, the differential geometry of surfaces deals with the differential geometry
of smooth surface
s with various additional structures, most often, a Riemannian metric
Surfaces have been extensively studied from various perspectives: ''extrinsically'', relating to their embedding
in Euclidean space
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
, first studied in depth by Carl Friedrich Gauss
, 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 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
s (in the spirit of the Erlangen program
), namely the symmetry group
s of the Euclidean plane
, the sphere
and the hyperbolic plane
. 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
: although Euler developed the one variable equations to understand geodesics
, defined independently of an embedding, one of Lagrange's main applications of the two variable equations was to minimal surfaces
, a concept that can only be defined in terms of an embedding.
The volumes of certain quadric surface
s of revolution
were calculated by Archimedes
. The development of calculus
in the seventeenth century provided a more systematic way of computing them. Curvature of general surfaces was first studied by Euler
. 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
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
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
of Gauss, established that the Gaussian curvature
is an intrinsic invariant, i.e. invariant under local isometries
. This point of view was extended to higher-dimensional spaces by Riemann
and led to what is known today as Riemannian geometry
. 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).
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
and general relativity
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
() 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
s, which are defined by the property that their mean curvature is zero at every point. The best-known examples are catenoid
s and helicoid
s, although many more have been discovered. Minimal surfaces can also be defined by properties to do with surface area
, with the consequence that they provide a mathematical model for the shape of soap film
s when stretched across a wire frame
s, which are surfaces that have at least one straight line running through every point; examples include the cylinder and the hyperboloid
of one sheet.
A surprising result of Carl Friedrich Gauss
, known as the theorema egregium
, 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
) 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
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
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
of the surface together with its surface area.
The notion of Riemannian manifold
and Riemann surface
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
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
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
, 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 this case, may have singularities such as cuspidal edge
s. Such surfaces are typically studied in singularity theory
. Other weakened forms of regular surfaces occur in computer-aided design
, 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
. The set is a torus of revolution
with radii and . It is a regular surface; local parametrizations can be given of the form
on two sheets is a regular surface; it can be covered by two Monge patches, with . The helicoid
appears in the theory of minimal surfaces
. 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 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 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, called 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
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
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
where is the Jacobian matrix
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
in . The Jacobian condition on and ensures, by the chain rule
, 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
be a regular surface,
an open subset of the plane and
a coordinate chart. If
, the space
can be identified with
identifies vector fields on
with vector fields on
. Taking standard variables and , a vector field has the form
, with and smooth functions. If
is a vector field and
is a smooth function, then
is also a smooth function. The first order differential operator
is a ''derivation'', i.e. it satisfies the Leibniz rule
For vector fields and it is simple to check that the operator
is a derivation corresponding to a vector field. It is called the Lie bracket