Differential geometry is a

mathematical Mathematics is a field of study that discovers and organizes methods, Mathematical theory, theories and theorems that are developed and Mathematical proof, proved for the needs of empirical sciences and mathematics itself. There are many ar ...

discipline that studies the

geometry Geometry (; ) is a branch of mathematics concerned with properties of space such as the distance, shape, size, and relative position of figures. Geometry is, along with arithmetic, one of the oldest branches of mathematics. A mathematician w ...

of smooth shapes and smooth spaces, otherwise known as

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 may ...

s. It uses the techniques of single variable calculus,

vector calculus Vector calculus or vector analysis is a branch of mathematics concerned with the differentiation and integration of vector fields, primarily in three-dimensional Euclidean space, \mathbb^3. The term ''vector calculus'' is sometimes used as a ...

linear algebra Linear algebra is the branch of mathematics concerning linear equations such as :a_1x_1+\cdots +a_nx_n=b, linear maps such as :(x_1, \ldots, x_n) \mapsto a_1x_1+\cdots +a_nx_n, and their representations in vector spaces and through matrix (mathemat ...

and

multilinear algebra Multilinear algebra is the study of Function (mathematics), functions with multiple vector space, vector-valued Argument of a function, arguments, with the functions being Linear map, linear maps with respect to each argument. It involves concept ...

. The field has its origins in the study of

spherical geometry 300px, A sphere with a spherical triangle on it. Spherical geometry or spherics () is the geometry of the two-dimensional surface of a sphere or the -dimensional surface of higher dimensional spheres. Long studied for its practical applicati ...

as far back as antiquity. It also relates to

astronomy Astronomy is a natural science that studies celestial objects and the phenomena that occur in the cosmos. It uses mathematics, physics, and chemistry in order to explain their origin and their overall evolution. Objects of interest includ ...

, the

geodesy Geodesy or geodetics is the science of measuring and representing the Figure of the Earth, geometry, Gravity of Earth, gravity, and Earth's rotation, spatial orientation of the Earth in Relative change, temporally varying Three-dimensional spac ...

of the

Earth Earth is the third planet from the Sun and the only astronomical object known to Planetary habitability, harbor life. This is enabled by Earth being an ocean world, the only one in the Solar System sustaining liquid surface water. Almost all ...

, and later the study of

hyperbolic geometry In mathematics, hyperbolic geometry (also called Lobachevskian geometry or János Bolyai, Bolyai–Nikolai Lobachevsky, Lobachevskian geometry) is a non-Euclidean geometry. The parallel postulate of Euclidean geometry is replaced with: :For a ...

by Lobachevsky. The simplest examples of smooth spaces are the plane and space curves and surfaces in the three-dimensional

Euclidean space Euclidean space is the fundamental space of geometry, intended to represent physical space. Originally, in Euclid's ''Elements'', it was the three-dimensional space of Euclidean geometry, but in modern mathematics there are ''Euclidean spaces ...

, and the study of these shapes formed the basis for development of modern differential geometry during the 18th and 19th centuries. Since the late 19th century, differential geometry has grown into a field concerned more generally with geometric structures on

differentiable 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 ...

s. A geometric structure is one which defines some notion of size, distance, shape, volume, or other rigidifying structure. For example, in

Riemannian geometry Riemannian geometry is the branch of differential geometry that studies Riemannian manifolds, defined as manifold, smooth manifolds with a ''Riemannian metric'' (an inner product on the tangent space at each point that varies smooth function, smo ...

distances and angles are specified, in

symplectic geometry Symplectic geometry is a branch of differential geometry and differential topology that studies symplectic manifolds; that is, differentiable manifolds equipped with a closed, nondegenerate 2-form. Symplectic geometry has its origins in the ...

volumes may be computed, in

conformal geometry In mathematics, conformal geometry is the study of the set of angle-preserving ( conformal) transformations on a space. In a real two dimensional space, conformal geometry is precisely the geometry of Riemann surfaces. In space higher than two di ...

only angles are specified, and in

gauge theory In physics, a gauge theory is a type of field theory in which the Lagrangian, and hence the dynamics of the system itself, does not change under local transformations according to certain smooth families of operations (Lie groups). Formally, t ...

certain fields are given over the space. Differential geometry is closely related to, and is sometimes taken to include,

differential topology In mathematics, differential topology is the field dealing with the topological properties and smooth properties of smooth manifolds. In this sense differential topology is distinct from the closely related field of differential geometry, which ...

, which concerns itself with properties of differentiable manifolds that do not rely on any additional geometric structure (see that article for more discussion on the distinction between the two subjects). Differential geometry is also related to the geometric aspects of the theory of differential equations, otherwise known as

geometric analysis Geometric analysis is a mathematical discipline where tools from differential equations, especially elliptic partial differential equations (PDEs), are used to establish new results in differential geometry and differential topology. The use of ...

. Differential geometry finds applications throughout mathematics and the

natural science Natural science or empirical science is one of the branches of science concerned with the description, understanding and prediction of natural phenomena, based on empirical evidence from observation and experimentation. Mechanisms such as peer ...

s. Most prominently the language of differential geometry was used by

Albert Einstein Albert Einstein (14 March 187918 April 1955) was a German-born theoretical physicist who is best known for developing the theory of relativity. Einstein also made important contributions to quantum mechanics. His mass–energy equivalence f ...

in his

theory of general relativity General relativity, also known as the general theory of relativity, and as 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 phys ...

, and subsequently by

physicists A physicist is a scientist who specializes in the field of physics, which encompasses the interactions of matter and energy at all length and time scales in the physical universe. Physicists generally are interested in the root or ultimate cau ...

in the development of

quantum field theory In theoretical physics, quantum field theory (QFT) is a theoretical framework that combines Field theory (physics), field theory and the principle of relativity with ideas behind quantum mechanics. QFT is used in particle physics to construct phy ...

and the standard model of particle physics. Outside of physics, differential geometry finds applications in

chemistry Chemistry is the scientific study of the properties and behavior of matter. It is a physical science within the natural sciences that studies the chemical elements that make up matter and chemical compound, compounds made of atoms, molecules a ...

economics Economics () is a behavioral science that studies the Production (economics), production, distribution (economics), distribution, and Consumption (economics), consumption of goods and services. Economics focuses on the behaviour and interac ...

engineering Engineering is the practice of using natural science, mathematics, and the engineering design process to Problem solving#Engineering, solve problems within technology, increase efficiency and productivity, and improve Systems engineering, s ...

control theory Control theory is a field of control engineering and applied mathematics that deals with the control system, control of dynamical systems in engineered processes and machines. The objective is to develop a model or algorithm governing the applic ...

computer graphics Computer graphics deals with generating images and art with the aid of computers. Computer graphics is a core technology in digital photography, film, video games, digital art, cell phone and computer displays, and many specialized applications. ...

and

computer vision Computer vision tasks include methods for image sensor, acquiring, Image processing, processing, Image analysis, analyzing, and understanding digital images, and extraction of high-dimensional data from the real world in order to produce numerical ...

, and recently in

machine learning Machine learning (ML) is a field of study in artificial intelligence concerned with the development and study of Computational statistics, statistical algorithms that can learn from data and generalise to unseen data, and thus perform Task ( ...

History and development

The history and development of differential geometry as a subject begins at least as far back as

classical antiquity Classical antiquity, also known as the classical era, classical period, classical age, or simply antiquity, is the period of cultural History of Europe, European history between the 8th century BC and the 5th century AD comprising the inter ...

. It is intimately linked to the development of geometry more generally, of the notion of space and shape, and of

topology Topology (from the Greek language, Greek words , and ) is the branch of mathematics concerned with the properties of a Mathematical object, geometric object that are preserved under Continuous function, continuous Deformation theory, deformat ...

, especially the study of

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 ...

s. In this section we focus primarily on the history of the application of

infinitesimal In mathematics, an infinitesimal number is a non-zero quantity that is closer to 0 than any non-zero real number is. The word ''infinitesimal'' comes from a 17th-century Modern Latin coinage ''infinitesimus'', which originally referred to the " ...

methods to geometry, and later to the ideas of

tangent space In mathematics, the tangent space of a manifold is a generalization of to curves in two-dimensional space and to surfaces in three-dimensional space in higher dimensions. In the context of physics the tangent space to a manifold at a point can be ...

s, and eventually the development of the modern formalism of the subject in terms of

tensor In mathematics, a tensor is an algebraic object that describes a multilinear relationship between sets of algebraic objects associated with a vector space. Tensors may map between different objects such as vectors, scalars, and even other ...

s and

tensor field In mathematics and physics, a tensor field is a function assigning a tensor to each point of a region of a mathematical space (typically a Euclidean space or manifold) or of the physical space. Tensor fields are used in differential geometry, ...

Classical antiquity until the Renaissance (300 BC1600 AD)

The study of differential geometry, or at least the study of the geometry of smooth shapes, can be traced back at least to

. In particular, much was known about the geometry of the

, a

, in the time of the

ancient Greek Ancient Greek (, ; ) includes the forms of the Greek language used in ancient Greece and the classical antiquity, ancient world from around 1500 BC to 300 BC. It is often roughly divided into the following periods: Mycenaean Greek (), Greek ...

mathematicians. Famously,

Eratosthenes Eratosthenes of Cyrene (; ; – ) was an Ancient Greek polymath: a Greek mathematics, mathematician, geographer, poet, astronomer, and music theory, music theorist. He was a man of learning, becoming the chief librarian at the Library of A ...

calculated the

circumference In geometry, the circumference () is the perimeter of a circle or ellipse. The circumference is the arc length of the circle, as if it were opened up and straightened out to a line segment. More generally, the perimeter is the curve length arou ...

of the Earth around 200 BC, and around 150 AD

Ptolemy Claudius Ptolemy (; , ; ; – 160s/170s AD) was a Greco-Roman mathematician, astronomer, astrologer, geographer, and music theorist who wrote about a dozen scientific treatises, three of which were important to later Byzantine science, Byzant ...

in his ''

Geography Geography (from Ancient Greek ; combining 'Earth' and 'write', literally 'Earth writing') is the study of the lands, features, inhabitants, and phenomena of Earth. Geography is an all-encompassing discipline that seeks an understanding o ...

'' introduced the

stereographic projection In mathematics, a stereographic projection is a perspective transform, perspective projection of the sphere, through a specific point (geometry), point on the sphere (the ''pole'' or ''center of projection''), onto a plane (geometry), plane (th ...

for the purposes of mapping the shape of the Earth.Struik, D. J. "Outline of a History of Differential Geometry: I." Isis, vol. 19, no. 1, 1933, pp. 92–120. JSTOR, www.jstor.org/stable/225188. Implicitly throughout this time principles that form the foundation of differential geometry and calculus were used in

, although in a much simplified form. Namely, as far back as

Euclid Euclid (; ; BC) was an ancient Greek mathematician active as a geometer and logician. Considered the "father of geometry", he is chiefly known for the '' Elements'' treatise, which established the foundations of geometry that largely domina ...

's '' Elements'' it was understood that a straight line could be defined by its property of providing the shortest distance between two points, and applying this same principle to the surface of the

leads to the conclusion that great circles, which are only locally similar to straight lines in a flat plane, provide the shortest path between two points on the Earth's surface. Indeed, the measurements of distance along such

geodesic In geometry, a geodesic () is a curve representing in some sense the locally 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 conn ...

paths by Eratosthenes and others can be considered a rudimentary measure of

arclength Arc length is the distance between two points along a section of a curve. Development of a formulation of arc length suitable for applications to mathematics and the sciences is a problem in vector calculus and in differential geometry. In the ...

of curves, a concept which did not see a rigorous definition in terms of calculus until the 1600s. Around this time there were only minimal overt applications of the theory of

s to the study of geometry, a precursor to the modern calculus-based study of the subject. In

's '' Elements'' the notion of tangency of a line to a circle is discussed, and

Archimedes Archimedes of Syracuse ( ; ) was an Ancient Greece, Ancient Greek Greek mathematics, mathematician, physicist, engineer, astronomer, and Invention, inventor from the ancient city of Syracuse, Sicily, Syracuse in History of Greek and Hellenis ...

applied the

method of exhaustion The method of exhaustion () is a method of finding the area of a shape by inscribing inside it a sequence of polygons (one at a time) whose areas converge to the area of the containing shape. If the sequence is correctly constructed, the differ ...

to compute the areas of smooth shapes such as the

circle A circle is a shape consisting of all point (geometry), points in a plane (mathematics), plane that are at a given distance from a given point, the Centre (geometry), centre. The distance between any point of the circle and the centre is cal ...

, and the volumes of smooth three-dimensional solids such as the sphere, cones, and cylinders. There was little development in the theory of differential geometry between antiquity and the beginning of the

Renaissance The Renaissance ( , ) is a Periodization, period of history and a European cultural movement covering the 15th and 16th centuries. It marked the transition from the Middle Ages to modernity and was characterized by an effort to revive and sur ...

. Before the development of calculus by Newton and

Leibniz Gottfried Wilhelm Leibniz (or Leibnitz; – 14 November 1716) was a German polymath active as a mathematician, philosopher, scientist and diplomat who is credited, alongside Sir Isaac Newton, with the creation of calculus in addition to many ...

, the most significant development in the understanding of differential geometry came from

Gerardus Mercator Gerardus Mercator (; 5 March 1512 – 2 December 1594) was a Flemish people, Flemish geographer, cosmographer and Cartography, cartographer. He is most renowned for creating the Mercator 1569 world map, 1569 world map based on a new Mercator pr ...

's development of the Mercator projection as a way of mapping the Earth. Mercator had an understanding of the advantages and pitfalls of his map design, and in particular was aware of the conformal nature of his projection, as well as the difference between ''praga'', the lines of shortest distance on the Earth, and the ''directio'', the straight line paths on his map. Mercator noted that the praga were ''oblique curvatur'' in this projection. This fact reflects the lack of a metric-preserving map of the Earth's surface onto a flat plane, a consequence of the later Theorema Egregium of

Gauss Johann Carl Friedrich Gauss (; ; ; 30 April 177723 February 1855) was a German mathematician, astronomer, Geodesy, geodesist, and physicist, who contributed to many fields in mathematics and science. He was director of the Göttingen Observat ...

After calculus (1600–1800)

The first systematic or rigorous treatment of geometry using the theory of infinitesimals and notions from

calculus Calculus 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 generalizations of arithmetic operations. Originally called infinitesimal calculus or "the ...

began around the 1600s when calculus was first developed by

Gottfried Leibniz Gottfried Wilhelm Leibniz (or Leibnitz; – 14 November 1716) was a German polymath active as a mathematician, philosopher, scientist and diplomat who is credited, alongside Isaac Newton, Sir Isaac Newton, with the creation of calculus in ad ...

and

Isaac Newton Sir Isaac Newton () was an English polymath active as a mathematician, physicist, astronomer, alchemist, theologian, and author. Newton was a key figure in the Scientific Revolution and the Age of Enlightenment, Enlightenment that followed ...

. At this time, the recent work of

René Descartes René Descartes ( , ; ; 31 March 1596 – 11 February 1650) was a French philosopher, scientist, and mathematician, widely considered a seminal figure in the emergence of modern philosophy and Modern science, science. Mathematics was paramou ...

introducing analytic coordinates to geometry allowed geometric shapes of increasing complexity to be described rigorously. In particular around this time

Pierre de Fermat Pierre de Fermat (; ; 17 August 1601 – 12 January 1665) was a French mathematician who is given credit for early developments that led to infinitesimal calculus, including his technique of adequality. In particular, he is recognized for his d ...

, Newton, and Leibniz began the study of

plane curve In mathematics, a plane curve is a curve in a plane that may be 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 plane c ...

s and the investigation of concepts such as points of

inflection In linguistic Morphology (linguistics), morphology, inflection (less commonly, inflexion) is a process of word formation in which a word is modified to express different grammatical category, grammatical categories such as grammatical tense, ...

and circles of osculation, which aid in the measurement of

curvature In mathematics, curvature is any of several strongly related concepts in geometry that intuitively measure the amount by which a curve deviates from being a straight line or by which a surface deviates from being a plane. If a curve or su ...

. Indeed, already in his first paper on the foundations of calculus, Leibniz notes that the infinitesimal condition

d^2 y = 0

indicates the existence of an inflection point. Shortly after this time the Bernoulli brothers,

Jacob Jacob, later known as Israel, is a Hebrew patriarch of the Abrahamic religions. He first appears in the Torah, where he is described in the Book of Genesis as a son of Isaac and Rebecca. Accordingly, alongside his older fraternal twin brother E ...

and

Johann Johann, typically a male given name, is the German form of ''Iohannes'', which is the Latin form of the Greek name ''Iōánnēs'' (), itself derived from Hebrew name '' Yochanan'' () in turn from its extended form (), meaning "Yahweh is Graciou ...

made important early contributions to the use of infinitesimals to study geometry. In lectures by Johann Bernoulli at the time, later collated by L'Hopital into the first textbook on differential calculus, the tangents to plane curves of various types are computed using the condition

dy=0

, and similarly points of inflection are calculated. At this same time the

orthogonality In mathematics, orthogonality is the generalization of the geometric notion of '' perpendicularity''. Although many authors use the two terms ''perpendicular'' and ''orthogonal'' interchangeably, the term ''perpendicular'' is more specifically ...

between the osculating circles of a plane curve and the tangent directions is realised, and the first analytical formula for the radius of an osculating circle, essentially the first analytical formula for the notion of

, is written down. In the wake of the development of analytic geometry and plane curves,

Alexis Clairaut Alexis Claude Clairaut (; ; 13 May 1713 – 17 May 1765) was a French mathematician, astronomer, and geophysicist. He was a prominent Newtonian whose work helped to establish the validity of the principles and results that Isaac Newton, Sir Isaa ...

began the study of

space curve In mathematics, a curve (also called a curved line in older texts) is an object similar to a line, but that does not have to be straight. Intuitively, a curve may be thought of as the trace left by a moving point. This is the definition that ...

s at just the age of 16. In his book Clairaut introduced the notion of tangent and subtangent directions to space curves in relation to the directions which lie along a

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 ...

on which the space curve lies. Thus Clairaut demonstrated an implicit understanding of the

of a surface and studied this idea using calculus for the first time. Importantly Clairaut introduced the terminology of ''curvature'' and ''double curvature'', essentially the notion of

principal curvature In differential geometry, the two principal curvatures at a given point of a surface (mathematics), 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 ...

s later studied by Gauss and others. Around this same time,

Leonhard Euler Leonhard Euler ( ; ; ; 15 April 170718 September 1783) was a Swiss polymath who was active as a mathematician, physicist, astronomer, logician, geographer, and engineer. He founded the studies of graph theory and topology and made influential ...

, originally a student of Johann Bernoulli, provided many significant contributions not just to the development of geometry, but to mathematics more broadly. In regards to differential geometry, Euler studied the notion of a

on a surface deriving the first analytical geodesic equation, and later introduced the first set of intrinsic coordinate systems on a surface, beginning the theory of ''intrinsic geometry'' upon which modern geometric ideas are based. Around this time Euler's study of mechanics in the ''

Mechanica ''Mechanica'' (; 1736) is a two-volume work published by mathematician Leonhard Euler which describes analytically the mathematics governing movement. Euler both developed the techniques of analysis and applied them to numerous problems in mec ...

'' lead to the realization that a mass traveling along a surface not under the effect of any force would traverse a geodesic path, an early precursor to the important foundational ideas of Einstein's

general relativity General relativity, also known as the general theory of relativity, and as Einstein's theory of gravity, is the differential geometry, geometric theory of gravitation published by Albert Einstein in 1915 and is the current description of grav ...

, and also to the Euler–Lagrange equations and the first theory of 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 Function (mathematics), functions and functional (mathematics), functionals, to find maxima and minima of f ...

, which underpins in modern differential geometry many techniques in

and

. This theory was used by

Lagrange Joseph-Louis Lagrange (born Giuseppe Luigi Lagrangiaminimal surface in terms of the Euler–Lagrange equation. In 1760 Euler proved a theorem expressing the curvature of a space curve on a surface in terms of the principal curvatures, known as

Euler's theorem In number theory, Euler's theorem (also known as the Fermat–Euler theorem or Euler's totient theorem) states that, if and are coprime positive integers, then a^ is congruent to 1 modulo , where \varphi denotes Euler's totient function; that ...

. Later in the 1700s, the new French school led by

Gaspard 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. Dur ...

began to make contributions to differential geometry. Monge made important contributions to the theory of plane curves, surfaces, and studied surfaces of revolution and

envelopes 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 sho ...

of plane curves and space curves. Several students of Monge made contributions to this same theory, and for example

Charles Dupin Baron Pierre Charles François Dupin (; 6 October 1784, Varzy, Nièvre – 18 January 1873, Paris, France) was a French Catholic mathematician, engineer, economist and politician, particularly known for work in the field of mathematics, where t ...

provided a new interpretation of Euler's theorem in terms of the principle curvatures, which is the modern form of the equation.

Intrinsic geometry and non-Euclidean geometry (1800–1900)

The field of differential geometry became an area of study considered in its own right, distinct from the more broad idea of analytic geometry, in the 1800s, primarily through the foundational work of

Carl Friedrich Gauss Johann Carl Friedrich Gauss (; ; ; 30 April 177723 February 1855) was a German mathematician, astronomer, geodesist, and physicist, who contributed to many fields in mathematics and science. He was director of the Göttingen Observatory and ...

and

Bernhard Riemann Georg Friedrich Bernhard Riemann (; ; 17September 182620July 1866) was a German mathematician who made profound contributions to analysis, number theory, and differential geometry. In the field of real analysis, he is mostly known for the f ...

, and also in the important contributions of

Nikolai Lobachevsky Nikolai Ivanovich Lobachevsky (; , ; – ) was a Russian mathematician and geometer, known primarily for his work on hyperbolic geometry, otherwise known as Lobachevskian geometry, and also for his fundamental study on Dirichlet integrals, kno ...

and

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 ge ...

and throughout the same period the development of

projective geometry In mathematics, projective geometry is the study of geometric properties that are invariant with respect to projective transformations. This means that, compared to elementary Euclidean geometry, projective geometry has a different setting (''p ...

. Dubbed the single most important work in the history of differential geometry,Spivak, M., 1975. A comprehensive introduction to differential geometry (Vol. 2). Publish or Perish, Incorporated. in 1827 Gauss produced the ''Disquisitiones generales circa superficies curvas'' detailing the general theory of curved surfaces.Gauss, C.F., 1828. Disquisitiones generales circa superficies curvas (Vol. 1). Typis Dieterichianis.Struik, D.J. "Outline of a History of Differential Geometry (II)." Isis, vol. 20, no. 1, 1933, pp. 161–191. JSTOR, www.jstor.org/stable/224886 In this work and his subsequent papers and unpublished notes on the theory of surfaces, Gauss has been dubbed the inventor of non-Euclidean geometry and the inventor of intrinsic differential geometry. In his fundamental paper Gauss introduced the Gauss map,

Gaussian curvature In differential geometry, the Gaussian curvature or Gauss curvature of a smooth Surface (topology), surface in three-dimensional space at a point is the product of the principal curvatures, and , at the given point: K = \kappa_1 \kappa_2. For ...

, first and second fundamental forms, proved the Theorema Egregium showing the intrinsic nature of the Gaussian curvature, and studied geodesics, computing the area of a geodesic triangle in various non-Euclidean geometries on surfaces. At this time Gauss was already of the opinion that the standard paradigm of

Euclidean geometry Euclidean geometry is a mathematical system attributed to ancient Greek mathematics, Greek mathematician Euclid, which he described in his textbook on geometry, ''Euclid's Elements, Elements''. Euclid's approach consists in assuming a small set ...

should be discarded, and was in possession of private manuscripts on non-Euclidean geometry which informed his study of geodesic triangles. Around this same time János Bolyai and Lobachevsky independently discovered

and thus demonstrated the existence of consistent geometries outside Euclid's paradigm. Concrete models of hyperbolic geometry were produced by Eugenio Beltrami later in the 1860s, and

Felix Klein Felix Christian Klein (; ; 25 April 1849 – 22 June 1925) was a German mathematician and Mathematics education, mathematics educator, known for his work in group theory, complex analysis, non-Euclidean geometry, and the associations betwe ...

coined the term non-Euclidean geometry in 1871, and through 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 na ...

put Euclidean and non-Euclidean geometries on the same footing. Implicitly, the

of the Earth that had been studied since antiquity was a non-Euclidean geometry, an

elliptic geometry Elliptic geometry is an example of a geometry in which Euclid's parallel postulate does not hold. Instead, as in spherical geometry, there are no parallel lines since any two lines must intersect. However, unlike in spherical geometry, two lines ...

. The development of intrinsic differential geometry in the language of Gauss was spurred on by his student,

in his Habilitationsschrift, ''On the hypotheses which lie at the foundation of geometry''. In this work Riemann introduced the notion of a

Riemannian metric In differential geometry, a Riemannian manifold is a geometric space on which many geometric notions such as distance, angles, length, volume, and curvature are defined. Euclidean space, the N-sphere, n-sphere, hyperbolic space, and smooth surf ...

and the Riemannian curvature tensor for the first time, and began the systematic study of differential geometry in higher dimensions. This intrinsic point of view in terms of the Riemannian metric, denoted by

ds^2

by Riemann, was the development of an idea of Gauss's about the linear element

ds

of a surface. At this time Riemann began to introduce the systematic use of

and

into the subject, making great use of the theory of

quadratic form In mathematics, a quadratic form is a polynomial with terms all of degree two (" form" is another name for a homogeneous polynomial). For example, 4x^2 + 2xy - 3y^2 is a quadratic form in the variables and . The coefficients usually belong t ...

s in his investigation of metrics and curvature. At this time Riemann did not yet develop the modern notion of a manifold, as even the notion of a

topological space In mathematics, a topological space is, roughly speaking, a Geometry, geometrical space in which Closeness (mathematics), closeness is defined but cannot necessarily be measured by a numeric Distance (mathematics), distance. More specifically, a to ...

had not been encountered, but he did propose that it might be possible to investigate or measure the properties of the metric of

spacetime In physics, spacetime, also called the space-time continuum, is a mathematical model that fuses the three dimensions of space and the one dimension of time into a single four-dimensional continuum. Spacetime diagrams are useful in visualiz ...

through the analysis of masses within spacetime, linking with the earlier observation of Euler that masses under the effect of no forces would travel along geodesics on surfaces, and predicting Einstein's fundamental observation of the

equivalence principle The equivalence principle is the hypothesis that the observed equivalence of gravitational and inertial mass is a consequence of nature. The weak form, known for centuries, relates to masses of any composition in free fall taking the same t ...

a full 60 years before it appeared in the scientific literature. In the wake of Riemann's new description, the focus of techniques used to study differential geometry shifted from the ad hoc and extrinsic methods of the study of curves and surfaces to a more systematic approach in terms of

tensor calculus In mathematics, a tensor is an algebraic object that describes a multilinear relationship between sets of algebraic objects associated with a vector space. Tensors may map between different objects such as vectors, scalars, and even other ...

and Klein's Erlangen program, and progress increased in the field. The notion of groups of transformations was developed by

Sophus Lie Marius Sophus Lie ( ; ; 17 December 1842 – 18 February 1899) was a Norwegian mathematician. He largely created the theory of continuous symmetry and applied it to the study of geometry and differential equations. He also made substantial cont ...

and Jean Gaston Darboux, leading to important results in the theory of

Lie groups In mathematics, a Lie group (pronounced ) is a group that is also a differentiable manifold, such that group multiplication and taking inverses are both differentiable. A manifold is a space that locally resembles Euclidean space, whereas ...

and

. The notion of differential calculus on curved spaces was studied by Elwin Christoffel, who introduced 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 surface (topology), surfaces or other manifolds endowed with a metri ...

which describe the

covariant derivative In mathematics and physics, covariance is a measure of how much two variables change together, and may refer to: Statistics * Covariance matrix, a matrix of covariances between a number of variables * Covariance or cross-covariance between ...

in 1868, and by others including Eugenio Beltrami who studied many analytic questions on manifolds. In 1899 Luigi Bianchi produced his ''Lectures on differential geometry'' which studied differential geometry from Riemann's perspective, and a year later

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 signifi ...

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 the ...

produced their textbook systematically developing the theory of absolute differential calculus and

. It was in this language that differential geometry was used by Einstein in the development of general relativity and pseudo-Riemannian geometry.

Modern differential geometry (1900–2000)

The subject of modern differential geometry emerged from the early 1900s in response to the foundational contributions of many mathematicians, including importantly the work of

Henri Poincaré Jules Henri Poincaré (, ; ; 29 April 185417 July 1912) was a French mathematician, Theoretical physics, theoretical physicist, engineer, and philosophy of science, philosopher of science. He is often described as a polymath, and in mathemati ...

on the foundations of

.Dieudonné, J., 2009. A history of algebraic and differential topology, 1900-1960. Springer Science & Business Media. At the start of the 1900s there was a major movement within mathematics to formalise the foundational aspects of the subject to avoid crises of rigour and accuracy, known as

Hilbert's program In mathematics, Hilbert's program, formulated by German mathematician David Hilbert in the early 1920s, was a proposed solution to the foundational crisis of mathematics, when early attempts to clarify the foundations of mathematics were found to ...

. As part of this broader movement, the notion of a

was distilled in by

Felix Hausdorff Felix Hausdorff ( , ; November 8, 1868 – January 26, 1942) was a German mathematician, pseudonym Paul Mongré (''à mogré' (Fr.) = "according to my taste"), who is considered to be one of the founders of modern topology and who contributed sig ...

in 1914, and by 1942 there were many different notions of manifold of a combinatorial and differential-geometric nature. Interest in the subject was also focused by the emergence of Einstein's theory of general relativity and the importance of the Einstein Field equations. Einstein's theory popularised the tensor calculus of Ricci and Levi-Civita and introduced the notation

g

for a Riemannian metric, and

\Gamma

for the Christoffel symbols, both coming from ''G'' in ''Gravitation''.

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 geometry. He ...

helped reformulate the foundations of the differential geometry of smooth manifolds in terms of exterior calculus and the theory of moving frames, leading in the world of physics to Einstein–Cartan theory.Fré, P.G., 2018. A Conceptual History of Space and Symmetry. Springer, Cham. Following this early development, many mathematicians contributed to the development of the modern theory, including Jean-Louis Koszul who introduced connections on vector bundles, Shiing-Shen Chern who introduced characteristic classes to the subject and began the study of

complex manifold In differential geometry and complex geometry, a complex manifold is a manifold with a ''complex structure'', that is an atlas (topology), atlas of chart (topology), charts to the open unit disc in the complex coordinate space \mathbb^n, such th ...

s, Sir William Vallance Douglas Hodge and

Georges de Rham Georges de Rham (; 10 September 1903 – 9 October 1990) was a Swiss mathematician, known for his contributions to differential topology. Biography Georges de Rham was born on 10 September 1903 in Roche, a small village in the canton of Vaud in ...

who expanded understanding of

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, ...

Charles Ehresmann Charles Ehresmann (19 April 1905 – 22 September 1979) was a German-born French mathematician who worked in differential topology and category theory. He was an early member of the Bourbaki group, and is known for his work on the differentia ...

who introduced the theory of fibre bundles and Ehresmann connections, and others. Of particular importance was

Hermann Weyl Hermann Klaus Hugo Weyl (; ; 9 November 1885 – 8 December 1955) was a German mathematician, theoretical physicist, logician and philosopher. Although much of his working life was spent in Zürich, Switzerland, and then Princeton, New Jersey, ...

who made important contributions to the foundations of general relativity, introduced the Weyl tensor providing insight into

, and first defined the notion of a gauge leading to the development of

in physics and

mathematics Mathematics is a field of study that discovers and organizes methods, Mathematical theory, theories and theorems that are developed and Mathematical proof, proved for the needs of empirical sciences and mathematics itself. There are many ar ...

. In the middle and late 20th century differential geometry as a subject expanded in scope and developed links to other areas of mathematics and physics. The development of

and

Yang–Mills theory Yang–Mills theory is a quantum field theory for nuclear binding devised by Chen Ning Yang and Robert Mills in 1953, as well as a generic term for the class of similar theories. The Yang–Mills theory is a gauge theory based on a special un ...

in physics brought bundles and connections into focus, leading to developments in

. Many analytical results were investigated including the proof of the Atiyah–Singer index theorem. The development of

complex geometry In mathematics, complex geometry is the study of geometry, geometric structures and constructions arising out of, or described by, the complex numbers. In particular, complex geometry is concerned with the study of space (mathematics), spaces su ...

was spurred on by parallel results in

algebraic geometry Algebraic geometry is a branch of mathematics which uses abstract algebraic techniques, mainly from commutative algebra, to solve geometry, geometrical problems. Classically, it studies zero of a function, zeros of multivariate polynomials; th ...

, and results in the geometry and global analysis of complex manifolds were proven by

Shing-Tung Yau Shing-Tung Yau (; ; born April 4, 1949) is a Chinese-American mathematician. He is the director of the Yau Mathematical Sciences Center at Tsinghua University and professor emeritus at Harvard University. Until 2022, Yau was the William Caspar ...

and others. In the latter half of the 20th century new analytic techniques were developed in regards to curvature flows such as the Ricci flow, which culminated in

Grigori Perelman Grigori Yakovlevich Perelman (, ; born 13June 1966) is a Russian mathematician and geometer who is known for his contributions to the fields of geometric analysis, Riemannian geometry, and geometric topology. In 2005, Perelman resigned from his ...

's proof of the

Poincaré conjecture In the mathematical field of geometric topology, the Poincaré conjecture (, , ) is a theorem about the characterization of the 3-sphere, which is the hypersphere that bounds the unit ball in four-dimensional space. Originally conjectured b ...

. During this same period primarily due to the influence of

Michael Atiyah Sir Michael Francis Atiyah (; 22 April 1929 – 11 January 2019) was a British-Lebanese mathematician specialising in geometry. His contributions include the Atiyah–Singer index theorem and co-founding topological K-theory. He was awarded the ...

, new links between

theoretical physics Theoretical physics is a branch of physics that employs mathematical models and abstractions of physical objects and systems to rationalize, explain, and predict List of natural phenomena, natural phenomena. This is in contrast to experimental p ...

and differential geometry were formed. Techniques from the study of the Yang–Mills equations and

were used by mathematicians to develop new invariants of smooth manifolds. Physicists such as

Edward Witten Edward Witten (born August 26, 1951) is an American theoretical physics, theoretical physicist known for his contributions to string theory, topological quantum field theory, and various areas of mathematics. He is a professor emeritus in the sc ...

, the only physicist to be awarded a

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 Mathematicians, International Congress of the International Mathematical Union (IMU), a meeting that takes place e ...

, made new impacts in mathematics by using

topological quantum field theory In gauge theory and mathematical physics, a topological quantum field theory (or topological field theory or TQFT) is a quantum field theory that computes topological invariants. While TQFTs were invented by physicists, they are also of mathemati ...

and

string theory In physics, string theory is a theoretical framework in which the point-like particles of particle physics are replaced by one-dimensional objects called strings. String theory describes how these strings propagate through space and intera ...

to make predictions and provide frameworks for new rigorous mathematics, which has resulted for example in the conjectural mirror symmetry and the Seiberg–Witten invariants.

Branches

Riemannian geometry

Riemannian geometry studies

Riemannian manifold In differential geometry, a Riemannian manifold is a geometric space on which many geometric notions such as distance, angles, length, volume, and curvature are defined. Euclidean space, the N-sphere, n-sphere, hyperbolic space, and smooth surf ...

s with a ''Riemannian metric''. This is a concept of distance expressed by means of a smooth positive definite

symmetric bilinear form In mathematics, a symmetric bilinear form on a vector space is a bilinear map from two copies of the vector space to the field of scalars such that the order of the two vectors does not affect the value of the map. In other words, it is a biline ...

defined on the tangent space at each point. Riemannian geometry generalizes

to spaces that are not necessarily flat, though they still resemble Euclidean space at each point infinitesimally, i.e. in the first order of approximation. Various concepts based on length, such as the

arc length Arc length is the distance between two points along a section of a curve. Development of a formulation of arc length suitable for applications to mathematics and the sciences is a problem in vector calculus and in differential geometry. In the ...

of curves,

area Area is the measure of a region's size on a surface. The area of a plane region or ''plane area'' refers to the area of a shape or planar lamina, while '' surface area'' refers to the area of an open surface or the boundary of a three-di ...

of plane regions, and

volume Volume is a measure of regions in three-dimensional space. It is often quantified numerically using SI derived units (such as the cubic metre and litre) or by various imperial or US customary units (such as the gallon, quart, cubic inch) ...

of solids all possess natural analogues in Riemannian geometry. The notion of a

directional derivative In multivariable calculus, the directional derivative measures the rate at which a function changes in a particular direction at a given point. The directional derivative of a multivariable differentiable (scalar) function along a given vect ...

of a function from

multivariable calculus Multivariable calculus (also known as multivariate calculus) is the extension of calculus in one variable to calculus with functions of several variables: the differentiation and integration of functions involving multiple variables ('' mult ...

is extended to the notion of a

of a

. Many concepts of analysis and differential equations have been generalized to the setting of Riemannian manifolds. A distance-preserving

diffeomorphism In mathematics, a diffeomorphism is an isomorphism of differentiable manifolds. It is an invertible function that maps one differentiable manifold to another such that both the function and its inverse are continuously differentiable. Definit ...

between Riemannian manifolds is called 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'' me ...

. This notion can also be defined ''locally'', i.e. for small neighborhoods of points. Any two regular curves are locally isometric. However, the Theorema Egregium of Carl Friedrich Gauss showed that for surfaces, the existence of a local isometry imposes that the

s at the corresponding points must be the same. In higher dimensions, the

Riemann curvature tensor Georg Friedrich Bernhard Riemann (; ; 17September 182620July 1866) was a German mathematician who made profound contributions to mathematical analysis, analysis, number theory, and differential geometry. In the field of real analysis, he is mos ...

is an important pointwise invariant associated with a Riemannian manifold that measures how close it is to being flat. An important class of Riemannian manifolds is the Riemannian symmetric spaces, whose curvature is not necessarily constant. These are the closest analogues to the "ordinary" plane and space considered in Euclidean and

Pseudo-Riemannian geometry

Pseudo-Riemannian geometry generalizes Riemannian geometry to the case in which the

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 allows ...

need not be positive-definite. A special case of this is a

Lorentzian manifold In mathematical physics, a pseudo-Riemannian manifold, also called a semi-Riemannian manifold, is a differentiable manifold with a metric tensor that is everywhere non-degenerate bilinear form, nondegenerate. This is a generalization of a Riema ...

, which is the mathematical basis of Einstein's general relativity theory of gravity.

Finsler geometry

Finsler geometry has ''Finsler manifolds'' as the main object of study. This is a differential manifold with a ''Finsler metric'', that is, a

Banach norm In mathematics, more specifically in functional analysis, a Banach space (, ) is a Complete metric space, complete normed vector space. Thus, a Banach space is a vector space with a Metric (mathematics), metric that allows the computation of Norm ( ...

defined on each tangent space. Riemannian manifolds are special cases of the more general Finsler manifolds. A Finsler structure on a manifold

M

is a function

F:\mathrmM\to[0,\infty)

such that: #

F(x,my)=mF(x,y)

for all

(x,y)

\mathrmM

and all

m\ge 0

, #

F

is infinitely differentiable in

\mathrmM\setminus\

, # The vertical Hessian of

F^2

is positive definite.

Symplectic geometry

Symplectic geometry is the study of

symplectic manifold In differential geometry, a subject of mathematics, a symplectic manifold is a smooth manifold, M , equipped with a closed nondegenerate differential 2-form \omega , called the symplectic form. The study of symplectic manifolds is called sy ...

s. An almost symplectic manifold is a differentiable manifold equipped with a smoothly varying non-degenerate skew-symmetric matrix, skew-symmetric

bilinear form In mathematics, a bilinear form is a bilinear map on a vector space (the elements of which are called '' vectors'') over a field ''K'' (the elements of which are called '' scalars''). In other words, a bilinear form is a function that is linea ...

on each tangent space, i.e., a nondegenerate 2- form ''ω'', called the ''symplectic form''. A symplectic manifold is an almost symplectic manifold for which the symplectic form ''ω'' is closed: . A diffeomorphism between two symplectic manifolds which preserves the symplectic form is called a

symplectomorphism In mathematics, a symplectomorphism or symplectic map is an isomorphism in the category of symplectic manifolds. In classical mechanics, a symplectomorphism represents a transformation of phase space that is volume-preserving and preserves the ...

. Non-degenerate skew-symmetric bilinear forms can only exist on even-dimensional vector spaces, so symplectic manifolds necessarily have even dimension. In dimension 2, a symplectic manifold is just a surface endowed with an area form and a symplectomorphism is an area-preserving diffeomorphism. The

phase space The phase space of a physical system is the set of all possible physical states of the system when described by a given parameterization. Each possible state corresponds uniquely to a point in the phase space. For mechanical systems, the p ...

of a mechanical system is a symplectic manifold and they made an implicit appearance already in the work of

Joseph Louis Lagrange Joseph-Louis Lagrange (born Giuseppe Luigi Lagrangiaanalytical mechanics In theoretical physics and mathematical physics, analytical mechanics, or theoretical mechanics is a collection of closely related formulations of classical mechanics. Analytical mechanics uses '' scalar'' properties of motion representing the sy ...

and later in

Carl Gustav Jacobi Carl Gustav Jacob Jacobi (; ; 10 December 1804 – 18 February 1851) was a German mathematician who made fundamental contributions to elliptic functions, dynamics, differential equations, determinants and number theory. Biography Jacobi was ...

's and

William Rowan Hamilton Sir William Rowan Hamilton (4 August 1805 – 2 September 1865) was an Irish astronomer, mathematician, and physicist who made numerous major contributions to abstract algebra, classical mechanics, and optics. His theoretical works and mathema ...

's formulations of classical mechanics. By contrast with Riemannian geometry, where the

provides a local invariant of Riemannian manifolds, Darboux's theorem states that all symplectic manifolds are locally isomorphic. The only invariants of a symplectic manifold are global in nature and topological aspects play a prominent role in symplectic geometry. The first result in symplectic topology is probably the Poincaré–Birkhoff theorem, conjectured by

and then proved by G.D. Birkhoff in 1912. It claims that if an area preserving map of an annulus twists each boundary component in opposite directions, then the map has at least two fixed points.

Contact geometry

Contact geometry deals with certain manifolds of odd dimension. It is close to symplectic geometry and like the latter, it originated in questions of classical mechanics. A ''contact structure'' on a -dimensional manifold ''M'' is given by a smooth hyperplane field ''H'' in the

tangent bundle A tangent bundle is the collection of all of the tangent spaces for all points on a manifold, structured in a way that it forms a new manifold itself. Formally, in differential geometry, the tangent bundle of a differentiable manifold M is ...

that is as far as possible from being associated with the level sets of a differentiable function on ''M'' (the technical term is "completely nonintegrable tangent hyperplane distribution"). Near each point ''p'', a hyperplane distribution is determined by a nowhere vanishing

1-form In differential geometry, a one-form (or covector field) on a differentiable manifold is a differential form of degree one, that is, a smooth section of the cotangent bundle. Equivalently, a one-form on a manifold M is a smooth mapping of the t ...

\alpha

, which is unique up to multiplication by a nowhere vanishing function: :

H_p = \ker\alpha_p\subset T_M.

A local 1-form on ''M'' is a ''contact form'' if the restriction of its

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 ...

to ''H'' is a non-degenerate two-form and thus induces a symplectic structure on ''H''_''p'' at each point. If the distribution ''H'' can be defined by a global one-form

\alpha

then this form is contact if and only if the top-dimensional form :

\alpha\wedge (d\alpha)^n

is a

volume form In mathematics, a volume form or top-dimensional form is a differential form of degree equal to the differentiable manifold dimension. Thus on a manifold M of dimension n, a volume form is an n-form. It is an element of the space of sections of t ...

on ''M'', i.e. does not vanish anywhere. A contact analogue of the Darboux theorem holds: all contact structures on an odd-dimensional manifold are locally isomorphic and can be brought to a certain local normal form by a suitable choice of the coordinate system.

Complex and Kähler geometry

''Complex differential geometry'' is the study of complex manifolds. An

almost complex manifold In mathematics, an almost complex manifold is a smooth manifold equipped with a smooth linear complex structure on each tangent space. Every complex manifold is an almost complex manifold, but there are almost complex manifolds that are not comple ...

is a ''real'' manifold

M

, endowed with a

of type (1, 1), i.e. a vector bundle endomorphism (called an '' almost complex structure'') :

J:TM\rightarrow TM

, such that

J^2=-1. \,

It follows from this definition that an almost complex manifold is even-dimensional. An almost complex manifold is called ''complex'' if

N_J=0

, where

N_J

is a tensor of type (2, 1) related to

J

, called the Nijenhuis tensor (or sometimes the ''torsion''). An almost complex manifold is complex if and only if it admits a holomorphic coordinate atlas. An '' almost Hermitian structure'' is given by an almost complex structure ''J'', along with a

''g'', satisfying the compatibility condition :

g(JX,JY)=g(X,Y). \,

An almost Hermitian structure defines naturally a differential two-form :

\omega_(X,Y):=g(JX,Y). \,

The following two conditions are equivalent: #

N_J=0\mboxd\omega=0 \,

\nabla J=0 \,

where

\nabla

is 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 that preserves the ( pseudo-) Riemannian ...

g

. In this case,

(J, g)

is called a '' Kähler structure'', and a ''Kähler manifold'' is a manifold endowed with a Kähler structure. In particular, a Kähler manifold is both a complex and a

. A large class of Kähler manifolds (the class of Hodge manifolds) is given by all the smooth complex projective varieties.

CR geometry

CR geometry is the study of the intrinsic geometry of boundaries of domains in

Conformal geometry

Conformal geometry In mathematics, conformal geometry is the study of the set of angle-preserving ( conformal) transformations on a space. In a real two dimensional space, conformal geometry is precisely the geometry of Riemann surfaces. In space higher than two di ...

is the study of the set of angle-preserving (conformal) transformations on a space.

Differential topology

Differential topology In mathematics, differential topology is the field dealing with the topological properties and smooth properties of smooth manifolds. In this sense differential topology is distinct from the closely related field of differential geometry, which ...

is the study of global geometric invariants without a metric or symplectic form. Differential topology starts from the natural operations such as Lie derivative of natural

vector bundle In mathematics, a vector bundle is a topological construction that makes precise the idea of a family of vector spaces parameterized by another space X (for example X could be a topological space, a manifold, or an algebraic variety): to eve ...

s and de Rham differential of forms. Beside

Lie algebroid In mathematics, a Lie algebroid is a vector bundle A \rightarrow M together with a Lie bracket on its space of sections \Gamma(A) and a vector bundle morphism \rho: A \rightarrow TM, satisfying a Leibniz rule. A Lie algebroid can thus be thought ...

s, also Courant algebroids start playing a more important role.

Lie groups

Lie group In mathematics, a Lie group (pronounced ) is a group (mathematics), group that is also a differentiable manifold, such that group multiplication and taking inverses are both differentiable. A manifold is a space that locally resembles Eucli ...

is a group in the category of smooth manifolds. Beside the algebraic properties this enjoys also differential geometric properties. The most obvious construction is that of a Lie algebra which is the tangent space at the unit endowed with the Lie bracket between left-invariant

vector field In vector calculus and physics, a vector field is an assignment of a vector to each point in a space, most commonly Euclidean space \mathbb^n. A vector field on a plane can be visualized as a collection of arrows with given magnitudes and dire ...

s. Beside the structure theory there is also the wide field of

representation theory Representation theory is a branch of mathematics that studies abstract algebra, abstract algebraic structures by ''representing'' their element (set theory), elements as linear transformations of vector spaces, and studies Module (mathematics), ...

Geometric analysis

Geometric analysis Geometric analysis is a mathematical discipline where tools from differential equations, especially elliptic partial differential equations (PDEs), are used to establish new results in differential geometry and differential topology. The use of ...

is a mathematical discipline where tools from differential equations, especially elliptic partial differential equations are used to establish new results in differential geometry and differential topology.

Gauge theory

Gauge theory is the study of connections on vector bundles and principal bundles, and arises out of problems in

mathematical physics Mathematical physics is the development of mathematics, mathematical methods for application to problems in physics. The ''Journal of Mathematical Physics'' defines the field as "the application of mathematics to problems in physics and the de ...

and physical

gauge theories In physics, a gauge theory is a type of field theory in which the Lagrangian, and hence the dynamics of the system itself, does not change under local transformations according to certain smooth families of operations (Lie groups). Formally, t ...

which underpin the standard model of particle physics. Gauge theory is concerned with the study of differential equations for connections on bundles, and the resulting geometric

moduli space In mathematics, in particular algebraic geometry, a moduli space is a geometric space (usually a scheme (mathematics), scheme or an algebraic stack) whose points represent algebro-geometric objects of some fixed kind, or isomorphism classes of suc ...

s of solutions to these equations as well as the invariants that may be derived from them. These equations often arise as the Euler–Lagrange equations describing the equations of motion of certain physical systems in

, and so their study is of considerable interest in physics.

Bundles and connections

The apparatus of

principal bundle In mathematics, a principal bundle is a mathematical object that formalizes some of the essential features of the Cartesian product X \times G of a space X with a group G. In the same way as with the Cartesian product, a principal bundle P is equ ...

s, and connections on bundles plays an extraordinarily important role in modern differential geometry. A smooth manifold always carries a natural vector bundle, the

. Loosely speaking, this structure by itself is sufficient only for developing analysis on the manifold, while doing geometry requires, in addition, some way to relate the tangent spaces at different points, i.e. a notion of

parallel transport In differential 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 ...

. An important example is provided by affine connections. For a surface in R³, tangent planes at different points can be identified using a natural path-wise parallelism induced by the ambient Euclidean space, which has a well-known standard definition of metric and parallelism. In

, the

serves a similar purpose. More generally, differential geometers consider spaces with a vector bundle and an arbitrary affine connection which is not defined in terms of a metric. In physics, the manifold may be

and the bundles and connections are related to various physical fields.

Intrinsic versus extrinsic

From the beginning and through the middle of the 19th century, differential geometry was studied from the ''extrinsic'' point of view: curves and surfaces were considered as lying in a Euclidean space of higher dimension (for example a surface in an ambient space of three dimensions). The simplest results are those in the differential geometry of curves and differential geometry of surfaces. Starting with the work of Riemann, the ''intrinsic'' point of view was developed, in which one cannot speak of moving "outside" the geometric object because it is considered to be given in a free-standing way. The fundamental result here is Gauss's theorema egregium, to the effect that

is an intrinsic invariant. The intrinsic point of view is more flexible. For example, it is useful in relativity where space-time cannot naturally be taken as extrinsic. However, there is a price to pay in technical complexity: the intrinsic definitions of curvature and connections become much less visually intuitive. These two points of view can be reconciled, i.e. the extrinsic geometry can be considered as a structure additional to the intrinsic one. (See the

Nash embedding theorem The Nash embedding theorems (or imbedding theorems), named after John Forbes Nash Jr., state that every Riemannian manifold can be isometrically embedding, embedded into some Euclidean space. Isometry, Isometric means preserving the length of ever ...

.) In the formalism of

geometric calculus In mathematics, geometric calculus extends geometric algebra to include differentiation and integration. The formalism is powerful and can be shown to reproduce other mathematical theories including vector calculus, differential geometry, an ...

both extrinsic and intrinsic geometry of a manifold can be characterized by a single bivector-valued one-form called the shape operator.

Applications

Below are some examples of how differential geometry is applied to other fields of science and mathematics. *In

physics Physics is the scientific study of matter, its Elementary particle, fundamental constituents, its motion and behavior through space and time, and the related entities of energy and force. "Physical science is that department of knowledge whi ...

, differential geometry has many applications, including: **Differential geometry is the language in which

's general theory of relativity is expressed. According to the theory, the universe is a smooth manifold equipped with a pseudo-Riemannian metric, which describes the curvature of

. Understanding this curvature is essential for the positioning of

satellites A satellite or an artificial satellite is an object, typically a spacecraft, placed into orbit around a celestial body. They have a variety of uses, including communication relay, weather forecasting, navigation ( GPS), broadcasting, scientif ...

into orbit around the Earth. Differential geometry is also indispensable in the study of

gravitational lensing A gravitational lens is matter, such as a galaxy cluster, cluster of galaxies or a point particle, that bends light from a distant source as it travels toward an observer. The amount of gravitational lensing is described by Albert Einstein's Ge ...

and

black holes A black hole is a massive, compact astronomical object so dense that its gravity prevents anything from escaping, even light. Albert Einstein's theory of general relativity predicts that a sufficiently compact mass will form a black hole. Th ...

. **

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, ...

are used in the study of

electromagnetism In physics, electromagnetism is an interaction that occurs between particles with electric charge via electromagnetic fields. The electromagnetic force is one of the four fundamental forces of nature. It is the dominant force in the interacti ...

. **Differential geometry has applications to both

Lagrangian mechanics In physics, Lagrangian mechanics is a formulation of classical mechanics founded on the d'Alembert principle of virtual work. It was introduced by the Italian-French mathematician and astronomer Joseph-Louis Lagrange in his presentation to the ...

and

Hamiltonian mechanics In physics, Hamiltonian mechanics is a reformulation of Lagrangian mechanics that emerged in 1833. Introduced by Sir William Rowan Hamilton, Hamiltonian mechanics replaces (generalized) velocities \dot q^i used in Lagrangian mechanics with (gener ...

. Symplectic manifolds in particular can be used to study Hamiltonian systems. **Riemannian geometry and contact geometry have been used to construct the formalism of geometrothermodynamics which has found applications in classical equilibrium

thermodynamics Thermodynamics is a branch of physics that deals with heat, Work (thermodynamics), work, and temperature, and their relation to energy, entropy, and the physical properties of matter and radiation. The behavior of these quantities is governed b ...

. *In

and

biophysics Biophysics is an interdisciplinary science that applies approaches and methods traditionally used in physics to study biological phenomena. Biophysics covers all scales of biological organization, from molecular to organismic and populations ...

when modelling cell membrane structure under varying pressure. *In

, differential geometry has applications to the field of

econometrics Econometrics is an application of statistical methods to economic data in order to give empirical content to economic relationships. M. Hashem Pesaran (1987). "Econometrics", '' The New Palgrave: A Dictionary of Economics'', v. 2, p. 8 p. 8 ...

. *

Geometric modeling __NOTOC__ Geometric modeling is a branch of applied mathematics and computational geometry that studies methods and algorithms for the mathematical description of shapes. The shapes studied in geometric modeling are mostly two- or three-dimensi ...

(including

) and

computer-aided geometric 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 c ...

draw on ideas from differential geometry. *In

, differential geometry can be applied to solve problems in

digital signal processing Digital signal processing (DSP) is the use of digital processing, such as by computers or more specialized digital signal processors, to perform a wide variety of signal processing operations. The digital signals processed in this manner are a ...

. *In

, differential geometry can be used to analyze nonlinear controllers, particularly geometric control * In

probability Probability is a branch of mathematics and statistics concerning events and numerical descriptions of how likely they are to occur. The probability of an event is a number between 0 and 1; the larger the probability, the more likely an e ...

statistics Statistics (from German language, German: ', "description of a State (polity), state, a country") is the discipline that concerns the collection, organization, analysis, interpretation, and presentation of data. In applying statistics to a s ...

, and

information theory Information theory is the mathematical study of the quantification (science), quantification, Data storage, storage, and telecommunications, communication of information. The field was established and formalized by Claude Shannon in the 1940s, ...

, one can interpret various structures as Riemannian manifolds, which yields the field of

information geometry Information geometry is an interdisciplinary field that applies the techniques of differential geometry to study probability theory and statistics. It studies statistical manifolds, which are Riemannian manifolds whose points correspond to proba ...

, particularly via the

Fisher information metric In information geometry, the Fisher information metric is a particular Riemannian metric which can be defined on a smooth statistical manifold, ''i.e.'', a smooth manifold whose points are probability distributions. It can be used to calculate the ...

. *In

structural geology Structural geology is the study of the three-dimensional distribution of rock units with respect to their deformational histories. The primary goal of structural geology is to use measurements of present-day rock geometries to uncover informati ...

, differential geometry is used to analyze and describe geologic structures. *In

, differential geometry is used to analyze shapes. *In

image processing An image or picture is a visual representation. An image can be two-dimensional, such as a drawing, painting, or photograph, or three-dimensional, such as a carving or sculpture. Images may be displayed through other media, including a pr ...

, differential geometry is used to process and analyse data on non-flat surfaces. *

's proof of the

using the techniques of Ricci flows demonstrated the power of the differential-geometric approach to questions in

and it highlighted the important role played by its analytic methods. * In wireless communications, Grassmannian manifolds are used for

beamforming Beamforming or spatial filtering is a signal processing technique used in sensor arrays for directional signal transmission or reception. This is achieved by combining elements in an antenna array in such a way that signals at particular angles ...

techniques in multiple antenna systems. * In

, for calculating distances and angles on the mean sea level surface of the

, modelled by an ellipsoid of revolution. * In

neuroimaging Neuroimaging is the use of quantitative (computational) techniques to study the neuroanatomy, structure and function of the central nervous system, developed as an objective way of scientifically studying the healthy human brain in a non-invasive ...

and brain-computer interface, symmetric positive definite manifolds are used to model functional, structural, or electrophysiological connectivity matrices.

References

External links

*
B. Conrad. Differential Geometry handouts, Stanford University

A Modern Course on Curves and Surfaces, Richard S Palais, 2003

Richard Palais's 3DXM Surfaces Gallery

N. J. Hicks, Notes on Differential Geometry, Van Nostrand.MIT OpenCourseWare: Differential Geometry, Fall 2008
{{DEFAULTSORT:Differential Geometry Geometry processing

History and development

Classical antiquity until the Renaissance (300 BC1600 AD)

After calculus (1600–1800)

Intrinsic geometry and non-Euclidean geometry (1800–1900)

Modern differential geometry (1900–2000)

Branches

Riemannian geometry

Pseudo-Riemannian geometry

Finsler geometry

Symplectic geometry

Contact geometry

Complex and Kähler geometry

CR geometry

Conformal geometry

Differential topology

Lie groups

Geometric analysis

Gauge theory

Bundles and connections

Intrinsic versus extrinsic

Applications

See also

References

Further reading

External links