, extrinsic properties relying on an embedding of a surface in Euclidean space have also been extensively studied. This is well illustrated by the non-linear Euler–Lagrange equations in the
calculus of variations The calculus of variations (or Variational Calculus) is a field of mathematical analysis that uses variations, which are small changes in functions and functionals, to find maxima and minima of functionals: mappings from a set of functions t ...
: although Euler developed the one variable equations to understand
geodesics In geometry, a geodesic () is a curve representing in some sense the shortest path ( arc) between two points in a surface, or more generally in a Riemannian manifold. The term also has meaning in any differentiable manifold with a connection. ...
, defined independently of an embedding, one of Lagrange's main applications of the two variable equations was to
minimal surfaces In mathematics, a minimal surface is a surface that locally minimizes its area. This is equivalent to having zero mean curvature (see definitions below). The term "minimal surface" is used because these surfaces originally arose as surfaces tha ...
, a concept that can only be defined in terms of an embedding.


History

The volumes of certain
quadric surface In mathematics, a quadric or quadric surface (quadric hypersurface in higher dimensions), is a generalization of conic sections ( ellipses, parabolas, and hyperbolas). It is a hypersurface (of dimension ''D'') in a -dimensional space, and it is ...
s of
revolution In political science, a revolution (Latin: ''revolutio'', "a turn around") is a fundamental and relatively sudden change in political power and political organization which occurs when the population revolts against the government, typically due ...
were calculated by
Archimedes Archimedes of Syracuse (;; ) was a Greek mathematician, physicist, engineer, astronomer, and inventor from the ancient city of Syracuse in Sicily. Although few details of his life are known, he is regarded as one of the leading scientis ...
. The development of
calculus Calculus, originally called infinitesimal calculus or "the calculus of infinitesimals", is the mathematics, mathematical study of continuous change, in the same way that geometry is the study of shape, and algebra is the study of generalizati ...
in the seventeenth century provided a more systematic way of computing them. Curvature of general surfaces was first studied by
Euler Leonhard Euler ( , ; 15 April 170718 September 1783) was a Swiss mathematician, physicist, astronomer, geographer, logician and engineer who founded the studies of graph theory and topology and made pioneering and influential discoveries in ...
. In 1760 he proved a formula for the curvature of a plane section of a surface and in 1771 he considered surfaces represented in a parametric form.
Monge Gaspard Monge, Comte de Péluse (9 May 1746 – 28 July 1818) was a French mathematician, commonly presented as the inventor of descriptive geometry, (the mathematical basis of) technical drawing, and the father of differential geometry. During ...
laid down the foundations of their theory in his classical memoir ''L'application de l'analyse à la géometrie'' which appeared in 1795. The defining contribution to the theory of surfaces was made by
Gauss Johann Carl Friedrich Gauss (; german: Gauß ; la, Carolus Fridericus Gauss; 30 April 177723 February 1855) was a German mathematician and physicist who made significant contributions to many fields in mathematics and science. Sometimes refer ...
in two remarkable papers written in 1825 and 1827. This marked a new departure from tradition because for the first time Gauss considered the ''intrinsic'' geometry of a surface, the properties which are determined only by the geodesic distances between points on the surface independently of the particular way in which the surface is located in the ambient Euclidean space. The crowning result, the
Theorema Egregium Gauss's ''Theorema Egregium'' (Latin for "Remarkable Theorem") is a major result of differential geometry, proved by Carl Friedrich Gauss in 1827, that concerns the curvature of surfaces. The theorem says that Gaussian curvature can be determi ...
of Gauss, established that the
Gaussian curvature In differential geometry, the Gaussian curvature or Gauss curvature of a surface at a point is the product of the principal curvatures, and , at the given point: K = \kappa_1 \kappa_2. The Gaussian radius of curvature is the reciprocal of . ...
is an intrinsic invariant, i.e. invariant under local
isometries In mathematics, an isometry (or congruence, or congruent transformation) is a distance-preserving transformation between metric spaces, usually assumed to be bijective. The word isometry is derived from the Ancient Greek: ἴσος ''isos'' mea ...
. This point of view was extended to higher-dimensional spaces by
Riemann Georg Friedrich Bernhard Riemann (; 17 September 1826 – 20 July 1866) was a German mathematician who made contributions to analysis, number theory, and differential geometry. In the field of real analysis, he is mostly known for the first rig ...
and led to what is known today as
Riemannian geometry Riemannian geometry is the branch of differential geometry that studies Riemannian manifolds, smooth manifolds with a ''Riemannian metric'', i.e. with an inner product on the tangent space at each point that varies smoothly from point to point ...
. The nineteenth century was the golden age for the theory of surfaces, from both the topological and the differential-geometric point of view, with most leading geometers devoting themselves to their study. Darboux collected many results in his four-volume treatise ''Théorie des surfaces'' (1887–1896).


Overview

It is intuitively quite familiar to say that the leaf of a plant, the surface of a glass, or the shape of a face, are curved in certain ways, and that all of these shapes, even after ignoring any distinguishing markings, have certain geometric features which distinguish one from another. The differential geometry of surfaces is concerned with a mathematical understanding of such phenomena. The study of this field, which was initiated in its modern form in the 1700s, has led to the development of higher-dimensional and abstract geometry, such as
Riemannian geometry Riemannian geometry is the branch of differential geometry that studies Riemannian manifolds, smooth manifolds with a ''Riemannian metric'', i.e. with an inner product on the tangent space at each point that varies smoothly from point to point ...
and
general relativity General relativity, also known as the general theory of relativity and Einstein's theory of gravity, is the geometric theory of gravitation published by Albert Einstein in 1915 and is the current description of gravitation in modern physics ...
. The essential mathematical object is that of a regular surface. Although conventions vary in their precise definition, these form a general class of subsets of three-dimensional
Euclidean space Euclidean space is the fundamental space of geometry, intended to represent physical space. Originally, that is, in Euclid's ''Elements'', it was the three-dimensional space of Euclidean geometry, but in modern mathematics there are Euclidea ...
() which capture part of the familiar notion of "surface." By analyzing the class of curves which lie on such a surface, and the degree to which the surfaces force them to curve in , one can associate to each point of the surface two numbers, called the principal curvatures. Their average is called the mean curvature of the surface, and their product is called the Gaussian curvature. There are many classic examples of regular surfaces, including: * familiar examples such as planes, cylinders, and spheres *
minimal surface In mathematics, a minimal surface is a surface that locally minimizes its area. This is equivalent to having zero mean curvature (see definitions below). The term "minimal surface" is used because these surfaces originally arose as surfaces tha ...
s, which are defined by the property that their mean curvature is zero at every point. The best-known examples are
catenoid In geometry, a catenoid is a type of surface, arising by rotating a catenary curve about an axis (a surface of revolution). It is a minimal surface, meaning that it occupies the least area when bounded by a closed space. It was formally descri ...
s and
helicoid The helicoid, also known as helical surface, after the plane and the catenoid, is the third minimal surface to be known. Description It was described by Euler in 1774 and by Jean Baptiste Meusnier in 1776. Its name derives from its similarity ...
s, although many more have been discovered. Minimal surfaces can also be defined by properties to do with
surface area The surface area of a solid object is a measure of the total area that the surface of the object occupies. The mathematical definition of surface area in the presence of curved surfaces is considerably more involved than the definition of ...
, with the consequence that they provide a mathematical model for the shape of
soap film Soap films are thin layers of liquid (usually water-based) surrounded by air. For example, if two soap bubbles come into contact, they merge and a thin film is created in between. Thus, foams are composed of a network of films connected by Platea ...
s when stretched across a wire frame *
ruled surface In geometry, a surface is ruled (also called a scroll) if through every point of there is a straight line that lies on . Examples include the plane, the lateral surface of a cylinder or cone, a conical surface with elliptical directrix, t ...
s, which are surfaces that have at least one straight line running through every point; examples include the cylinder and the
hyperboloid In geometry, a hyperboloid of revolution, sometimes called a circular hyperboloid, is the surface generated by rotating a hyperbola around one of its principal axes. A hyperboloid is the surface obtained from a hyperboloid of revolution by def ...
of one sheet. A surprising result of
Carl Friedrich Gauss Johann Carl Friedrich Gauss (; german: Gauß ; la, Carolus Fridericus Gauss; 30 April 177723 February 1855) was a German mathematician and physicist who made significant contributions to many fields in mathematics and science. Sometimes refer ...
, known as the
theorema egregium Gauss's ''Theorema Egregium'' (Latin for "Remarkable Theorem") is a major result of differential geometry, proved by Carl Friedrich Gauss in 1827, that concerns the curvature of surfaces. The theorem says that Gaussian curvature can be determi ...
, showed that the Gaussian curvature of a surface, which by its definition has to do with how curves on the surface change directions in three dimensional space, can actually be measured by the lengths of curves lying on the surfaces together with the angles made when two curves on the surface intersect. Terminologically, this says that the Gaussian curvature can be calculated from the first fundamental form (also called
metric tensor In the mathematical field of differential geometry, a metric tensor (or simply metric) is an additional structure on a manifold (such as a surface) that allows defining distances and angles, just as the inner product on a Euclidean space allow ...
) of the surface. The second fundamental form, by contrast, is an object which encodes how lengths and angles of curves on the surface are distorted when the curves are pushed off of the surface. Despite measuring different aspects of length and angle, the first and second fundamental forms are not independent from one another, and they satisfy certain constraints called the Gauss-Codazzi equations. A major theorem, often called the fundamental theorem of the differential geometry of surfaces, asserts that whenever two objects satisfy the Gauss-Codazzi constraints, they will arise as the first and second fundamental forms of a regular surface. Using the first fundamental form, it is possible to define new objects on a regular surface. Geodesics are curves on the surface which satisfy a certain second-order
ordinary differential equation In mathematics, an ordinary differential equation (ODE) is a differential equation whose unknown(s) consists of one (or more) function(s) of one variable and involves the derivatives of those functions. The term ''ordinary'' is used in contrast ...
which is specified by the first fundamental form. They are very directly connected to the study of lengths of curves; a geodesic of sufficiently short length will always be the curve of ''shortest'' length on the surface which connects its two endpoints. Thus, geodesics are fundamental to the optimization problem of determining the shortest path between two given points on a regular surface. One can also define parallel transport along any given curve, which gives a prescription for how to deform a tangent vector to the surface at one point of the curve to tangent vectors at all other points of the curve. The prescription is determined by a first-order
ordinary differential equation In mathematics, an ordinary differential equation (ODE) is a differential equation whose unknown(s) consists of one (or more) function(s) of one variable and involves the derivatives of those functions. The term ''ordinary'' is used in contrast ...
which is specified by the first fundamental form. The above concepts are essentially all to do with multivariable calculus. The Gauss-Bonnet theorem is a more global result, which relates the Gaussian curvature of a surface together with its topological type. It asserts that the average value of the Gaussian curvature is completely determined by the
Euler characteristic In mathematics, and more specifically in algebraic topology and polyhedral combinatorics, the Euler characteristic (or Euler number, or Euler–Poincaré characteristic) is a topological invariant, a number that describes a topological spac ...
of the surface together with its surface area. The notion of
Riemannian manifold In differential geometry, a Riemannian manifold or Riemannian space , so called after the German mathematician Bernhard Riemann, is a real, smooth manifold ''M'' equipped with a positive-definite inner product ''g'p'' on the tangent space ...
and
Riemann surface In mathematics, particularly in complex analysis, a Riemann surface is a connected one-dimensional complex manifold. These surfaces were first studied by and are named after Bernhard Riemann. Riemann surfaces can be thought of as deformed ver ...
are two generalizations of the regular surfaces discussed above. In particular, essentially all of the theory of regular surfaces as discussed here has a generalization in the theory of Riemannian manifolds. This is not the case for Riemann surfaces, although every regular surface gives an example of a Riemann surface.


Regular surfaces in Euclidean space


Definition

It is intuitively clear that a sphere is smooth, while a cone or a pyramid, due to their vertex or edges, are not. The notion of a "regular surface" is a formalization of the notion of a smooth surface. The definition utilizes the local representation of a surface via maps between
Euclidean space Euclidean space is the fundamental space of geometry, intended to represent physical space. Originally, that is, in Euclid's ''Elements'', it was the three-dimensional space of Euclidean geometry, but in modern mathematics there are Euclidea ...
s. There is a standard notion of smoothness for such maps; a map between two open subsets of Euclidean space is smooth if its partial derivatives of every order exist at every point of the domain. The following gives three equivalent ways to present the definition; the middle definition is perhaps the most visually intuitive, as it essentially says that a regular surface is a subset of which is locally the graph of a smooth function (whether over a region in the plane, the plane, or the plane). The homeomorphisms appearing in the first definition are known as local parametrizations or local coordinate systems or local charts on . The equivalence of the first two definitions asserts that, around any point on a regular surface, there always exist local parametrizations of the form , , or , known as Monge patches. Functions as in the third definition are called local defining functions. The equivalence of all three definitions follows from the implicit function theorem. Given any two local parametrizations and of a regular surface, the composition is necessarily smooth as a map between open subsets of . This shows that any regular surface naturally has the structure of a
smooth manifold In mathematics, a differentiable manifold (also differential manifold) is a type of manifold that is locally similar enough to a vector space to allow one to apply calculus. Any manifold can be described by a collection of charts (atlas). One ma ...
, with a smooth atlas being given by the inverses of local parametrizations. In the classical theory of differential geometry, surfaces are usually studied only in the regular case. It is, however, also common to study non-regular surfaces, in which the two partial derivatives and of a local parametrization may fail to be
linearly independent In the theory of vector spaces, a set of vectors is said to be if there is a nontrivial linear combination of the vectors that equals the zero vector. If no such linear combination exists, then the vectors are said to be . These concepts are ...
. In this case, may have singularities such as
cuspidal edge In mathematics, a cusp, sometimes called spinode in old texts, is a point on a curve where a moving point must reverse direction. A typical example is given in the figure. A cusp is thus a type of singular point of a curve. For a plane curve def ...
s. Such surfaces are typically studied in
singularity theory In mathematics, singularity theory studies spaces that are almost manifolds, but not quite. A string can serve as an example of a one-dimensional manifold, if one neglects its thickness. A singularity can be made by balling it up, dropping it ...
. Other weakened forms of regular surfaces occur in
computer-aided design Computer-aided design (CAD) is the use of computers (or ) to aid in the creation, modification, analysis, or optimization of a design. This software is used to increase the productivity of the designer, improve the quality of design, improve co ...
, where a surface is broken apart into disjoint pieces, with the derivatives of local parametrizations failing to even be continuous along the boundaries. Simple examples. A simple example of a regular surface is given by the 2-sphere ; this surface can be covered by six Monge patches (two of each of the three types given above), taking . It can also be covered by two local parametrizations, using
stereographic projection In mathematics, a stereographic projection is a perspective projection of the sphere, through a specific point on the sphere (the ''pole'' or ''center of projection''), onto a plane (the ''projection plane'') perpendicular to the diameter thro ...
. The set is a
torus of revolution In geometry, a torus (plural tori, colloquially donut or doughnut) is a surface of revolution generated by revolving a circle in three-dimensional space about an axis that is coplanar with the circle. If the axis of revolution does not tou ...
with radii and . It is a regular surface; local parametrizations can be given of the form :f(s,t)=\big((R \cos s +r)\cos t, (R \cos s +r) \sin t, R\sin s\big). The
hyperboloid In geometry, a hyperboloid of revolution, sometimes called a circular hyperboloid, is the surface generated by rotating a hyperbola around one of its principal axes. A hyperboloid is the surface obtained from a hyperboloid of revolution by def ...
on two sheets is a regular surface; it can be covered by two Monge patches, with . The
helicoid The helicoid, also known as helical surface, after the plane and the catenoid, is the third minimal surface to be known. Description It was described by Euler in 1774 and by Jean Baptiste Meusnier in 1776. Its name derives from its similarity ...
appears in the theory of
minimal surfaces In mathematics, a minimal surface is a surface that locally minimizes its area. This is equivalent to having zero mean curvature (see definitions below). The term "minimal surface" is used because these surfaces originally arose as surfaces tha ...
. It is covered by a single local parametrization, .


Tangent vectors and normal vectors

Let be a regular surface in , and let be an element of . Using any of the above definitions, one can single out certain vectors in as being tangent to at , and certain vectors in as being orthogonal to at . One sees that the ''
tangent space In mathematics, the tangent space of a manifold generalizes to higher dimensions the notion of '' tangent planes'' to surfaces in three dimensions and ''tangent lines'' to curves in two dimensions. In the context of physics the tangent space to a ...
'' or ''
tangent plane In geometry, the tangent line (or simply tangent) to a plane curve at a given point is the straight line that "just touches" the curve at that point. Leibniz defined it as the line through a pair of infinitely close points on the curve. More ...
'' to at , which is defined to consist of all tangent vectors to at , is a two-dimensional linear subspace of ; it is often denoted by . The ''
normal space In topology and related branches of mathematics, a normal space is a topological space ''X'' that satisfies Axiom T4: every two disjoint closed sets of ''X'' have disjoint open neighborhoods. A normal Hausdorff space is also called a T4 space. T ...
'' to at , which is defined to consist of all normal vectors to at , is a one-dimensional linear subspace of which is orthogonal to the tangent space . As such, at each point of , there are two normal vectors of unit length (unit normal vectors). It is useful to note that the unit normal vectors at can be given in terms of local parametrizations, Monge patches, or local defining functions, via the formulas :\pm\left.\frac\_,\qquad \pm\left.\frac\_,\qquad\text\qquad\pm\frac, following the same notations as in the previous definitions. It is also useful to note an "intrinsic" definition of tangent vectors, which is typical of the generalization of regular surface theory to the setting of
smooth manifold In mathematics, a differentiable manifold (also differential manifold) is a type of manifold that is locally similar enough to a vector space to allow one to apply calculus. Any manifold can be described by a collection of charts (atlas). One ma ...
s. It defines the tangent space as an abstract two-dimensional real vector space, rather than as a linear subspace of . In this definition, one says that a tangent vector to at is an assignment, to each local parametrization with , of two numbers and , such that for any other local parametrization with (and with corresponding numbers and ), one has :\beginX^1\\ X^2\end=A_\begin(X')^1\\ (X')^2\end, where is the
Jacobian matrix In vector calculus, the Jacobian matrix (, ) of a vector-valued function of several variables is the matrix of all its first-order partial derivatives. When this matrix is square, that is, when the function takes the same number of variable ...
of the mapping , evaluated at the point . The collection of tangent vectors to at naturally has the structure of a two-dimensional vector space. A tangent vector in this sense corresponds to a tangent vector in the previous sense by considering the vector :X^1\frac+X^2\frac. in . The Jacobian condition on and ensures, by the
chain rule In calculus, the chain rule is a formula that expresses the derivative of the composition of two differentiable functions and in terms of the derivatives of and . More precisely, if h=f\circ g is the function such that h(x)=f(g(x)) for every , ...
, that this vector does not depend on . For smooth functions on a surface, vector fields (i.e. tangent vector fields) have an important interpretation as first order operators or derivations. Let S be a regular surface, U an open subset of the plane and f:U\rightarrow S a coordinate chart. If V=f(U), the space C^\infty(U) can be identified with C^\infty(V). Similarly f identifies vector fields on U with vector fields on V. Taking standard variables and , a vector field has the form X= a\partial_u + b\partial_v, with and smooth functions. If X is a vector field and g is a smooth function, then Xg is also a smooth function. The first order differential operator X is a ''derivation'', i.e. it satisfies the Leibniz rule X(gh)= (Xg) h + g (Xh). For vector fields and it is simple to check that the operator ,YXY-YX is a derivation corresponding to a vector field. It is called the
Lie bracket In mathematics, a Lie algebra (pronounced ) is a vector space \mathfrak g together with an operation called the Lie bracket, an alternating bilinear map \mathfrak g \times \mathfrak g \rightarrow \mathfrak g, that satisfies the Jacobi identi ...
,Y/math>. It is skew-symmetric ,Y- ,X/math> and satisfies the Jacobi identity: : X,YZ] + Y,ZX] + Z,XY]=0. In summary, vector fields on U or V form a
Lie algebra In mathematics, a Lie algebra (pronounced ) is a vector space \mathfrak g together with an operation called the Lie bracket, an alternating bilinear map \mathfrak g \times \mathfrak g \rightarrow \mathfrak g, that satisfies the Jacobi identi ...
under the Lie bracket.


First and second fundamental forms, the shape operator, and the curvature

Let be a regular surface in . Given a local parametrization and a unit normal vector field to , one defines the following objects as real-valued or matrix-valued functions on . The first fundamental form depends only on , and not on . The fourth column records the way in which these functions depend on , by relating the functions etc., arising for a different choice of local parametrization, , to those arising for . Here denotes the
Jacobian matrix In vector calculus, the Jacobian matrix (, ) of a vector-valued function of several variables is the matrix of all its first-order partial derivatives. When this matrix is square, that is, when the function takes the same number of variable ...
of . The key relation in establishing the formulas of the fourth column is then :\begin\frac\\ \frac\end=A\begin\frac\\ \frac\end, as follows by the
chain rule In calculus, the chain rule is a formula that expresses the derivative of the composition of two differentiable functions and in terms of the derivatives of and . More precisely, if h=f\circ g is the function such that h(x)=f(g(x)) for every , ...
. By a direct calculation with the matrix defining the shape operator, it can be checked that the Gaussian curvature is the
determinant In mathematics, the determinant is a scalar value that is a function of the entries of a square matrix. It characterizes some properties of the matrix and the linear map represented by the matrix. In particular, the determinant is nonzero if a ...
of the shape operator, the mean curvature is half of the trace of the shape operator, and the principal curvatures are the
eigenvalue In linear algebra, an eigenvector () or characteristic vector of a linear transformation is a nonzero vector that changes at most by a scalar factor when that linear transformation is applied to it. The corresponding eigenvalue, often denote ...
s of the shape operator; moreover the Gaussian curvature is the product of the principal curvatures and the mean curvature is their sum. These observations can also be formulated as definitions of these objects. These observations also make clear that the last three rows of the fourth column follow immediately from the previous row, as
similar matrices In linear algebra, two ''n''-by-''n'' matrices and are called similar if there exists an invertible ''n''-by-''n'' matrix such that B = P^ A P . Similar matrices represent the same linear map under two (possibly) different bases, with being ...
have identical determinant, trace, and eigenvalues. It is fundamental to note , , and are all necessarily positive. This ensures that the matrix inverse in the definition of the shape operator is well-defined, and that the principal curvatures are real numbers. Note also that a negation of the choice of unit normal vector field will negate the second fundamental form, the shape operator, the mean curvature, and the principal curvatures, but will leave the Gaussian curvature unchanged. In summary, this has shown that, given a regular surface , the Gaussian curvature of can be regarded as a real-valued function on ; relative to a choice of unit normal vector field on all of , the two principal curvatures and the mean curvature are also real-valued functions on . Geometrically, the first and second fundamental forms can be viewed as giving information on how moves around in as moves around in . In particular, the first fundamental form encodes how quickly moves, while the second fundamental form encodes the extent to which its motion is in the direction of the normal vector . In other words, the second fundamental form at a point encodes the length of the orthogonal projection from to the tangent plane to at ; in particular it gives the quadratic function which best approximates this length. This thinking can be made precise by the formulas :\begin \lim_\frac &= 0\\ \lim_\frac &= 0, \end as follows directly from the definitions of the fundamental forms and
Taylor's theorem In calculus, Taylor's theorem gives an approximation of a ''k''-times differentiable function around a given point by a polynomial of degree ''k'', called the ''k''th-order Taylor polynomial. For a smooth function, the Taylor polynomial is th ...
in two dimensions. The principal curvatures can be viewed in the following way. At a given point of , consider the collection of all planes which contain the orthogonal line to . Each such plane has a curve of intersection with , which can be regarded as a
plane curve In mathematics, a plane curve is a curve in a plane that may be either a Euclidean plane, an affine plane or a projective plane. The most frequently studied cases are smooth plane curves (including piecewise smooth plane curves), and algebraic ...
inside of the plane itself. The two principal curvatures at are the maximum and minimum possible values of the curvature of this plane curve at , as the plane under consideration rotates around the normal line. The following summarizes the calculation of the above quantities relative to a Monge patch . Here and denote the two partial derivatives of , with analogous notation for the second partial derivatives. The second fundamental form and all subsequent quantities are calculated relative to the given choice of unit normal vector field.


Christoffel symbols, Gauss–Codazzi equations, and the Theorema Egregium

Let be a regular surface in . The
Christoffel symbols In mathematics and physics, the Christoffel symbols are an array of numbers describing a metric connection. The metric connection is a specialization of the affine connection to surfaces or other manifolds endowed with a metric, allowing distanc ...
assign, to each local parametrization , eight functions on , defined by :\begin\Gamma_^1&\Gamma_^1&\Gamma_^1&\Gamma_^1\\ \Gamma_^2&\Gamma_^2&\Gamma_^2&\Gamma_^2\end=\beginE&F\\ F&G\end^\begin\frac\frac&\frac\frac&\frac\frac &\frac-\frac\frac\\ \frac-\frac\frac&\frac\frac&\frac\frac&\frac\frac\end. They can also be defined by the following formulas, in which is a unit normal vector field along and are the corresponding components of the second fundamental form: :\begin \frac&=\Gamma_^1\frac+\Gamma_^2\frac+Ln\\ \frac&=\Gamma_^1\frac+\Gamma_^2\frac+Mn\\ \frac&=\Gamma_^1\frac+\Gamma_^2\frac+Nn. \end The key to this definition is that , , and form a basis of at each point, relative to which each of the three equations uniquely specifies the Christoffel symbols as coordinates of the second partial derivatives of . The choice of unit normal has no effect on the Christoffel symbols, since if is exchanged for its negation, then the components of the second fundamental form are also negated, and so the signs of are left unchanged. The second definition shows, in the context of local parametrizations, that the Christoffel symbols are geometrically natural. Although the formulas in the first definition appear less natural, they have the importance of showing that the Christoffel symbols can be calculated from the first fundamental form, which is not immediately apparent from the second definition. The equivalence of the definitions can be checked by directly substituting the first definition into the second, and using the definitions of . The Codazzi equations assert that :\begin \frac-\frac&=L\Gamma_^1 + M(\Gamma_^2-\Gamma_^1) - N\Gamma_^2\\ \frac-\frac&=L\Gamma_^1 + M(\Gamma_^2-\Gamma_^1) - N\Gamma_^2. \end These equations can be directly derived from the second definition of Christoffel symbols given above; for instance, the first Codazzi equation is obtained by differentiating the first equation with respect to , the second equation with respect to , subtracting the two, and taking the dot product with . The Gauss equation asserts that :\begin KE&=\frac-\frac+\Gamma_^2\Gamma_^1+\Gamma_^2\Gamma_^2-\Gamma_^2\Gamma_^1-\Gamma_^2\Gamma_^2\\ KF&=\frac-\frac+\Gamma_^2\Gamma_^1-\Gamma_^2\Gamma_^1\\ KG&=\frac-\frac+\Gamma_^1\Gamma_^1+\Gamma_^1\Gamma_^2-\Gamma_^1\Gamma_^1-\Gamma_^1\Gamma_^2 \end These can be similarly derived as the Codazzi equations, with one using the
Weingarten equations The Weingarten equations give the expansion of the derivative of the unit normal vector to a surface in terms of the first derivatives of the position vector of a point on the surface. These formulas were established in 1861 by the German mathematic ...
instead of taking the dot product with . Although these are written as three separate equations, they are identical when the definitions of the Christoffel symbols, in terms of the first fundamental form, are substituted in. There are many ways to write the resulting expression, one of them derived in 1852 by Brioschi using a skillful use of determinants: :K = \frac\det\begin-\frac + \frac - \frac & \frac & \frac - \frac\\ \frac - \frac & E & F \\ \frac& F & G\end-\frac\det\begin 0 & \frac & \frac\\ \frac & E & F \\ \frac& F & G\end. When the Christoffel symbols are considered as being defined by the first fundamental form, the Gauss and Codazzi equations represent certain constraints between the first and second fundamental forms. The Gauss equation is particularly noteworthy, as it shows that the Gaussian curvature can be computed directly from the first fundamental form, without the need for any other information; equivalently, this says that can actually be written as a function of , even though the individual components cannot. This is known as the
theorema egregium Gauss's ''Theorema Egregium'' (Latin for "Remarkable Theorem") is a major result of differential geometry, proved by Carl Friedrich Gauss in 1827, that concerns the curvature of surfaces. The theorem says that Gaussian curvature can be determi ...
, and was a major discovery of
Carl Friedrich Gauss Johann Carl Friedrich Gauss (; german: Gauß ; la, Carolus Fridericus Gauss; 30 April 177723 February 1855) was a German mathematician and physicist who made significant contributions to many fields in mathematics and science. Sometimes refer ...
. It is particularly striking when one recalls the geometric definition of the Gaussian curvature of as being defined by the maximum and minimum radii of osculating circles; they seem to be fundamentally defined by the geometry of how bends within . Nevertheless, the theorem shows that their product can be determined from the "intrinsic" geometry of , having only to do with the lengths of curves along and the angles formed at their intersections. As said by Marcel Berger: The Gauss-Codazzi equations can also be succinctly expressed and derived in the language of
connection form In mathematics, and specifically differential geometry, a connection form is a manner of organizing the data of a connection using the language of moving frames and differential forms. Historically, connection forms were introduced by Élie Carta ...
s due to
Élie Cartan Élie Joseph Cartan (; 9 April 1869 – 6 May 1951) was an influential French mathematician who did fundamental work in the theory of Lie groups, differential systems (coordinate-free geometric formulation of PDEs), and differential geometr ...
. In the language of
tensor calculus In mathematics, tensor calculus, tensor analysis, or Ricci calculus is an extension of vector calculus to tensor fields (tensors that may vary over a manifold, e.g. in spacetime). Developed by Gregorio Ricci-Curbastro and his student Tullio Levi ...
, making use of natural metrics and connections on tensor bundles, the Gauss equation can be written as and the two Codazzi equations can be written as and ; the complicated expressions to do with Christoffel symbols and the first fundamental form are completely absorbed into the definitions of the covariant tensor derivative and the
scalar curvature In the mathematical field of Riemannian geometry, the scalar curvature (or the Ricci scalar) is a measure of the curvature of a Riemannian manifold. To each point on a Riemannian manifold, it assigns a single real number determined by the geome ...
. Pierre Bonnet proved that two quadratic forms satisfying the Gauss-Codazzi equations always uniquely determine an embedded surface locally. For this reason the Gauss-Codazzi equations are often called the fundamental equations for embedded surfaces, precisely identifying where the intrinsic and extrinsic curvatures come from. They admit generalizations to surfaces embedded in more general
Riemannian manifold In differential geometry, a Riemannian manifold or Riemannian space , so called after the German mathematician Bernhard Riemann, is a real, smooth manifold ''M'' equipped with a positive-definite inner product ''g'p'' on the tangent space ...
s.


Isometries

A diffeomorphism \varphi between open sets U and V in a regular surface S is said to be an
isometry In mathematics, an isometry (or congruence, or congruent transformation) is a distance-preserving transformation between metric spaces, usually assumed to be bijective. The word isometry is derived from the Ancient Greek: ἴσος ''isos'' ...
if it preserves the metric, i.e. the first fundamental form. Thus for every point p in U and tangent vectors w_1,\,\, w_2 at p, there are equalities : E(p) w_1\cdot w_1 + 2F(p) w_1\cdot w_2 + G(p) w_2\cdot w_2= E(\varphi(p)) \varphi^\prime(w_1)\cdot \varphi^\prime(w_1) +2F(\varphi(p)) \varphi^\prime(w_1)\cdot \varphi^\prime(w_2) + G (\varphi(p)) \varphi^\prime(w_1)\cdot \varphi^\prime(w_2). In terms of the inner product coming from the first fundamental form, this can be rewritten as :(w_1,w_2)_p=(\varphi^\prime(w_1),\varphi^\prime(w_2))_. On the other hand, the length of a parametrized curve \gamma(t)=(x(t),y(t)) can be calculated as :L(\gamma)=\int_a^b \sqrt \, dt and, if the curve lies in U, the rules for change of variables show that :L(\varphi\circ \gamma) = L(\gamma). Conversely if \varphi preserves the lengths of all parametrized in curves then \varphi is an isometry. Indeed, for suitable choices of \gamma, the tangent vectors \dot and \dot give arbitrary tangent vectors w_1 and w_2. The equalities must hold for all choice of tangent vectors w_1 and w_2 as well as \varphi^\prime(w_1) and \varphi^\prime(w_2), so that (\varphi^\prime(w_1),\varphi^\prime(w_2))_ = (w_1,w_1)_p. A simple example of an isometry is provided by two parametrizations f_1 and f_2 of an open set U into regular surfaces S_1 and S_2. If E_1=E_2, F_1=F_2 and G_1=G_2, then \varphi=f_2\circ f_1^ is an isometry of f_1(U) onto f_2(U). The cylinder and the plane give examples of surfaces that are locally isometric but which cannot be extended to an isometry for topological reasons. As another example, the
catenoid In geometry, a catenoid is a type of surface, arising by rotating a catenary curve about an axis (a surface of revolution). It is a minimal surface, meaning that it occupies the least area when bounded by a closed space. It was formally descri ...
and
helicoid The helicoid, also known as helical surface, after the plane and the catenoid, is the third minimal surface to be known. Description It was described by Euler in 1774 and by Jean Baptiste Meusnier in 1776. Its name derives from its similarity ...
are locally isometric.


Covariant derivatives

A tangential vector field on assigns, to each in , a tangent vector to at . According to the "intrinsic" definition of tangent vectors given above, a tangential vector field then assigns, to each local parametrization , two real-valued functions and on , so that :X_p=X^1\big(f^(p)\big)\frac\Big, _+X^2\big(f^(p)\big)\frac\Big, _ for each in . One says that is smooth if the functions and are smooth, for any choice of . According to the other definitions of tangent vectors given above, one may also regard a tangential vector field on as a map such that is contained in the tangent space for each in . As is common in the more general situation of
smooth manifold In mathematics, a differentiable manifold (also differential manifold) is a type of manifold that is locally similar enough to a vector space to allow one to apply calculus. Any manifold can be described by a collection of charts (atlas). One ma ...
s, tangential vector fields can also be defined as certain differential operators on the space of smooth functions on . The covariant derivatives (also called "tangential derivatives") of
Tullio Levi-Civita Tullio Levi-Civita, (, ; 29 March 1873 – 29 December 1941) was an Italian mathematician, most famous for his work on absolute differential calculus ( tensor calculus) and its applications to the theory of relativity, but who also made signi ...
and
Gregorio Ricci-Curbastro Gregorio Ricci-Curbastro (; 12January 1925) was an Italian mathematician. He is most famous as the discoverer of tensor calculus. With his former student Tullio Levi-Civita, he wrote his most famous single publication, a pioneering work on th ...
provide a means of differentiating smooth tangential vector fields. Given a tangential vector field and a tangent vector to at , the covariant derivative is a certain tangent vector to at . Consequently, if and are both tangential vector fields, then can also be regarded as a tangential vector field; iteratively, if , , and are tangential vector fields, the one may compute , which will be another tangential vector field. There are a few ways to define the covariant derivative; the first below uses the Christoffel symbols and the "intrinsic" definition of tangent vectors, and the second is more manifestly geometric. Given a tangential vector field and a tangent vector to at , one defines to be the tangent vector to which assigns to a local parametrization the two numbers :(\nabla_YX)^k=D_X^k\Big, _+\sum_^2\sum_^2\big(\Gamma_^kX^j\big)\Big, _Y^i,\qquad(k=1,2) where is the
directional derivative In mathematics, the directional derivative of a multivariable differentiable (scalar) function along a given vector v at a given point x intuitively represents the instantaneous rate of change of the function, moving through x with a velocity ...
.Do Carmo, page 242 This is often abbreviated in the less cumbersome form , making use of
Einstein notation In mathematics, especially the usage of linear algebra in Mathematical physics, Einstein notation (also known as the Einstein summation convention or Einstein summation notation) is a notational convention that implies summation over a set of ...
and with the locations of function evaluation being implicitly understood. This follows a standard prescription in
Riemannian geometry Riemannian geometry is the branch of differential geometry that studies Riemannian manifolds, smooth manifolds with a ''Riemannian metric'', i.e. with an inner product on the tangent space at each point that varies smoothly from point to point ...
for obtaining a connection from a
Riemannian metric In differential geometry, a Riemannian manifold or Riemannian space , so called after the German mathematician Bernhard Riemann, is a real, smooth manifold ''M'' equipped with a positive-definite inner product ''g'p'' on the tangent space '' ...
. It is a fundamental fact that the vector :(\nabla_YX)^1\frac+(\nabla_YX)^2\frac in is independent of the choice of local parametization , although this is rather tedious to check. One can also define the covariant derivative by the following geometric approach, which does not make use of Christoffel symbols or local parametrizations. Let be a vector field on , viewed as a function . Given any curve , one may consider the composition . As a map between Euclidean spaces, it can be differentiated at any input value to get an element of . The
orthogonal projection In linear algebra and functional analysis, a projection is a linear transformation P from a vector space to itself (an endomorphism) such that P\circ P=P. That is, whenever P is applied twice to any vector, it gives the same result as if it wer ...
of this vector onto defines the covariant derivative . Although this is a very geometrically clean definition, it is necessary to show that the result only depends on and , and not on and ; local parametrizations can be used for this small technical argument. It is not immediately apparent from the second definition that covariant differentiation depends only on the first fundamental form of ; however, this is immediate from the first definition, since the Christoffel symbols can be defined directly from the first fundamental form. It is straightforward to check that the two definitions are equivalent. The key is that when one regards as a -valued function, its differentiation along a curve results in second partial derivatives ; the Christoffel symbols enter with orthogonal projection to the tangent space, due to the formulation of the Christoffel symbols as the tangential components of the second derivatives of relative to the basis , , . This is discussed in the above section. The right-hand side of the three Gauss equations can be expressed using covariant differentiation. For instance, the right-hand side :\frac-\frac+\Gamma_^2\Gamma_^1+\Gamma_^2\Gamma_^2-\Gamma_^2\Gamma_^1-\Gamma_^2\Gamma_^2 can be recognized as the second coordinate of :\nabla_\nabla_\frac-\nabla_\nabla_\frac relative to the basis , , as can be directly verified using the definition of covariant differentiation by Christoffel symbols. In the language of
Riemannian geometry Riemannian geometry is the branch of differential geometry that studies Riemannian manifolds, smooth manifolds with a ''Riemannian metric'', i.e. with an inner product on the tangent space at each point that varies smoothly from point to point ...
, this observation can also be phrased as saying that the right-hand sides of the Gauss equations are various components of the
Ricci curvature In differential geometry, the Ricci curvature tensor, named after Gregorio Ricci-Curbastro, is a geometric object which is determined by a choice of Riemannian or pseudo-Riemannian metric on a manifold. It can be considered, broadly, as a measur ...
of the
Levi-Civita connection In Riemannian or pseudo Riemannian geometry (in particular the Lorentzian geometry of general relativity), the Levi-Civita connection is the unique affine connection on the tangent bundle of a manifold (i.e. affine connection) that preserves ...
of the first fundamental form, when interpreted as a
Riemannian metric In differential geometry, a Riemannian manifold or Riemannian space , so called after the German mathematician Bernhard Riemann, is a real, smooth manifold ''M'' equipped with a positive-definite inner product ''g'p'' on the tangent space '' ...
.


Examples


Surfaces of revolution

A surface of revolution is obtained by rotating a curve in the -plane about the -axis. Such surfaces include spheres, cylinders, cones, tori, and the
catenoid In geometry, a catenoid is a type of surface, arising by rotating a catenary curve about an axis (a surface of revolution). It is a minimal surface, meaning that it occupies the least area when bounded by a closed space. It was formally descri ...
. The general
ellipsoid An ellipsoid is a surface that may be obtained from a sphere by deforming it by means of directional scalings, or more generally, of an affine transformation. An ellipsoid is a quadric surface;  that is, a surface that may be defined as th ...
s,
hyperboloid In geometry, a hyperboloid of revolution, sometimes called a circular hyperboloid, is the surface generated by rotating a hyperbola around one of its principal axes. A hyperboloid is the surface obtained from a hyperboloid of revolution by def ...
s, and
paraboloid In geometry, a paraboloid is a quadric surface that has exactly one axis of symmetry and no center of symmetry. The term "paraboloid" is derived from parabola, which refers to a conic section that has a similar property of symmetry. Every plan ...
s are not. Suppose that the curve is parametrized by : x= c_1(s),\,\, z=c_2(s) with drawn from an interval . If is never zero, if and are never both equal to zero, and if and are both smooth, then the corresponding surface of revolution :S=\Big\ will be a regular surface in . A local parametrization is given by :f(s,t)=\big(c_1(s)\cos t, c_1(s)\sin t,c_2(s)\big). Relative to this parametrization, the geometric data is: In the special case that the original curve is parametrized by arclength, i.e. , one can differentiate to find . On substitution into the Gaussian curvature, one has the simplified :K=-\frac\qquad\text\qquad H=c_1'(s)c_2''(s)-c_2'(s)c_1''(s)+\frac. The simplicity of this formula makes it particularly easy to study the class of rotationally symmetric surfaces with constant Gaussian curvature. By reduction to the alternative case that , one can study the rotationally symmetric minimal surfaces, with the result that any such surface is part of a plane or a scaled catenoid. Each constant- curve on can be parametrized as a geodesic; a constant- curve on can be parametrized as a geodesic if and only if is equal to zero. Generally, geodesics on are governed by
Clairaut's relation In classical differential geometry, Clairaut's relation, named after Alexis Claude de Clairaut, is a formula that characterizes the great circle paths on the unit sphere. The formula states that if γ is a parametrization of a great circle th ...
.


Quadric surfaces

Consider the quadric surface defined by : + +=1. This surface admits a parametrization :x=\sqrt,\,\, y=\sqrt, \,\, z=\sqrt. The Gaussian curvature and mean curvature are given by :K= ,\,\,K_m=-(u+v)\sqrt.


Ruled surfaces

A ruled surface is one which can be generated by the motion of a straight line in . Choosing a ''directrix'' on the surface, i.e. a smooth unit speed curve orthogonal to the straight lines, and then choosing to be unit vectors along the curve in the direction of the lines, the velocity vector and satisfy :u\cdot v=0, \,\,\, u\, =1,\,\,\, v\, =1. The surface consists of points :c(t) + s\cdot u(t) as and vary. Then, if :a=\, u_t\, , \,\, b=u_t\cdot v, \,\, \alpha=-\frac, \,\, \beta=\frac, the Gaussian and mean curvature are given by :K=- ,\,\, K_m=-. The Gaussian curvature of the ruled surface vanishes if and only if and are proportional, This condition is equivalent to the surface being the
envelope An envelope is a common packaging item, usually made of thin, flat material. It is designed to contain a flat object, such as a letter or card. Traditional envelopes are made from sheets of paper cut to one of three shapes: a rhombus, a ...
of the planes along the curve containing the tangent vector and the orthogonal vector , i.e. to the surface being developable along the curve. More generally a surface in has vanishing Gaussian curvature near a point if and only if it is developable near that point. (An equivalent condition is given below in terms of the metric.)


Minimal surfaces

In 1760
Lagrange Joseph-Louis Lagrange (born Giuseppe Luigi Lagrangiacalculus of variations The calculus of variations (or Variational Calculus) is a field of mathematical analysis that uses variations, which are small changes in functions and functionals, to find maxima and minima of functionals: mappings from a set of functions t ...
involving integrals in one variable to two variables. He had in mind the following problem: Such a surface is called a minimal surface. In 1776
Jean Baptiste Meusnier Jean Baptiste Marie Charles Meusnier de la Place (Tours, 19 June 1754 — le Pont de Cassel, near Mainz, 13 June 1793) was a French mathematician, engineer and Revolutionary general. He is best known for Meusnier's theorem on the curvature ...
showed that the differential equation derived by Lagrange was equivalent to the vanishing of the mean curvature of the surface: Minimal surfaces have a simple interpretation in real life: they are the shape a soap film will assume if a wire frame shaped like the curve is dipped into a soap solution and then carefully lifted out. The question as to whether a minimal surface with given boundary exists is called
Plateau's problem In mathematics, Plateau's problem is to show the existence of a minimal surface with a given boundary, a problem raised by Joseph-Louis Lagrange in 1760. However, it is named after Joseph Plateau who experimented with soap films. The problem ...
after the Belgian physicist
Joseph Plateau Joseph Antoine Ferdinand Plateau (14 October 1801 – 15 September 1883) was a Belgian physicist and mathematician. He was one of the first people to demonstrate the illusion of a moving image. To do this, he used counterrotating disks with repe ...
who carried out experiments on soap films in the mid-nineteenth century. In 1930
Jesse Douglas Jesse Douglas (3 July 1897 – 7 September 1965) was an American mathematician and Fields Medalist known for his general solution to Plateau's problem. Life and career He was born to a Jewish family in New York City, the son of Sarah (née ...
and
Tibor Radó Tibor Radó (June 2, 1895 – December 29, 1965) was a Hungarian mathematician who moved to the United States after World War I. Biography Radó was born in Budapest and between 1913 and 1915 attended the Polytechnic Institute, studying ...
gave an affirmative answer to Plateau's problem (Douglas was awarded one of the first
Fields medal The Fields Medal is a prize awarded to two, three, or four mathematicians under 40 years of age at the International Congress of the International Mathematical Union (IMU), a meeting that takes place every four years. The name of the award h ...
s for this work in 1936). Many explicit examples of minimal surface are known explicitly, such as the
catenoid In geometry, a catenoid is a type of surface, arising by rotating a catenary curve about an axis (a surface of revolution). It is a minimal surface, meaning that it occupies the least area when bounded by a closed space. It was formally descri ...
, the
helicoid The helicoid, also known as helical surface, after the plane and the catenoid, is the third minimal surface to be known. Description It was described by Euler in 1774 and by Jean Baptiste Meusnier in 1776. Its name derives from its similarity ...
, the
Scherk surface In mathematics, a Scherk surface (named after Heinrich Scherk) is an example of a minimal surface. Scherk described two complete embedded minimal surfaces in 1834; his first surface is a doubly periodic surface, his second surface is singly peri ...
and the
Enneper surface In differential geometry and algebraic geometry, the Enneper surface is a self-intersecting surface that can be described parametrically by: \begin x &= \tfrac u \left(1 - \tfracu^2 + v^2\right), \\ y &= \tfrac v \left(1 - \tfracv^2 + u^2\righ ...
. There has been extensive research in this area, summarised in . In particular a result of Osserman shows that if a minimal surface is non-planar, then its image under the Gauss map is dense in .


Surfaces of constant Gaussian curvature

If a surface has constant Gaussian curvature, it is called a surface of constant curvature. *The unit
sphere A sphere () is a geometrical object that is a three-dimensional analogue to a two-dimensional circle. A sphere is the set of points that are all at the same distance from a given point in three-dimensional space.. That given point is the c ...
in has constant Gaussian curvature +1. *The Euclidean plane and the
cylinder A cylinder (from ) has traditionally been a three-dimensional solid, one of the most basic of curvilinear geometric shapes. In elementary geometry, it is considered a prism with a circle as its base. A cylinder may also be defined as an ...
both have constant Gaussian curvature 0. *The surfaces of revolution with have constant Gaussian curvature –1. Particular cases are obtained by taking , and . The latter case is the classical pseudosphere generated by rotating a
tractrix In geometry, a tractrix (; plural: tractrices) is the curve along which an object moves, under the influence of friction, when pulled on a horizontal plane by a line segment attached to a pulling point (the ''tractor'') that moves at a right ...
around a central axis. In 1868 Eugenio Beltrami showed that the geometry of the pseudosphere was directly related to that of the
hyperbolic plane In mathematics, hyperbolic geometry (also called Lobachevskian geometry or Bolyai– Lobachevskian geometry) is a non-Euclidean geometry. The parallel postulate of Euclidean geometry is replaced with: :For any given line ''R'' and point ' ...
, discovered independently by
Lobachevsky Nikolai Ivanovich Lobachevsky ( rus, Никола́й Ива́нович Лобаче́вский, p=nʲikɐˈlaj ɪˈvanəvʲɪtɕ ləbɐˈtɕɛfskʲɪj, a=Ru-Nikolai_Ivanovich_Lobachevsky.ogg; – ) was a Russian mathematician and geometer, kn ...
(1830) and Bolyai (1832). Already in 1840, F. Minding, a student of Gauss, had obtained trigonometric formulas for the pseudosphere identical to those for the hyperbolic plane. The intrinsic geometry of this surface is now better understood in terms of the Poincaré metric on the
upper half plane In mathematics, the upper half-plane, \,\mathcal\,, is the set of points in the Cartesian plane with > 0. Complex plane Mathematicians sometimes identify the Cartesian plane with the complex plane, and then the upper half-plane corresponds to t ...
or the
unit disc In mathematics, the open unit disk (or disc) around ''P'' (where ''P'' is a given point in the plane), is the set of points whose distance from ''P'' is less than 1: :D_1(P) = \.\, The closed unit disk around ''P'' is the set of points whose d ...
, and has been described by other models such as the
Klein model Klein may refer to: People * Klein (surname) *Klein (musician) Places * Klein (crater), a lunar feature * Klein, Montana, United States *Klein, Texas, United States *Klein (Ohm), a river of Hesse, Germany, tributary of the Ohm *Klein River, a ri ...
or the
hyperboloid model In geometry, the hyperboloid model, also known as the Minkowski model after Hermann Minkowski, is a model of ''n''-dimensional hyperbolic geometry in which points are represented by points on the forward sheet ''S''+ of a two-sheeted hyperbo ...
, obtained by considering the two-sheeted hyperboloid in three-dimensional
Minkowski space In mathematical physics, Minkowski space (or Minkowski spacetime) () is a combination of three-dimensional Euclidean space and time into a four-dimensional manifold where the spacetime interval between any two events is independent of the iner ...
, where . Each of these surfaces of constant curvature has a transitive
Lie group In mathematics, a Lie group (pronounced ) is a group that is also a differentiable manifold. A manifold is a space that locally resembles Euclidean space, whereas groups define the abstract concept of a binary operation along with the addi ...
of symmetries. This group theoretic fact has far-reaching consequences, all the more remarkable because of the central role these special surfaces play in the geometry of surfaces, due to Poincaré's
uniformization theorem In mathematics, the uniformization theorem says that every simply connected Riemann surface is conformally equivalent to one of three Riemann surfaces: the open unit disk, the complex plane, or the Riemann sphere. The theorem is a generalization ...
(see below). Other examples of surfaces with Gaussian curvature 0 include
cones A cone is a three-dimensional geometric shape that tapers smoothly from a flat base (frequently, though not necessarily, circular) to a point called the apex or vertex. A cone is formed by a set of line segments, half-lines, or lines conn ...
,
tangent developable In the mathematical study of the differential geometry of surfaces, a tangent developable is a particular kind of developable surface obtained from a curve in Euclidean space as the surface swept out by the tangent lines to the curve. Such a surf ...
s, and more generally any developable surface.


Local metric structure

For any surface embedded in Euclidean space of dimension 3 or higher, it is possible to measure the length of a curve on the surface, the angle between two curves and the area of a region on the surface. This structure is encoded infinitesimally in a
Riemannian metric In differential geometry, a Riemannian manifold or Riemannian space , so called after the German mathematician Bernhard Riemann, is a real, smooth manifold ''M'' equipped with a positive-definite inner product ''g'p'' on the tangent space '' ...
on the surface through ''line elements'' and ''area elements''. Classically in the nineteenth and early twentieth centuries only surfaces embedded in were considered and the metric was given as a 2×2
positive definite matrix In mathematics, a symmetric matrix M with real entries is positive-definite if the real number z^\textsfMz is positive for every nonzero real column vector z, where z^\textsf is the transpose of More generally, a Hermitian matrix (that is, a ...
varying smoothly from point to point in a local parametrization of the surface. The idea of local parametrization and change of coordinate was later formalized through the current abstract notion of a
manifold In mathematics, a manifold is a topological space that locally resembles Euclidean space near each point. More precisely, an n-dimensional manifold, or ''n-manifold'' for short, is a topological space with the property that each point has a n ...
, a topological space where the
smooth structure In mathematics, a smooth structure on a manifold allows for an unambiguous notion of smooth function. In particular, a smooth structure allows one to perform mathematical analysis on the manifold. Definition A smooth structure on a manifold M is ...
is given by local charts on the manifold, exactly as the planet Earth is mapped by
atlas An atlas is a collection of maps; it is typically a bundle of maps of Earth or of a region of Earth. Atlases have traditionally been bound into book form, but today many atlases are in multimedia formats. In addition to presenting geogra ...
es today. Changes of coordinates between different charts of the same region are required to be smooth. Just as contour lines on real-life maps encode changes in elevation, taking into account local distortions of the Earth's surface to calculate true distances, so the Riemannian metric describes distances and areas "in the small" in each local chart. In each local chart a Riemannian metric is given by smoothly assigning a 2×2 positive definite matrix to each point; when a different chart is taken, the matrix is transformed according to the
Jacobian matrix In vector calculus, the Jacobian matrix (, ) of a vector-valued function of several variables is the matrix of all its first-order partial derivatives. When this matrix is square, that is, when the function takes the same number of variable ...
of the coordinate change. The manifold then has the structure of a 2-dimensional
Riemannian manifold In differential geometry, a Riemannian manifold or Riemannian space , so called after the German mathematician Bernhard Riemann, is a real, smooth manifold ''M'' equipped with a positive-definite inner product ''g'p'' on the tangent space ...
.


Shape operator

The differential of the
Gauss map In differential geometry, the Gauss map (named after Carl F. Gauss) maps a surface in Euclidean space R3 to the unit sphere ''S''2. Namely, given a surface ''X'' lying in R3, the Gauss map is a continuous map ''N'': ''X'' → ''S''2 such that ' ...
can be used to define a type of extrinsic curvature, known as the shape operator or Weingarten map. This operator first appeared implicitly in the work of
Wilhelm Blaschke Wilhelm Johann Eugen Blaschke (13 September 1885 – 17 March 1962) was an Austrian mathematician working in the fields of differential and integral geometry. Education and career Blaschke was the son of mathematician Josef Blaschke, who taugh ...
and later explicitly in a treatise by Burali-Forti and Burgati. Since at each point of the surface, the tangent space is an
inner product space In mathematics, an inner product space (or, rarely, a Hausdorff pre-Hilbert space) is a real vector space or a complex vector space with an operation called an inner product. The inner product of two vectors in the space is a scalar, often ...
, the shape operator can be defined as a linear operator on this space by the formula : (S_x v, w) =(dn(v), w) for tangent vectors , (the inner product makes sense because and both lie in ). The right hand side is symmetric in and , so the shape operator is
self-adjoint In mathematics, and more specifically in abstract algebra, an element ''x'' of a *-algebra is self-adjoint if x^*=x. A self-adjoint element is also Hermitian, though the reverse doesn't necessarily hold. A collection ''C'' of elements of a st ...
on the tangent space. The eigenvalues of are just the principal curvatures and at . In particular the
determinant In mathematics, the determinant is a scalar value that is a function of the entries of a square matrix. It characterizes some properties of the matrix and the linear map represented by the matrix. In particular, the determinant is nonzero if a ...
of the shape operator at a point is the Gaussian curvature, but it also contains other information, since the
mean curvature In mathematics, the mean curvature H of a surface S is an ''extrinsic'' measure of curvature that comes from differential geometry and that locally describes the curvature of an embedded surface in some ambient space such as Euclidean space. The ...
is half the trace of the shape operator. The mean curvature is an extrinsic invariant. In intrinsic geometry, a cylinder is developable, meaning that every piece of it is intrinsically indistinguishable from a piece of a plane since its Gauss curvature vanishes identically. Its mean curvature is not zero, though; hence extrinsically it is different from a plane. Equivalently, the shape operator can be defined as a linear operator on tangent spaces, ''S''''p'': ''T''''p''''M''→''T''''p''''M''. If ''n'' is a unit normal field to ''M'' and ''v'' is a tangent vector then :S(v)=\pm \nabla_n (there is no standard agreement whether to use + or − in the definition). In general, the eigenvectors and eigenvalues of the shape operator at each point determine the directions in which the surface bends at each point. The eigenvalues correspond to the
principal curvatures In differential geometry, the two principal curvatures at a given point of a surface are the maximum and minimum values of the curvature as expressed by the eigenvalues of the shape operator at that point. They measure how the surface bends by ...
of the surface and the eigenvectors are the corresponding principal directions. The principal directions specify the directions that a curve embedded in the surface must travel to have maximum and minimum curvature, these being given by the principal curvatures.


Geodesic curves on a surface

Curves on a surface which minimize length between the endpoints are called
geodesic In geometry, a geodesic () is a curve representing in some sense the shortest path ( arc) between two points in a surface, or more generally in a Riemannian manifold. The term also has meaning in any differentiable manifold with a connecti ...
s; they are the shape that an elastic band stretched between the two points would take. Mathematically they are described using
ordinary differential equation In mathematics, an ordinary differential equation (ODE) is a differential equation whose unknown(s) consists of one (or more) function(s) of one variable and involves the derivatives of those functions. The term ''ordinary'' is used in contrast ...
s and the
calculus of variations The calculus of variations (or Variational Calculus) is a field of mathematical analysis that uses variations, which are small changes in functions and functionals, to find maxima and minima of functionals: mappings from a set of functions t ...
. The differential geometry of surfaces revolves around the study of geodesics. It is still an open question whether every Riemannian metric on a 2-dimensional local chart arises from an embedding in 3-dimensional Euclidean space: the theory of geodesics has been used to show this is true in the important case when the components of the metric are
analytic Generally speaking, analytic (from el, ἀναλυτικός, ''analytikos'') refers to the "having the ability to analyze" or "division into elements or principles". Analytic or analytical can also have the following meanings: Chemistry * ...
.


Geodesics

Given a piecewise smooth path in the chart for in , its ''length'' is defined by : L(c) = \int_a^b (E\dot^2 + 2F \dot\dot + G \dot^2)^\, dt and ''energy'' by : E(c) = \int_a^b (E\dot^2 + 2F \dot\dot + G \dot^2)\, dt. The length is independent of the parametrization of a path. By the Euler–Lagrange equations, if is a path minimising length, ''parametrized by arclength'', it must satisfy the
Euler equations 200px, Leonhard Euler (1707–1783) In mathematics and physics, many topics are named in honor of Swiss mathematician Leonhard Euler (1707–1783), who made many important discoveries and innovations. Many of these items named after Euler include ...
:\ddot + \Gamma_^1 \dot^2 + 2\Gamma_^1 \dot\dot+ \Gamma_^1\dot^2 =0 :\ddot+ \Gamma_^2 \dot^2 + 2\Gamma_^2 \dot\dot+ \Gamma_^2 \dot^2 =0 where the
Christoffel symbols In mathematics and physics, the Christoffel symbols are an array of numbers describing a metric connection. The metric connection is a specialization of the affine connection to surfaces or other manifolds endowed with a metric, allowing distanc ...
are given by :\Gamma_^k = \tfrac12 \sum_m g^ (\partial_j g_ + \partial_i g_ - \partial_m g_) where , , and is the inverse matrix to . A path satisfying the Euler equations is called a
geodesic In geometry, a geodesic () is a curve representing in some sense the shortest path ( arc) between two points in a surface, or more generally in a Riemannian manifold. The term also has meaning in any differentiable manifold with a connecti ...
. By the
Cauchy–Schwarz inequality The Cauchy–Schwarz inequality (also called Cauchy–Bunyakovsky–Schwarz inequality) is considered one of the most important and widely used inequalities in mathematics. The inequality for sums was published by . The corresponding inequality f ...
a path minimising energy is just a geodesic parametrised by arc length; and, for any geodesic, the parameter is proportional to arclength.


Geodesic curvature

The geodesic curvature at a point of a curve , parametrised by arc length, on an oriented surface is defined to be :k_g= \ddot(t)\cdot \mathbf(t). where is the "principal" unit normal to the curve in the surface, constructed by rotating the unit tangent vector through an angle of +90°. *The geodesic curvature at a point is an intrinsic invariant depending only on the metric near the point. *A unit speed curve on a surface is a geodesic if and only if its geodesic curvature vanishes at all points on the curve. *A unit speed curve in an embedded surface is a geodesic if and only if its acceleration vector is normal to the surface. The geodesic curvature measures in a precise way how far a curve on the surface is from being a geodesic.


Orthogonal coordinates

When throughout a coordinate chart, such as with the geodesic polar coordinates discussed below, the images of lines parallel to the - and -axes are
orthogonal In mathematics, orthogonality is the generalization of the geometric notion of '' perpendicularity''. By extension, orthogonality is also used to refer to the separation of specific features of a system. The term also has specialized meanings in ...
and provide
orthogonal coordinates In mathematics, orthogonal coordinates are defined as a set of ''d'' coordinates q = (''q''1, ''q''2, ..., ''q'd'') in which the coordinate hypersurfaces all meet at right angles (note: superscripts are indices, not exponents). A coordinate su ...
. If , then the Gaussian curvature is given by : K=- \left partial_x\left(\frac\right) +\partial_y\left(\frac\right)\right If in addition , so that , then the angle at the intersection between geodesic and the line = constant is given by the equation :\tan \varphi = H\cdot \frac. The derivative of is given by a classical derivative formula of Gauss: : \dot = -H_x \cdot \dot.


Geodesic polar coordinates

Once a metric is given on a surface and a base point is fixed, there is a unique geodesic connecting the base point to each sufficiently nearby point. The direction of the geodesic at the base point and the distance uniquely determine the other endpoint. These two bits of data, a direction and a magnitude, thus determine a tangent vector at the base point. The map from tangent vectors to endpoints smoothly sweeps out a neighbourhood of the base point and defines what is called the "exponential map", defining a local coordinate chart at that base point. The neighbourhood swept out has similar properties to balls in Euclidean space, namely any two points in it are joined by a unique geodesic. This property is called "geodesic convexity" and the coordinates are called "normal coordinates". The explicit calculation of normal coordinates can be accomplished by considering the differential equation satisfied by geodesics. The convexity properties are consequences of Gauss's lemma and its generalisations. Roughly speaking this lemma states that geodesics starting at the base point must cut the spheres of fixed radius centred on the base point at right angles. Geodesic polar coordinates are obtained by combining the exponential map with polar coordinates on tangent vectors at the base point. The Gaussian curvature of the surface is then given by the second order deviation of the metric at the point from the Euclidean metric. In particular the Gaussian curvature is an invariant of the metric, Gauss's celebrated ''
Theorema Egregium Gauss's ''Theorema Egregium'' (Latin for "Remarkable Theorem") is a major result of differential geometry, proved by Carl Friedrich Gauss in 1827, that concerns the curvature of surfaces. The theorem says that Gaussian curvature can be determi ...
''. A convenient way to understand the curvature comes from an ordinary differential equation, first considered by Gauss and later generalized by Jacobi, arising from the change of normal coordinates about two different points. The Gauss–Jacobi equation provides another way of computing the Gaussian curvature. Geometrically it explains what happens to geodesics from a fixed base point as the endpoint varies along a small curve segment through data recorded in the Jacobi field, a vector field along the geodesic. One and a quarter centuries after Gauss and Jacobi,
Marston Morse Harold Calvin Marston Morse (March 24, 1892 – June 22, 1977) was an American mathematician best known for his work on the ''calculus of variations in the large'', a subject where he introduced the technique of differential topology now known a ...
gave a more conceptual interpretation of the Jacobi field in terms of second derivatives of the energy function on the infinite-dimensional
Hilbert manifold In mathematics, a Hilbert manifold is a manifold modeled on Hilbert spaces. Thus it is a separable Hausdorff space in which each point has a neighbourhood homeomorphic to an infinite dimensional Hilbert space. The concept of a Hilbert manifold pro ...
of paths.


Exponential map

The theory of
ordinary differential equation In mathematics, an ordinary differential equation (ODE) is a differential equation whose unknown(s) consists of one (or more) function(s) of one variable and involves the derivatives of those functions. The term ''ordinary'' is used in contrast ...
s shows that if is smooth then the differential equation with initial condition has a unique solution for sufficiently small and the solution depends smoothly on and . This implies that for sufficiently small
tangent vector In mathematics, a tangent vector is a vector that is tangent to a curve or surface at a given point. Tangent vectors are described in the differential geometry of curves in the context of curves in R''n''. More generally, tangent vectors are e ...
s at a given point , there is a geodesic defined on with and . Moreover, if , then . The exponential map is defined by : (1) and gives a diffeomorphism between a disc and a neighbourhood of ; more generally the map sending to gives a local diffeomorphism onto a neighbourhood of . The exponential map gives
geodesic normal coordinates In differential geometry, normal coordinates at a point ''p'' in a differentiable manifold equipped with a symmetric affine connection are a local coordinate system in a neighborhood of ''p'' obtained by applying the exponential map to the tange ...
near .


Computation of normal coordinates

There is a standard technique (see for example ) for computing the change of variables to normal coordinates , at a point as a formal Taylor series expansion. If the coordinates , at (0,0) are locally orthogonal, write : : where , are quadratic and , cubic
homogeneous polynomial In mathematics, a homogeneous polynomial, sometimes called quantic in older texts, is a polynomial whose nonzero terms all have the same degree. For example, x^5 + 2 x^3 y^2 + 9 x y^4 is a homogeneous polynomial of degree 5, in two variables; ...
s in and . If and are fixed, and can be considered as formal
power series In mathematics, a power series (in one variable) is an infinite series of the form \sum_^\infty a_n \left(x - c\right)^n = a_0 + a_1 (x - c) + a_2 (x - c)^2 + \dots where ''an'' represents the coefficient of the ''n''th term and ''c'' is a con ...
solutions of the Euler equations: this uniquely determines , , , , and .


Gauss's lemma

In these coordinates the matrix satisfies and the lines are geodesics through 0. Euler's equations imply the matrix equation :, a key result, usually called the Gauss lemma. Geometrically it states that : Taking
polar coordinates In mathematics, the polar coordinate system is a two-dimensional coordinate system in which each point on a plane is determined by a distance from a reference point and an angle from a reference direction. The reference point (analogous to th ...
, it follows that the metric has the form :. In geodesic coordinates, it is easy to check that the geodesics through zero minimize length. The topology on the Riemannian manifold is then given by a distance function , namely the
infimum In mathematics, the infimum (abbreviated inf; plural infima) of a subset S of a partially ordered set P is a greatest element in P that is less than or equal to each element of S, if such an element exists. Consequently, the term ''greatest lo ...
of the lengths of piecewise smooth paths between and . This distance is realised locally by geodesics, so that in normal coordinates . If the radius is taken small enough, a slight sharpening of the Gauss lemma shows that the image of the disc under the exponential map is
geodesically convex In mathematics — specifically, in Riemannian geometry — geodesic convexity is a natural generalization of convexity for sets and functions to Riemannian manifolds. It is common to drop the prefix "geodesic" and refer simply to "convex ...
, i.e. any two points in are joined by a unique geodesic lying entirely inside .


Theorema Egregium

Gauss's
Theorema Egregium Gauss's ''Theorema Egregium'' (Latin for "Remarkable Theorem") is a major result of differential geometry, proved by Carl Friedrich Gauss in 1827, that concerns the curvature of surfaces. The theorem says that Gaussian curvature can be determi ...
, the "Remarkable Theorem", shows that the Gaussian curvature of a surface can be computed solely in terms of the metric and is thus an intrinsic invariant of the surface, independent of any
isometric embedding In mathematics, an embedding (or imbedding) is one instance of some mathematical structure contained within another instance, such as a group that is a subgroup. When some object X is said to be embedded in another object Y, the embedding is giv ...
in and unchanged under coordinate transformations. In particular isometries of surfaces preserve Gaussian curvature. This theorem can expressed in terms of the power series expansion of the metric, , is given in normal coordinates as :.


Gauss–Jacobi equation

Taking a coordinate change from normal coordinates at to normal coordinates at a nearby point , yields the Sturm–Liouville equation satisfied by , discovered by Gauss and later generalised by
Jacobi Jacobi may refer to: * People with the surname Jacobi Mathematics: * Jacobi sum, a type of character sum * Jacobi method, a method for determining the solutions of a diagonally dominant system of linear equations * Jacobi eigenvalue algorithm, ...
, : The
Jacobian In mathematics, a Jacobian, named for Carl Gustav Jacob Jacobi, may refer to: * Jacobian matrix and determinant * Jacobian elliptic functions * Jacobian variety *Intermediate Jacobian In mathematics, the intermediate Jacobian of a compact Kähle ...
of this coordinate change at is equal to . This gives another way of establishing the intrinsic nature of Gaussian curvature. Because can be interpreted as the length of the line element in the direction, the Gauss–Jacobi equation shows that the Gaussian curvature measures the spreading of geodesics on a geometric surface as they move away from a point.


Laplace–Beltrami operator

On a surface with local metric : ds^2 = E \, dx^2 + 2F \, dx \, dy + G \, dy^2 and Laplace–Beltrami operator :\Delta f = \left(\partial_x \partial_x f - \partial_x \partial_y f -\partial_y \partial_x f + \partial_y \partial_yf\right), where , the Gaussian curvature at a point is given by the formula : K=- 3 \lim_ \Delta (\log r), where denotes the geodesic distance from the point. In
isothermal coordinates In mathematics, specifically in differential geometry, isothermal coordinates on a Riemannian manifold are local coordinates where the metric is conformal to the Euclidean metric. This means that in isothermal coordinates, the Riemannian metric l ...
, first considered by Gauss, the metric is required to be of the special form :ds^2 = e^\varphi (dx^2+dy^2). \, In this case the Laplace–Beltrami operator is given by :\Delta = e^ \left(\frac + \frac\right) and satisfies
Liouville's equation :''For Liouville's equation in dynamical systems, see Liouville's theorem (Hamiltonian).'' : ''For Liouville's equation in quantum mechanics, see Von Neumann equation.'' : ''For Liouville's equation in Euclidean space, see Liouville–Bratu–Gelf ...
:\Delta \varphi=-2K. \, Isothermal coordinates are known to exist in a neighbourhood of any point on the surface, although all proofs to date rely on non-trivial results on
partial differential equation In mathematics, a partial differential equation (PDE) is an equation which imposes relations between the various partial derivatives of a multivariable function. The function is often thought of as an "unknown" to be solved for, similarly to h ...
s. There is an elementary proof for minimal surfaces.


Gauss–Bonnet theorem

On a
sphere A sphere () is a geometrical object that is a three-dimensional analogue to a two-dimensional circle. A sphere is the set of points that are all at the same distance from a given point in three-dimensional space.. That given point is the c ...
or a
hyperboloid In geometry, a hyperboloid of revolution, sometimes called a circular hyperboloid, is the surface generated by rotating a hyperbola around one of its principal axes. A hyperboloid is the surface obtained from a hyperboloid of revolution by def ...
, the area of a
geodesic triangle In geometry, a geodesic () is a curve representing in some sense the shortest path ( arc) between two points in a surface, or more generally in a Riemannian manifold. The term also has meaning in any differentiable manifold with a connectio ...
, i.e. a triangle all the sides of which are geodesics, is proportional to the difference of the sum of the interior angles and . The constant of proportionality is just the Gaussian curvature, a constant for these surfaces. For the torus, the difference is zero, reflecting the fact that its Gaussian curvature is zero. These are standard results in spherical, hyperbolic and high school trigonometry (see below). Gauss generalised these results to an arbitrary surface by showing that the integral of the Gaussian curvature over the interior of a geodesic triangle is also equal to this angle difference or excess. His formula showed that the Gaussian curvature could be calculated near a point as the limit of area over angle excess for geodesic triangles shrinking to the point. Since any closed surface can be decomposed up into geodesic triangles, the formula could also be used to compute the integral of the curvature over the whole surface. As a special case of what is now called the
Gauss–Bonnet theorem In the mathematical field of differential geometry, the Gauss–Bonnet theorem (or Gauss–Bonnet formula) is a fundamental formula which links the curvature of a surface to its underlying topology. In the simplest application, the case of a t ...
, Gauss proved that this integral was remarkably always 2π times an integer, a topological invariant of the surface called the
Euler characteristic In mathematics, and more specifically in algebraic topology and polyhedral combinatorics, the Euler characteristic (or Euler number, or Euler–Poincaré characteristic) is a topological invariant, a number that describes a topological spac ...
. This invariant is easy to compute combinatorially in terms of the number of vertices, edges, and faces of the triangles in the decomposition, also called a
triangulation In trigonometry and geometry, triangulation is the process of determining the location of a point by forming triangles to the point from known points. Applications In surveying Specifically in surveying, triangulation involves only angle me ...
. This interaction between analysis and topology was the forerunner of many later results in geometry, culminating in the Atiyah-Singer index theorem. In particular properties of the curvature impose restrictions on the topology of the surface.


Geodesic triangles

Gauss proved that, if is a geodesic triangle on a surface with angles , and at vertices , and , then :\int_\Delta K\,dA = \alpha + \beta + \gamma - \pi. In fact taking geodesic polar coordinates with origin and , the radii at polar angles 0 and : :\begin \int_\Delta K\,dA &= \int_\Delta KH\,dr\,d\theta = - \int_0^\alpha \int_0^ \! H_\,dr\,d\theta \\ &= \int_0^\alpha 1 -H_r(r_\theta,\theta)\,d\theta = \int_0^\alpha d\theta + \int_^\gamma \!\! d\varphi \\ &= \alpha + \beta + \gamma - \pi, \end where the second equality follows from the Gauss–Jacobi equation and the fourth from Gauss' derivative formula in the orthogonal coordinates . Gauss' formula shows that the curvature at a point can be calculated as the limit of ''angle excess'' over ''area'' for successively smaller geodesic triangles near the point. Qualitatively a surface is positively or negatively curved according to the sign of the angle excess for arbitrarily small geodesic triangles.


Gauss–Bonnet theorem

Since every compact oriented 2-manifold can be triangulated by small geodesic triangles, it follows that : \int_M K dA = 2\pi\,\chi(M) where denotes the
Euler characteristic In mathematics, and more specifically in algebraic topology and polyhedral combinatorics, the Euler characteristic (or Euler number, or Euler–Poincaré characteristic) is a topological invariant, a number that describes a topological spac ...
of the surface. In fact if there are faces, edges and vertices, then and the left hand side equals . This is the celebrated
Gauss–Bonnet theorem In the mathematical field of differential geometry, the Gauss–Bonnet theorem (or Gauss–Bonnet formula) is a fundamental formula which links the curvature of a surface to its underlying topology. In the simplest application, the case of a t ...
: it shows that the integral of the Gaussian curvature is a topological invariant of the manifold, namely the Euler characteristic. This theorem can be interpreted in many ways; perhaps one of the most far-reaching has been as the index theorem for an
elliptic differential operator In the theory of partial differential equations, elliptic operators are differential operators that generalize the Laplace operator. They are defined by the condition that the coefficients of the highest-order derivatives be positive, which imp ...
on , one of the simplest cases of the Atiyah-Singer index theorem. Another related result, which can be proved using the Gauss–Bonnet theorem, is the Poincaré-Hopf index theorem for vector fields on which vanish at only a finite number of points: the sum of the indices at these points equals the Euler characteristic, where the ''index'' of a point is defined as follows: on a small circle round each isolated zero, the vector field defines a map into the unit circle; the index is just the
winding number In mathematics, the winding number or winding index of a closed curve in the plane around a given point is an integer representing the total number of times that curve travels counterclockwise around the point, i.e., the curve's number of t ...
of this map.)


Curvature and embeddings

If the Gaussian curvature of a surface is everywhere positive, then the Euler characteristic is positive so is homeomorphic (and therefore diffeomorphic) to . If in addition the surface is isometrically embedded in , the Gauss map provides an explicit diffeomorphism. As Hadamard observed, in this case the surface is
convex Convex or convexity may refer to: Science and technology * Convex lens, in optics Mathematics * Convex set, containing the whole line segment that joins points ** Convex polygon, a polygon which encloses a convex set of points ** Convex polytop ...
; this criterion for convexity can be viewed as a 2-dimensional generalisation of the well-known second derivative criterion for convexity of plane curves.
Hilbert David Hilbert (; ; 23 January 1862 – 14 February 1943) was a German mathematician, one of the most influential mathematicians of the 19th and early 20th centuries. Hilbert discovered and developed a broad range of fundamental ideas in many ...
proved that every isometrically embedded closed surface must have a point of positive curvature. Thus a closed Riemannian 2-manifold of non-positive curvature can never be embedded isometrically in ; however, as Adriano Garsia showed using the
Beltrami equation In mathematics, the Beltrami equation, named after Eugenio Beltrami, is the partial differential equation : = \mu . for ''w'' a complex distribution of the complex variable ''z'' in some open set ''U'', with derivatives that are locally ''L''2 ...
for
quasiconformal mapping In mathematical complex analysis, a quasiconformal mapping, introduced by and named by , is a homeomorphism between plane domains which to first order takes small circles to small ellipses of bounded eccentricity. Intuitively, let ''f'' : ''D' ...
s, this is always possible for some conformally equivalent metric.;


Surfaces of constant curvature

The
simply connected In topology, a topological space is called simply connected (or 1-connected, or 1-simply connected) if it is path-connected and every path between two points can be continuously transformed (intuitively for embedded spaces, staying within the spa ...
surfaces of constant curvature 0, +1 and –1 are the Euclidean plane, the unit sphere in , and the
hyperbolic plane In mathematics, hyperbolic geometry (also called Lobachevskian geometry or Bolyai– Lobachevskian geometry) is a non-Euclidean geometry. The parallel postulate of Euclidean geometry is replaced with: :For any given line ''R'' and point ' ...
. Each of these has a transitive three-dimensional
Lie group In mathematics, a Lie group (pronounced ) is a group that is also a differentiable manifold. A manifold is a space that locally resembles Euclidean space, whereas groups define the abstract concept of a binary operation along with the addi ...
of orientation preserving
isometries In mathematics, an isometry (or congruence, or congruent transformation) is a distance-preserving transformation between metric spaces, usually assumed to be bijective. The word isometry is derived from the Ancient Greek: ἴσος ''isos'' mea ...
, which can be used to study their geometry. Each of the two non-compact surfaces can be identified with the quotient where is a
maximal compact subgroup In mathematics, a maximal compact subgroup ''K'' of a topological group ''G'' is a subgroup ''K'' that is a compact space, in the subspace topology, and maximal amongst such subgroups. Maximal compact subgroups play an important role in the class ...
of . Here is isomorphic to . Any other closed Riemannian 2-manifold of constant Gaussian curvature, after scaling the metric by a constant factor if necessary, will have one of these three surfaces as its
universal covering space A covering of a topological space X is a continuous map \pi : E \rightarrow X with special properties. Definition Let X be a topological space. A covering of X is a continuous map : \pi : E \rightarrow X such that there exists a discrete spa ...
. In the orientable case, the
fundamental group In the mathematical field of algebraic topology, the fundamental group of a topological space is the group of the equivalence classes under homotopy of the loops contained in the space. It records information about the basic shape, or holes, o ...
of can be identified with a torsion-free uniform subgroup of and can then be identified with the
double coset space A double is a look-alike or doppelgänger; one person or being that resembles another. Double, The Double or Dubble may also refer to: Film and television * Double (filmmaking), someone who substitutes for the credited actor of a character * ...
. In the case of the sphere and the Euclidean plane, the only possible examples are the sphere itself and tori obtained as quotients of by discrete rank 2 subgroups. For closed surfaces of
genus Genus ( plural genera ) is a taxonomic rank used in the biological classification of living and fossil organisms as well as viruses. In the hierarchy of biological classification, genus comes above species and below family. In binomial nom ...
, the
moduli space In mathematics, in particular algebraic geometry, a moduli space is a geometric space (usually a scheme or an algebraic stack) whose points represent algebro-geometric objects of some fixed kind, or isomorphism classes of such objects. Such sp ...
of Riemann surfaces obtained as varies over all such subgroups, has real dimension . By Poincaré's
uniformization theorem In mathematics, the uniformization theorem says that every simply connected Riemann surface is conformally equivalent to one of three Riemann surfaces: the open unit disk, the complex plane, or the Riemann sphere. The theorem is a generalization ...
, any orientable closed 2-manifold is conformally equivalent to a surface of constant curvature 0, +1 or –1. In other words, by multiplying the metric by a positive scaling factor, the Gaussian curvature can be made to take exactly one of these values (the sign of the
Euler characteristic In mathematics, and more specifically in algebraic topology and polyhedral combinatorics, the Euler characteristic (or Euler number, or Euler–Poincaré characteristic) is a topological invariant, a number that describes a topological spac ...
of ).


Euclidean geometry

In the case of the Euclidean plane, the symmetry group is the Euclidean motion group, the
semidirect product In mathematics, specifically in group theory, the concept of a semidirect product is a generalization of a direct product. There are two closely related concepts of semidirect product: * an ''inner'' semidirect product is a particular way in wh ...
of the two dimensional group of translations by the group of rotations. Geodesics are straight lines and the geometry is encoded in the elementary formulas of
trigonometry Trigonometry () is a branch of mathematics that studies relationships between side lengths and angles of triangles. The field emerged in the Hellenistic world during the 3rd century BC from applications of geometry to astronomical studies. ...
, such as the
cosine rule In trigonometry, the law of cosines (also known as the cosine formula, cosine rule, or al-Kashi's theorem) relates the lengths of the sides of a triangle to the cosine of one of its angles. Using notation as in Fig. 1, the law of cosines states ...
for a triangle with sides , , and angles , , : : c^2 = a^2 +b^2 -2ab \,\cos \gamma. Flat tori can be obtained by taking the quotient of by a
lattice Lattice may refer to: Arts and design * Latticework, an ornamental criss-crossed framework, an arrangement of crossing laths or other thin strips of material * Lattice (music), an organized grid model of pitch ratios * Lattice (pastry), an orna ...
, i.e. a free Abelian subgroup of rank 2. These closed surfaces have no isometric embeddings in . They do nevertheless admit isometric embeddings in ; in the easiest case this follows from the fact that the torus is a product of two circles and each circle can be isometrically embedded in .


Spherical geometry

The isometry group of the unit sphere in is the orthogonal group , with the rotation group as the subgroup of isometries preserving orientation. It is the direct product of with the antipodal map, sending to . The group acts transitively on . The
stabilizer subgroup In mathematics, a group action on a space is a group homomorphism of a given group into the group of transformations of the space. Similarly, a group action on a mathematical structure is a group homomorphism of a group into the automorphis ...
of the unit vector (0,0,1) can be identified with , so that . The geodesics between two points on the sphere are the
great circle In mathematics, a great circle or orthodrome is the circular intersection of a sphere and a plane passing through the sphere's center point. Any arc of a great circle is a geodesic of the sphere, so that great circles in spherical geome ...
arcs with these given endpoints. If the points are not antipodal, there is a unique shortest geodesic between the points. The geodesics can also be described group theoretically: each geodesic through the North pole (0,0,1) is the orbit of the subgroup of rotations about an axis through antipodal points on the equator. A
spherical triangle Spherical trigonometry is the branch of spherical geometry that deals with the metrical relationships between the sides and angles of spherical triangles, traditionally expressed using trigonometric functions. On the sphere, geodesics are gre ...
is a geodesic triangle on the sphere. It is defined by points , , on the sphere with sides , , formed from great circle arcs of length less than . If the lengths of the sides are , , and the angles between the sides , , , then the spherical cosine law states that :\cos c = \cos a \, \cos b + \sin a\, \sin b \,\cos \gamma. The area of the triangle is given by :. Using
stereographic projection In mathematics, a stereographic projection is a perspective projection of the sphere, through a specific point on the sphere (the ''pole'' or ''center of projection''), onto a plane (the ''projection plane'') perpendicular to the diameter thro ...
from the North pole, the sphere can be identified with the
extended complex plane In mathematics, the Riemann sphere, named after Bernhard Riemann, is a model of the extended complex plane: the complex plane plus one point at infinity. This extended plane represents the extended complex numbers, that is, the complex numbers ...
. The explicit map is given by :\pi(x,y,z)=\equiv u + iv. Under this correspondence every rotation of corresponds to a
Möbius transformation In geometry and complex analysis, a Möbius transformation of the complex plane is a rational function of the form f(z) = \frac of one complex variable ''z''; here the coefficients ''a'', ''b'', ''c'', ''d'' are complex numbers satisfying ''ad' ...
in , unique up to sign. With respect to the coordinates in the complex plane, the spherical metric becomes : ds^2 = . The unit sphere is the unique closed orientable surface with constant curvature +1. The quotient can be identified with the
real projective plane In mathematics, the real projective plane is an example of a compact non-orientable two-dimensional manifold; in other words, a one-sided surface. It cannot be embedded in standard three-dimensional space without intersecting itself. It has b ...
. It is non-orientable and can be described as the quotient of by the antipodal map (multiplication by −1). The sphere is simply connected, while the real projective plane has fundamental group . The finite subgroups of , corresponding to the finite subgroups of and the symmetry groups of the
platonic solids In geometry, a Platonic solid is a convex, regular polyhedron in three-dimensional Euclidean space. Being a regular polyhedron means that the faces are congruent (identical in shape and size) regular polygons (all angles congruent and all edges c ...
, do not act freely on , so the corresponding quotients are not 2-manifolds, just
orbifold In the mathematical disciplines of topology and geometry, an orbifold (for "orbit-manifold") is a generalization of a manifold. Roughly speaking, an orbifold is a topological space which is locally a finite group quotient of a Euclidean space. D ...
s.


Hyperbolic geometry

Non-Euclidean geometry In mathematics, non-Euclidean geometry consists of two geometries based on axioms closely related to those that specify Euclidean geometry. As Euclidean geometry lies at the intersection of metric geometry and affine geometry, non-Euclidean g ...
was first discussed in letters of Gauss, who made extensive computations at the turn of the nineteenth century which, although privately circulated, he decided not to put into print. In 1830
Lobachevsky Nikolai Ivanovich Lobachevsky ( rus, Никола́й Ива́нович Лобаче́вский, p=nʲikɐˈlaj ɪˈvanəvʲɪtɕ ləbɐˈtɕɛfskʲɪj, a=Ru-Nikolai_Ivanovich_Lobachevsky.ogg; – ) was a Russian mathematician and geometer, kn ...
and independently in 1832 Bolyai, the son of one Gauss' correspondents, published synthetic versions of this new geometry, for which they were severely criticized. However it was not until 1868 that Beltrami, followed by Klein in 1871 and Poincaré in 1882, gave concrete analytic models for what Klein dubbed
hyperbolic geometry In mathematics, hyperbolic geometry (also called Lobachevskian geometry or Bolyai–Lobachevskian geometry) is a non-Euclidean geometry. The parallel postulate of Euclidean geometry is replaced with: :For any given line ''R'' and point ''P ...
. The four models of 2-dimensional hyperbolic geometry that emerged were: *the Beltrami-Klein model; *the Poincaré disk; *the Poincaré upper half-plane; *the
hyperboloid model In geometry, the hyperboloid model, also known as the Minkowski model after Hermann Minkowski, is a model of ''n''-dimensional hyperbolic geometry in which points are represented by points on the forward sheet ''S''+ of a two-sheeted hyperbo ...
of Wilhelm Killing in 3-dimensional
Minkowski space In mathematical physics, Minkowski space (or Minkowski spacetime) () is a combination of three-dimensional Euclidean space and time into a four-dimensional manifold where the spacetime interval between any two events is independent of the iner ...
. The first model, based on a disk, has the advantage that geodesics are actually line segments (that is, intersections of Euclidean lines with the open unit disk). The last model has the advantage that it gives a construction which is completely parallel to that of the unit sphere in 3-dimensional Euclidean space. Because of their application in complex analysis and geometry, however, the models of Poincaré are the most widely used: they are interchangeable thanks to the Möbius transformations between the disk and the upper half-plane. Let :D=\ be the Poincaré disk in the complex plane with Poincaré metric :ds^2= . In polar coordinates the metric is given by : ds^2= . The length of a curve is given by the formula :\ell(\gamma)=\int_a^b . The group given by :G=\left\ acts transitively by
Möbius transformation In geometry and complex analysis, a Möbius transformation of the complex plane is a rational function of the form f(z) = \frac of one complex variable ''z''; here the coefficients ''a'', ''b'', ''c'', ''d'' are complex numbers satisfying ''ad' ...
s on and the
stabilizer subgroup In mathematics, a group action on a space is a group homomorphism of a given group into the group of transformations of the space. Similarly, a group action on a mathematical structure is a group homomorphism of a group into the automorphis ...
of 0 is the rotation group : K=\left\. The quotient group is the group of orientation-preserving isometries of . Any two points , in are joined by a unique geodesic, given by the portion of the circle or straight line passing through and and orthogonal to the boundary circle. The distance between and is given by :d(z,w)=2 \tanh^ \frac. In particular and is the geodesic through 0 along the real axis, parametrized by arclength. The topology defined by this metric is equivalent to the usual Euclidean topology, although as a metric space is complete. A hyperbolic triangle is a geodesic triangle for this metric: any three points in are vertices of a hyperbolic triangle. If the sides have length , , with corresponding angles , , , then the hyperbolic cosine rule states that :\cosh c = \cosh a\, \cosh b - \sinh a \,\sinh b \,\cos \gamma. The area of the hyperbolic triangle is given by :. The unit disk and the upper half-plane :H=\ are conformally equivalent by the Möbius transformations : w=i ,\,\, z=. Under this correspondence the action of by Möbius transformations on corresponds to that of on . The metric on becomes : ds^2 = . Since lines or circles are preserved under Möbius transformations, geodesics are again described by lines or circles orthogonal to the real axis. The unit disk with the Poincaré metric is the unique simply connected oriented 2-dimensional Riemannian manifold with constant curvature −1. Any oriented closed surface with this property has as its universal covering space. Its
fundamental group In the mathematical field of algebraic topology, the fundamental group of a topological space is the group of the equivalence classes under homotopy of the loops contained in the space. It records information about the basic shape, or holes, o ...
can be identified with a torsion-free concompact subgroup of , in such a way that : M= \Gamma\backslash G /K. In this case is a
finitely presented group In mathematics, a presentation is one method of specifying a group. A presentation of a group ''G'' comprises a set ''S'' of generators—so that every element of the group can be written as a product of powers of some of these generators—and ...
. The generators and relations are encoded in a geodesically convex fundamental geodesic polygon in (or ) corresponding geometrically to closed geodesics on . Examples. * the
Bolza surface In mathematics, the Bolza surface, alternatively, complex algebraic Bolza curve (introduced by ), is a compact Riemann surface of genus (mathematics), genus 2 with the highest possible order of the conformal map, conformal automorphism group in thi ...
of genus 2; * the
Klein quartic In hyperbolic geometry, the Klein quartic, named after Felix Klein, is a compact Riemann surface of genus with the highest possible order automorphism group for this genus, namely order orientation-preserving automorphisms, and automorphisms ...
of genus 3; * the Macbeath surface of genus 7; * the
First Hurwitz triplet In the mathematical theory of Riemann surfaces, the first Hurwitz triplet is a triple of distinct Hurwitz surfaces with the identical automorphism group of the lowest possible genus, namely 14 (genera 3 and 7 each admit a unique Hurwitz surface, re ...
of genus 14.


Uniformization

Given an oriented closed surface with Gaussian curvature , the metric on can be changed conformally by scaling it by a factor . The new Gaussian curvature is then given by :K^\prime(x)= e^ (K(x) - \Delta u), where is the Laplacian for the original metric. Thus to show that a given surface is conformally equivalent to a metric with constant curvature it suffices to solve the following variant of
Liouville's equation :''For Liouville's equation in dynamical systems, see Liouville's theorem (Hamiltonian).'' : ''For Liouville's equation in quantum mechanics, see Von Neumann equation.'' : ''For Liouville's equation in Euclidean space, see Liouville–Bratu–Gelf ...
: :\Delta u = K^\prime e^ + K(x). When has Euler characteristic 0, so is diffeomorphic to a
torus In geometry, a torus (plural tori, colloquially donut or doughnut) is a surface of revolution generated by revolving a circle in three-dimensional space about an axis that is coplanar with the circle. If the axis of revolution does n ...
, , so this amounts to solving : \Delta u = K(x). By standard elliptic theory, this is possible because the integral of over is zero, by the Gauss–Bonnet theorem. When has negative Euler characteristic, , so the equation to be solved is: :\Delta u = -e^ + K(x). Using the continuity of the exponential map on
Sobolev space In mathematics, a Sobolev space is a vector space of functions equipped with a norm that is a combination of ''Lp''-norms of the function together with its derivatives up to a given order. The derivatives are understood in a suitable weak sense ...
due to
Neil Trudinger Neil Sidney Trudinger (born 20 June 1942) is an Australian mathematician, known particularly for his work in the field of nonlinear elliptic partial differential equations. After completing his B.Sc at the University of New England (Australia) ...
, this non-linear equation can always be solved. Finally in the case of the 2-sphere, and the equation becomes: :\Delta u = e^ + K(x). So far this non-linear equation has not been analysed directly, although classical results such as the
Riemann–Roch theorem The Riemann–Roch theorem is an important theorem in mathematics, specifically in complex analysis and algebraic geometry, for the computation of the dimension of the space of meromorphic functions with prescribed zeros and allowed poles. It rel ...
imply that it always has a solution. The method of
Ricci flow In the mathematical fields of differential geometry and geometric analysis, the Ricci flow ( , ), sometimes also referred to as Hamilton's Ricci flow, is a certain partial differential equation for a Riemannian metric. It is often said to be an ...
, developed by Richard S. Hamilton, gives another proof of existence based on non-linear
partial differential equations In mathematics, a partial differential equation (PDE) is an equation which imposes relations between the various partial derivatives of a multivariable function. The function is often thought of as an "unknown" to be solved for, similarly to ...
to prove existence. In fact the Ricci flow on conformal metrics on is defined on functions by : u_t = 4\pi - K'(x,t) = 4\pi -e^ (K(x) - \Delta u). After finite time, Chow showed that becomes positive; previous results of Hamilton could then be used to show that converges to +1. Prior to these results on Ricci flow, had given an alternative and technically simpler approach to uniformization based on the flow on Riemannian metrics defined by . A proof using elliptic operators, discovered in 1988, can be found in . Let be the
Green's function In mathematics, a Green's function is the impulse response of an inhomogeneous linear differential operator defined on a domain with specified initial conditions or boundary conditions. This means that if \operatorname is the linear differenti ...
on satisfying , where is the point measure at a fixed point of . The equation , has a smooth solution , because the right hand side has integral 0 by the Gauss–Bonnet theorem. Thus satisfies away from . It follows that is a complete metric of constant curvature 0 on the complement of , which is therefore isometric to the plane. Composing with
stereographic projection In mathematics, a stereographic projection is a perspective projection of the sphere, through a specific point on the sphere (the ''pole'' or ''center of projection''), onto a plane (the ''projection plane'') perpendicular to the diameter thro ...
, it follows that there is a smooth function such that has Gaussian curvature +1 on the complement of . The function automatically extends to a smooth function on the whole of .


Riemannian connection and parallel transport

The classical approach of Gauss to the differential geometry of surfaces was the standard elementary approach which predated the emergence of the concepts of
Riemannian manifold In differential geometry, a Riemannian manifold or Riemannian space , so called after the German mathematician Bernhard Riemann, is a real, smooth manifold ''M'' equipped with a positive-definite inner product ''g'p'' on the tangent space ...
initiated by
Bernhard Riemann Georg Friedrich Bernhard Riemann (; 17 September 1826 – 20 July 1866) was a German mathematician who made contributions to analysis, number theory, and differential geometry. In the field of real analysis, he is mostly known for the first ...
in the mid-nineteenth century and of connection developed by
Tullio Levi-Civita Tullio Levi-Civita, (, ; 29 March 1873 – 29 December 1941) was an Italian mathematician, most famous for his work on absolute differential calculus ( tensor calculus) and its applications to the theory of relativity, but who also made signi ...
,
Élie Cartan Élie Joseph Cartan (; 9 April 1869 – 6 May 1951) was an influential French mathematician who did fundamental work in the theory of Lie groups, differential systems (coordinate-free geometric formulation of PDEs), and differential geometr ...
and
Hermann Weyl Hermann Klaus Hugo Weyl, (; 9 November 1885 – 8 December 1955) was a German mathematician, theoretical physicist and philosopher. Although much of his working life was spent in Zürich, Switzerland, and then Princeton, New Jersey, he is asso ...
in the early twentieth century. The notion of connection,
covariant derivative In mathematics, the covariant derivative is a way of specifying a derivative along tangent vectors of a manifold. Alternatively, the covariant derivative is a way of introducing and working with a connection on a manifold by means of a differe ...
and
parallel transport In geometry, parallel transport (or parallel translation) is a way of transporting geometrical data along smooth curves in a manifold. If the manifold is equipped with an affine connection (a covariant derivative or connection on the tangent b ...
gave a more conceptual and uniform way of understanding curvature, which not only allowed generalisations to higher dimensional manifolds but also provided an important tool for defining new geometric invariants, called
characteristic classes In mathematics, a characteristic class is a way of associating to each principal bundle of ''X'' a cohomology class of ''X''. The cohomology class measures the extent the bundle is "twisted" and whether it possesses sections. Characteristic classes ...
. The approach using covariant derivatives and connections is nowadays the one adopted in more advanced textbooks.


Covariant derivative

Connections on a surface can be defined from various equivalent but equally important points of view. The Riemannian connection or
Levi-Civita connection In Riemannian or pseudo Riemannian geometry (in particular the Lorentzian geometry of general relativity), the Levi-Civita connection is the unique affine connection on the tangent bundle of a manifold (i.e. affine connection) that preserves ...
. is perhaps most easily understood in terms of lifting vector fields, considered as first order
differential operator In mathematics, a differential operator is an operator defined as a function of the differentiation operator. It is helpful, as a matter of notation first, to consider differentiation as an abstract operation that accepts a function and retur ...
s acting on functions on the manifold, to differential operators on the
tangent bundle In differential geometry, the tangent bundle of a differentiable manifold M is a manifold TM which assembles all the tangent vectors in M . As a set, it is given by the disjoint unionThe disjoint union ensures that for any two points and of ...
or
frame bundle In mathematics, a frame bundle is a principal fiber bundle F(''E'') associated to any vector bundle ''E''. The fiber of F(''E'') over a point ''x'' is the set of all ordered bases, or ''frames'', for ''E'x''. The general linear group acts nat ...
. In the case of an embedded surface, the lift to an operator on vector fields, called the covariant derivative, is very simply described in terms of orthogonal projection. Indeed, a vector field on a surface embedded in can be regarded as a function from the surface into . Another vector field acts as a differential operator component-wise. The resulting vector field will not be tangent to the surface, but this can be corrected taking its orthogonal projection onto the tangent space at each point of the surface. As
Ricci Ricci () is an Italian surname, derived from the adjective "riccio", meaning curly. Notable Riccis Arts and entertainment * Antonio Ricci (painter) (c.1565–c.1635), Spanish Baroque painter of Italian origin * Christina Ricci (born 1980), Ameri ...
and
Levi-Civita Tullio Levi-Civita, (, ; 29 March 1873 – 29 December 1941) was an Italian mathematician, most famous for his work on absolute differential calculus (tensor calculus) and its applications to the theory of relativity, but who also made signific ...
realised at the turn of the twentieth century, this process depends only on the metric and can be locally expressed in terms of the Christoffel symbols.


Parallel transport

Parallel transport In geometry, parallel transport (or parallel translation) is a way of transporting geometrical data along smooth curves in a manifold. If the manifold is equipped with an affine connection (a covariant derivative or connection on the tangent b ...
of tangent vectors along a curve in the surface was the next major advance in the subject, due to
Levi-Civita Tullio Levi-Civita, (, ; 29 March 1873 – 29 December 1941) was an Italian mathematician, most famous for his work on absolute differential calculus (tensor calculus) and its applications to the theory of relativity, but who also made signific ...
. It is related to the earlier notion of covariant derivative, because it is the
monodromy In mathematics, monodromy is the study of how objects from mathematical analysis, algebraic topology, algebraic geometry and differential geometry behave as they "run round" a singularity. As the name implies, the fundamental meaning of ''mono ...
of the
ordinary differential equation In mathematics, an ordinary differential equation (ODE) is a differential equation whose unknown(s) consists of one (or more) function(s) of one variable and involves the derivatives of those functions. The term ''ordinary'' is used in contrast ...
on the curve defined by the covariant derivative with respect to the velocity vector of the curve. Parallel transport along geodesics, the "straight lines" of the surface, can also easily be described directly. A vector in the tangent plane is transported along a geodesic as the unique vector field with constant length and making a constant angle with the velocity vector of the geodesic. For a general curve, this process has to be modified using the geodesic curvature, which measures how far the curve departs from being a geodesic. A vector field along a unit speed curve , with geodesic curvature , is said to be parallel along the curve if * it has constant length * the angle that it makes with the velocity vector satisfies :\dot(t) = - k_g(t) This recaptures the rule for parallel transport along a geodesic or piecewise geodesic curve, because in that case , so that the angle should remain constant on any geodesic segment. The existence of parallel transport follows because can be computed as the
integral In mathematics, an integral assigns numbers to functions in a way that describes displacement, area, volume, and other concepts that arise by combining infinitesimal data. The process of finding integrals is called integration. Along with ...
of the geodesic curvature. Since it therefore depends continuously on the norm of , it follows that parallel transport for an arbitrary curve can be obtained as the limit of the parallel transport on approximating piecewise geodesic curves. The connection can thus be described in terms of lifting paths in the manifold to paths in the tangent or orthonormal frame bundle, thus formalising the classical theory of the "
moving frame In mathematics, a moving frame is a flexible generalization of the notion of an ordered basis of a vector space often used to study the extrinsic differential geometry of smooth manifolds embedded in a homogeneous space. Introduction In lay te ...
", favoured by French authors. Lifts of loops about a point give rise to the
holonomy group In differential geometry, the holonomy of a connection on a smooth manifold is a general geometrical consequence of the curvature of the connection measuring the extent to which parallel transport around closed loops fails to preserve the geomet ...
at that point. The Gaussian curvature at a point can be recovered from parallel transport around increasingly small loops at the point. Equivalently curvature can be calculated directly at an infinitesimal level in terms of Lie brackets of lifted vector fields.


Connection 1-form

The approach of Cartan and Weyl, using connection 1-forms on the
frame bundle In mathematics, a frame bundle is a principal fiber bundle F(''E'') associated to any vector bundle ''E''. The fiber of F(''E'') over a point ''x'' is the set of all ordered bases, or ''frames'', for ''E'x''. The general linear group acts nat ...
of , gives a third way to understand the Riemannian connection. They noticed that parallel transport dictates that a path in the surface be lifted to a path in the frame bundle so that its tangent vectors lie in a special subspace of codimension one in the three-dimensional tangent space of the frame bundle. The projection onto this subspace is defined by a differential 1-form on the orthonormal frame bundle, the
connection form In mathematics, and specifically differential geometry, a connection form is a manner of organizing the data of a connection using the language of moving frames and differential forms. Historically, connection forms were introduced by Élie Carta ...
. This enabled the curvature properties of the surface to be encoded in
differential forms In mathematics, differential forms provide a unified approach to define integrands over curves, surfaces, solids, and higher-dimensional manifolds. The modern notion of differential forms was pioneered by Élie Cartan. It has many applications, ...
on the frame bundle and formulas involving their
exterior derivative On a differentiable manifold, the exterior derivative extends the concept of the differential of a function to differential forms of higher degree. The exterior derivative was first described in its current form by Élie Cartan in 1899. The re ...
s. This approach is particularly simple for an embedded surface. Thanks to a result of , the connection 1-form on a surface embedded in Euclidean space is just the pullback under the Gauss map of the connection 1-form on . Using the identification of with the
homogeneous space In mathematics, particularly in the theories of Lie groups, algebraic groups and topological groups, a homogeneous space for a group ''G'' is a non-empty manifold or topological space ''X'' on which ''G'' acts transitively. The elements ...
, the connection 1-form is just a component of the Maurer–Cartan 1-form on .


Global differential geometry of surfaces

Although the characterisation of curvature involves only the local geometry of a surface, there are important global aspects such as the
Gauss–Bonnet theorem In the mathematical field of differential geometry, the Gauss–Bonnet theorem (or Gauss–Bonnet formula) is a fundamental formula which links the curvature of a surface to its underlying topology. In the simplest application, the case of a t ...
, the
uniformization theorem In mathematics, the uniformization theorem says that every simply connected Riemann surface is conformally equivalent to one of three Riemann surfaces: the open unit disk, the complex plane, or the Riemann sphere. The theorem is a generalization ...
, the von Mangoldt-Hadamard theorem, and the embeddability theorem. There are other important aspects of the global geometry of surfaces. These include: *
Injectivity radius This is a glossary of some terms used in Riemannian geometry and metric geometry — it doesn't cover the terminology of differential topology. The following articles may also be useful; they either contain specialised vocabulary or prov ...
, defined as the largest such that two points at a distance less than are joined by a unique geodesic. Wilhelm Klingenberg proved in 1959 that the injectivity radius of a closed surface is bounded below by the minimum of and the length of its smallest closed geodesic. This improved a theorem of Bonnet who showed in 1855 that the diameter of a closed surface of positive Gaussian curvature is always bounded above by ; in other words a geodesic realising the metric distance between two points cannot have length greater than . *Rigidity. In 1927 Cohn-Vossen proved that two ovaloids – closed surfaces with positive Gaussian curvature – that are isometric are necessarily congruent by an isometry of . Moreover, a closed embedded surface with positive Gaussian curvature and constant mean curvature is necessarily a sphere; likewise a closed embedded surface of constant Gaussian curvature must be a sphere (Liebmann 1899).
Heinz Hopf Heinz Hopf (19 November 1894 – 3 June 1971) was a German mathematician who worked on the fields of topology and geometry. Early life and education Hopf was born in Gräbschen, Germany (now , part of Wrocław, Poland), the son of Eliza ...
showed in 1950 that a closed embedded surface with constant mean curvature and genus 0, i.e. homeomorphic to a sphere, is necessarily a sphere; five years later Alexandrov removed the topological assumption. In the 1980s, Wente constructed immersed tori of constant mean curvature in Euclidean 3-space. *
Carathéodory conjecture In differential geometry, the Carathéodory conjecture is a mathematical conjecture attributed to Constantin Carathéodory by Hans Ludwig Hamburger in a session of the Berlin Mathematical Society in 1924.''Sitzungsberichte der Berliner Mathematisc ...
: This conjecture states that a closed convex three times differentiable surface admits at least two
umbilic point In the differential geometry of surfaces in three dimensions, umbilics or umbilical points are points on a surface that are locally spherical. At such points the normal curvatures in all directions are equal, hence, both principal curvatures are eq ...
s. The first work on this conjecture was in 1924 by
Hans Hamburger Hans Ludwig Hamburger (5 August 1889, Berlin – 14 August 1956, Cologne) was a German mathematician. He was a professor at universities in Berlin, Cologne and Ankara.. Biography Hans was the elder son of Karl Hamburger and Margarethe Levy. He ...
, who noted that it follows from the following stronger claim: the half-integer valued index of the principal curvature foliation of an isolated umbilic is at most one. *Zero Gaussian curvature: a complete surface in with zero Gaussian curvature must be a cylinder or a plane. *Hilbert's theorem (1901): no complete surface with constant negative curvature can be immersed isometrically in . *The
Willmore conjecture In differential geometry, the Willmore conjecture is a lower bound on the Willmore energy of a torus. It is named after the English mathematician Tom Willmore, who conjectured it in 1965. A proof by Fernando Codá Marques and André Neves ...
. This conjecture states that the integral of the square of the mean curvature of a torus immersed in should be bounded below by . It is known that the integral is Moebius invariant. It was solved in 2012 by
Fernando Codá Marques Fernando Codá dos Santos Cavalcanti Marques (born 8 October 1979) is a Brazilian mathematician working mainly in geometry, topology, partial differential equations and Morse theory. He is a professor at Princeton University. In 2012, together ...
and André Neves. * Isoperimetric inequalities. In 1939 Schmidt proved that the classical isoperimetric inequality for curves in the Euclidean plane is also valid on the sphere or in the hyperbolic plane: namely he showed that among all closed curves bounding a domain of fixed area, the perimeter is minimized by when the curve is a circle for the metric. In one dimension higher, it is known that among all closed surfaces in arising as the boundary of a bounded domain of unit volume, the surface area is minimized for a Euclidean ball. * Systolic inequalities for curves on surfaces. Given a closed surface, its
systole Systole ( ) is the part of the cardiac cycle during which some chambers of the heart contract after refilling with blood. The term originates, via New Latin, from Ancient Greek (''sustolē''), from (''sustéllein'' 'to contract'; from ...
is defined to be the smallest length of any non-contractible closed curve on the surface. In 1949 Loewner proved a torus inequality for metrics on the torus, namely that the area of the torus over the square of its systole is bounded below by , with equality in the flat (constant curvature) case. A similar result is given by Pu's inequality for the real projective plane from 1952, with a lower bound of also attained in the constant curvature case. For the
Klein bottle In topology, a branch of mathematics, the Klein bottle () is an example of a non-orientable surface; it is a two-dimensional manifold against which a system for determining a normal vector cannot be consistently defined. Informally, it is a ...
, Blatter and Bavard later obtained a lower bound of . For a closed surface of genus , Hebda and Burago showed that the ratio is bounded below by . Three years later Mikhail Gromov found a lower bound given by a constant times , although this is not optimal. Asymptotically sharp upper and lower bounds given by constant times are due to Gromov and Buser-Sarnak, and can be found in . There is also a version for metrics on the sphere, taking for the systole the length of the smallest
closed geodesic In differential geometry and dynamical systems, a closed geodesic on a Riemannian manifold is a geodesic that returns to its starting point with the same tangent direction. It may be formalized as the projection of a closed orbit of the geodesic flo ...
. Gromov conjectured a lower bound of in 1980: the best result so far is the lower bound of obtained by Regina Rotman in 2006.Rotman, R. (2006) "The length of a shortest closed geodesic and the area of a 2-dimensional sphere", Proc. Amer. Math. Soc. 134: 3041-3047. Previous lower bounds had been obtained by Croke, Rotman-Nabutovsky and Sabourau.


Reading guide

One of the most comprehensive introductory surveys of the subject, charting the historical development from before Gauss to modern times, is by . Accounts of the classical theory are given in , and ; the more modern copiously illustrated undergraduate textbooks by , and might be found more accessible. An accessible account of the classical theory can be found in . More sophisticated graduate-level treatments using the Riemannian connection on a surface can be found in , and .


See also

*
Flatness (mathematics) In mathematics, the flatness (symbol: ⏥) of a surface is the degree to which it approximates a mathematical plane. The term is often generalized for higher-dimensional manifolds to describe the degree to which they approximate the Euclidean spa ...
*
Tangent vector In mathematics, a tangent vector is a vector that is tangent to a curve or surface at a given point. Tangent vectors are described in the differential geometry of curves in the context of curves in R''n''. More generally, tangent vectors are e ...
*
Zoll surface In mathematics, particularly in differential geometry, a Zoll surface, named after Otto Zoll, is a surface homeomorphic to the 2-sphere, equipped with a Riemannian metric all of whose geodesics are closed and of equal length. While the usua ...


Notes


References

* *; translated from the Russian by K. Vogtmann and A. Weinstein. * * * * *; translated from 2nd edition of '' Leçons sur la géométrie des espaces de Riemann'' (1951) by James Glazebrook. * ; translated from Russian by V. V. Goldberg with a foreword by S. S. Chern. * * * * *
Volume I (1887)Volume_II_(1915)_[1889
/nowiki>.html" ;"title="889">Volume II (1915) [1889
/nowiki>">889">Volume II (1915) [1889
/nowiki>br>Volume III (1894)Volume IV (1896)
* * * * *. *. * translated by A.M. Hiltebeitel and J.C. Morehead
"Disquisitiones generales circa superficies curvas"
''Commentationes Societatis Regiae Scientiarum Gottingesis Recentiores'' Vol. VI (1827), pp. 99–146. **. **. * * * * * * * * * * *, * * * * * * * * * Ian R. Porteous (2001) ''Geometric Differentiation: for the intelligence of curves and surfaces'', Cambridge University Press . * * * * * * * * * * *
Full text of book
* * *


External links

* {{DEFAULTSORT:Differential Geometry Of Surfaces