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Differential geometry is a mathematical discipline that studies the
geometry Geometry (; ) is, with arithmetic, one of the oldest branches of mathematics. It is concerned with properties of space such as the distance, shape, size, and relative position of figures. A mathematician who works in the field of geometry is c ...
of smooth shapes and smooth spaces, otherwise known as smooth manifolds. It uses the techniques of
differential calculus In mathematics, differential calculus is a subfield of calculus that studies the rates at which quantities change. It is one of the two traditional divisions of calculus, the other being integral calculus—the study of the area beneath a curve ...
, integral calculus,
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 matrice ...
and multilinear algebra. The field has its origins in the study of spherical geometry as far back as
antiquity Antiquity or Antiquities may refer to: Historical objects or periods Artifacts *Antiquities, objects or artifacts surviving from ancient cultures Eras Any period before the European Middle Ages (5th to 15th centuries) but still within the histo ...
. It also relates to
astronomy Astronomy () is a natural science that studies celestial objects and phenomena. It uses mathematics, physics, and chemistry in order to explain their origin and evolution. Objects of interest include planets, moons, stars, nebulae, g ...
, the geodesy of the
Earth Earth is the third planet from the Sun and the only astronomical object known to harbor life. While large volumes of water can be found throughout the Solar System, only Earth sustains liquid surface water. About 71% of Earth's sur ...
, and later the study of hyperbolic geometry 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, that is, in Euclid's ''Elements'', it was the three-dimensional space of Euclidean geometry, but in modern mathematics there are Euclidea ...
, 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 manifolds. A geometric structure is one which defines some notion of size, distance, shape, volume, or other rigidifying structure. For example, in Riemannian geometry distances and angles are specified, in symplectic geometry volumes may be computed, in conformal geometry only angles are specified, and in gauge theory certain fields are given over the space. Differential geometry is closely related to, and is sometimes taken to include, differential topology, which concerns itself with properties of differentiable manifolds which 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 equation In mathematics, a differential equation is an equation that relates one or more unknown functions and their derivatives. In applications, the functions generally represent physical quantities, the derivatives represent their rates of change, ...
s, otherwise known as geometric analysis. Differential geometry finds applications throughout mathematics and the natural sciences. Most prominently the language of differential geometry was used by
Albert Einstein Albert Einstein ( ; ; 14 March 1879 – 18 April 1955) was a German-born theoretical physicist, widely acknowledged to be one of the greatest and most influential physicists of all time. Einstein is best known for developing the theor ...
in his theory of general relativity, and subsequently by physicists in the development of quantum field theory 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 natural science that covers the elements that make up matter to the compounds made of atoms, molecules and ions: their composition, structure, proper ...
,
economics Economics () is the social science that studies the production, distribution, and consumption of goods and services. Economics focuses on the behaviour and interactions of economic agents and how economies work. Microeconomics anal ...
,
engineering Engineering is the use of scientific principles to design and build machines, structures, and other items, including bridges, tunnels, roads, vehicles, and buildings. The discipline of engineering encompasses a broad range of more speciali ...
,
control theory Control theory is a field of 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 application of system inputs to drive ...
,
computer graphics Computer graphics deals with generating images with the aid of computers. Today, computer graphics is a core technology in digital photography, film, video games, cell phone and computer displays, and many specialized applications. A great de ...
and computer vision, and recently in
machine learning Machine learning (ML) is a field of inquiry devoted to understanding and building methods that 'learn', that is, methods that leverage data to improve performance on some set of tasks. It is seen as a part of artificial intelligence. Machine ...
.


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 the classical era, classical period or classical age) is the period of cultural history between the 8th century BC and the 5th century AD centred on the Mediterranean Sea, comprising the interlocking civilizations of ...
. It is intimately linked to the development of geometry more generally, of the notion of space and shape, and of
topology In mathematics, topology (from the Greek words , and ) is concerned with the properties of a geometric object that are preserved under continuous deformations, such as stretching, twisting, crumpling, and bending; that is, without closing ...
, especially the study of manifolds. In this section we focus primarily on the history of the application of infinitesimal methods to geometry, and later to the ideas of tangent spaces, and eventually the development of the modern formalism of the subject in terms of tensors and tensor fields.


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
classical antiquity Classical antiquity (also the classical era, classical period or classical age) is the period of cultural history between the 8th century BC and the 5th century AD centred on the Mediterranean Sea, comprising the interlocking civilizations of ...
. In particular, much was known about the geometry of the
Earth Earth is the third planet from the Sun and the only astronomical object known to harbor life. While large volumes of water can be found throughout the Solar System, only Earth sustains liquid surface water. About 71% of Earth's sur ...
, a spherical geometry, in the time of the
ancient Greek Ancient Greek includes the forms of the Greek language used in ancient Greece and the ancient world from around 1500 BC to 300 BC. It is often roughly divided into the following periods: Mycenaean Greek (), Dark Ages (), the Archaic pe ...
mathematicians. Famously, Eratosthenes calculated the
circumference In geometry, the circumference (from Latin ''circumferens'', meaning "carrying around") is the perimeter of a circle or ellipse. That is, the circumference would be the arc length of the circle, as if it were opened up and straightened out ...
of the Earth around 200 BC, and around 150 AD
Ptolemy Claudius Ptolemy (; grc-gre, Πτολεμαῖος, ; la, Claudius Ptolemaeus; AD) was a mathematician, astronomer, astrologer, geographer, and music theorist, who wrote about a dozen scientific treatises, three of which were of importanc ...
in his ''
Geography Geography (from Greek: , ''geographia''. Combination of Greek words ‘Geo’ (The Earth) and ‘Graphien’ (to describe), literally "earth description") is a field of science devoted to the study of the lands, features, inhabitants, an ...
'' introduced the stereographic projection 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 geodesy, although in a much simplified form. Namely, as far back as
Euclid Euclid (; grc-gre, Εὐκλείδης; 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 ...
'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
Earth Earth is the third planet from the Sun and the only astronomical object known to harbor life. While large volumes of water can be found throughout the Solar System, only Earth sustains liquid surface water. About 71% of Earth's sur ...
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 paths by Eratosthenes and others can be considered a rudimentary measure of arclength 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 infinitesimals to the study of geometry, a precursor to the modern calculus-based study of the subject. In
Euclid Euclid (; grc-gre, Εὐκλείδης; 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 ...
's '' Elements'' the notion of
tangency In geometry, the tangent line (or simply tangent) to a plane curve at a given point is the straight line that "just touches" the curve at that point. Leibniz defined it as the line through a pair of infinitely close points on the curve. Mo ...
of a line to a circle is discussed, and Archimedes 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 whose areas converge to the area of the containing shape. If the sequence is correctly constructed, the difference in are ...
to compute the areas of smooth shapes such as the
circle A circle is a shape consisting of all points in a plane that are at a given distance from a given point, the centre. Equivalently, it is the curve traced out by a point that moves in a plane so that its distance from a given point is cons ...
, 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 ( , ) , from , with the same meanings. is a period in European history marking the transition from the Middle Ages to modernity and covering the 15th and 16th centuries, characterized by an effort to revive and surpass ide ...
. Before the development of calculus by Newton and
Leibniz Gottfried Wilhelm (von) Leibniz . ( – 14 November 1716) was a German polymath active as a mathematician, philosopher, scientist and diplomat. He is one of the most prominent figures in both the history of philosophy and the history of ma ...
, the most significant development in the understanding of differential geometry came from
Gerardus Mercator Gerardus Mercator (; 5 March 1512 – 2 December 1594) was a 16th-century geographer, cosmographer and cartographer from the County of Flanders. He is most renowned for creating the 1569 world map based on a new projection which represented ...
'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.


After calculus (1600–1800)

The first systematic or rigorous treatment of geometry using the theory of infinitesimals and notions from
calculus Calculus, originally called infinitesimal calculus or "the calculus of infinitesimals", is the mathematics, mathematical study of continuous change, in the same way that geometry is the study of shape, and algebra is the study of generalizati ...
began around the 1600s when calculus was first developed by Gottfried Leibniz and
Isaac Newton Sir Isaac Newton (25 December 1642 – 20 March 1726/27) was an English mathematician, physicist, astronomer, alchemist, Theology, theologian, and author (described in his time as a "natural philosophy, natural philosopher"), widely ...
. At this time, the recent work of
René Descartes René Descartes ( or ; ; Latinized: Renatus Cartesius; 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 science. Ma ...
introducing analytic coordinates to geometry allowed geometric shapes of increasing complexity to be described rigorously. In particular around this time Pierre de Fermat, Newton, and Leibniz began the study of plane curves and the investigation of concepts such as points of
inflection In linguistic morphology, inflection (or inflexion) is a process of word formation in which a word is modified to express different grammatical categories such as tense, case, voice, aspect, person, number, gender, mood, animacy, and ...
and circles of osculation, which aid in the measurement of curvature. 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 (; ; ar, يَعْقُوب, Yaʿqūb; gr, Ἰακώβ, Iakṓb), later given the name Israel, is regarded as a patriarch of the Israelites and is an important figure in Abrahamic religions, such as Judaism, Christianity, and Islam. ...
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 Gracio ...
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 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 curvature, is written down. In the wake of the development of analytic geometry and plane curves, Alexis Clairaut 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 In geometry, the subtangent and related terms are certain line segments defined using the line tangent to a curve at a given point and the coordinate axes. The terms are somewhat archaic today but were in common use until the early part of the 20 ...
directions to space curves in relation to the directions which lie along a surface on which the space curve lies. Thus Clairaut demonstrated an implicit understanding of the tangent space 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 are the maximum and minimum values of the curvature as expressed by the eigenvalues of the shape operator at that point. They measure how the surface bends b ...
s later studied by Gauss and others. Around this same time,
Leonhard Euler Leonhard Euler ( , ; 15 April 170718 September 1783) was a Swiss mathematician, physicist, astronomer, geographer, logician and engineer who founded the studies of graph theory and topology and made pioneering and influential discoveries ...
, 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 geodesic on a surface deriving the first analytical
geodesic equation In geometry, a geodesic () is a curve representing in some sense the shortest path ( arc) between two points in a surface, or more generally in a Riemannian manifold. The term also has meaning in any differentiable manifold with a connection. ...
, 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'' 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 Einstein's theory of gravity, is the geometric theory of gravitation published by Albert Einstein in 1915 and is the current description of gravitation in modern physics ...
, 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 functions and functionals, to find maxima and minima of functionals: mappings from a set of functions t ...
, which underpins in modern differential geometry many techniques in symplectic geometry and geometric analysis. This theory was used by Lagrange, a co-developer of the calculus of variations, to derive the first differential equation describing a minimal 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. 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. During ...
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 sh ...
of plane curves and space curves. Several students of Monge made contributions to this same theory, and for example Charles Dupin 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 and Bernhard Riemann, and also in the important contributions of
Nikolai Lobachevsky Nikolai Ivanovich Lobachevsky ( rus, Никола́й Ива́нович Лобаче́вский, p=nʲikɐˈlaj ɪˈvanəvʲɪtɕ ləbɐˈtɕɛfskʲɪj, a=Ru-Nikolai_Ivanovich_Lobachevsky.ogg; – ) was a Russian mathematician and geometer, ...
on hyperbolic geometry and non-Euclidean geometry and throughout the same period the development of projective geometry. 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, 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 geometry, a geodesic () is a curve representing in some sense the shortest path ( arc) between two points in a surface, or more generally in a Riemannian manifold. The term also has meaning in any differentiable manifold with a connectio ...
in various non-Euclidean geometries on surfaces. At this time Gauss was already of the opinion that the standard paradigm of Euclidean geometry 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 János Bolyai (; 15 December 1802 – 27 January 1860) or Johann Bolyai, was a Hungarian mathematician, who developed absolute geometry—a geometry that includes both Euclidean geometry and hyperbolic geometry. The discovery of a consis ...
and Lobachevsky independently discovered hyperbolic geometry 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 coined the term non-Euclidean geometry in 1871, and through the Erlangen program put Euclidean and non-Euclidean geometries on the same footing. Implicitly, the spherical geometry of the Earth that had been studied since antiquity was a non-Euclidean geometry, an elliptic geometry. The development of intrinsic differential geometry in the language of Gauss was spurred on by his student, Bernhard Riemann in his Habilitationsschrift, ''On the hypotheses which lie at the foundation of geometry''. In this work Riemann introduced the notion of a Riemannian metric 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' about the linear element ds of a surface. At this time Riemann began to introduce the systematic use of
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 matrice ...
and multilinear algebra into the subject, making great use of the theory of quadratic forms 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 geometrical space in which closeness is defined but cannot necessarily be measured by a numeric distance. More specifically, a topological space is a set whose elements are called poin ...
had not been encountered, but he did propose that it might be possible to investigate or measure the properties of the metric of spacetime 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 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 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. Life and career Marius S ...
and
Jean Gaston Darboux Jean-Gaston Darboux FAS MIF FRS FRSE (14 August 1842 – 23 February 1917) was a French mathematician. Life According this birth certificate he was born in Nîmes in France on 14 August 1842, at 1 am. However, probably due to the midnig ...
, leading to important results in the theory of Lie groups and symplectic geometry. The notion of differential calculus on curved spaces was studied by
Elwin Christoffel Elwin Bruno Christoffel (; 10 November 1829 – 15 March 1900) was a German mathematician and physicist. He introduced fundamental concepts of differential geometry, opening the way for the development of tensor calculus, which would later pro ...
, who introduced the Christoffel symbols which describe the covariant derivative in 1868, and by others including Eugenio Beltrami who studied many analytic questions on manifolds. In 1899
Luigi Bianchi Luigi Bianchi (18 January 1856 – 6 June 1928) was an Italian mathematician. He was born in Parma, Emilia-Romagna, and died in Pisa. He was a leading member of the vigorous geometric school which flourished in Italy during the later years of ...
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 signi ...
and Gregorio Ricci-Curbastro produced their textbook systematically developing the theory of
absolute differential calculus In mathematics, Ricci calculus constitutes the rules of index notation and manipulation for tensors and tensor fields on a differentiable manifold, with or without a metric tensor or connection. It is also the modern name for what used to be ...
and tensor calculus. It was in this language that differential geometry was used by Einstein in the development of general relativity and
pseudo-Riemannian geometry In differential geometry, a pseudo-Riemannian manifold, also called a semi-Riemannian manifold, is a differentiable manifold with a metric tensor that is everywhere nondegenerate. This is a generalization of a Riemannian manifold in which th ...
.


Modern differential geometry (1900–2000)

The subject of modern differential geometry emerged out of the early 1900s in response to the foundational contributions of many mathematicians, including importantly the work of Henri Poincaré on the foundations of
topology In mathematics, topology (from the Greek words , and ) is concerned with the properties of a geometric object that are preserved under continuous deformations, such as stretching, twisting, crumpling, and bending; that is, without closing ...
.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. As part of this broader movement, the notion of a
topological space In mathematics, a topological space is, roughly speaking, a geometrical space in which closeness is defined but cannot necessarily be measured by a numeric distance. More specifically, a topological space is a set whose elements are called poin ...
was distilled in by Felix Hausdorff 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 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 In theoretical physics, the Einstein–Cartan theory, also known as the Einstein–Cartan–Sciama–Kibble theory, is a classical theory of gravitation similar to general relativity. The theory was first proposed by Élie Cartan in 1922. Einst ...
.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 In mathematics, a characteristic class is a way of associating to each principal bundle of ''X'' a cohomology class of ''X''. The cohomology class measures the extent the bundle is "twisted" and whether it possesses sections. Characteristic classe ...
to the subject and began the study of complex manifolds, Sir William Vallance Douglas Hodge and Georges de Rham who expanded understanding of differential forms,
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 differential ...
who introduced the theory of fibre bundles and
Ehresmann connection In differential geometry, an Ehresmann connection (after the French mathematician Charles Ehresmann who first formalized this concept) is a version of the notion of a connection, which makes sense on any smooth fiber bundle. In particular, it do ...
s, and others. Of particular importance was Hermann Weyl who made important contributions to the foundations of general relativity, introduced the
Weyl tensor In differential geometry, the Weyl curvature tensor, named after Hermann Weyl, is a measure of the curvature of spacetime or, more generally, a pseudo-Riemannian manifold. Like the Riemann curvature tensor, the Weyl tensor expresses the tida ...
providing insight into conformal geometry, and first defined the notion of a
gauge Gauge ( or ) may refer to: Measurement * Gauge (instrument), any of a variety of measuring instruments * Gauge (firearms) * Wire gauge, a measure of the size of a wire ** American wire gauge, a common measure of nonferrous wire diameter, es ...
leading to the development of gauge theory in physics and
mathematics Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...
. 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 gauge theory and
Yang–Mills theory In mathematical physics, Yang–Mills theory is a gauge theory based on a special unitary group SU(''N''), or more generally any compact, reductive Lie algebra. Yang–Mills theory seeks to describe the behavior of elementary particles using t ...
in physics brought bundles and connections into focus, leading to developments in gauge theory. Many analytical results were investigated including the proof of the Atiyah–Singer index theorem. The development of complex geometry was spurred on by parallel results in algebraic geometry, and results in the geometry and global analysis of complex manifolds were proven by Shing-Tung Yau 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 In the mathematical fields of differential geometry and geometric analysis, the Ricci flow ( , ), sometimes also referred to as Hamilton's Ricci flow, is a certain partial differential equation for a Riemannian metric. It is often said to be an ...
, which culminated in Grigori Perelman'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 ...
. During this same period primarily due to the influence of Michael Atiyah, new links between theoretical physics and differential geometry were formed. Techniques from the study of the Yang–Mills equations and gauge theory were used by mathematicians to develop new invariants of smooth manifolds. Physicists such as Edward Witten, the only physicist to be awarded a Fields medal, made new impacts in mathematics by using topological quantum field theory and string theory 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 manifolds, smooth manifolds with a ''Riemannian metric''. This is a concept of distance expressed by means of a smooth
positive definite In mathematics, positive definiteness is a property of any object to which a bilinear form or a sesquilinear form may be naturally associated, which is positive-definite. See, in particular: * Positive-definite bilinear form * Positive-definite fu ...
symmetric bilinear form defined on the tangent space at each point. Riemannian geometry generalizes Euclidean geometry 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 may refer to: Business * Aircraft Radio Corporation, a major avionics manufacturer from the 1920s to the '50s * Airlines Reporting Corporation, an airline-owned company that provides ticket distribution, reporting, and settlement services * ...
of curves,
area Area is the quantity that expresses the extent of a region on the plane or on a curved 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 op ...
of plane regions, and
volume Volume is a measure of occupied 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 of a function from
multivariable calculus Multivariable calculus (also known as multivariate calculus) is the extension of calculus in one Variable (mathematics), variable to calculus with Function of several real variables, functions of several variables: the Differential calculus, di ...
is extended to the notion of a covariant derivative of a tensor. Many concepts of analysis and differential equations have been generalized to the setting of Riemannian manifolds. A distance-preserving diffeomorphism 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'' ...
. 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 Gaussian curvatures at the corresponding points must be the same. In higher dimensions, the Riemann curvature tensor 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 non-Euclidean geometry.


Pseudo-Riemannian geometry

Pseudo-Riemannian geometry In differential geometry, a pseudo-Riemannian manifold, also called a semi-Riemannian manifold, is a differentiable manifold with a metric tensor that is everywhere nondegenerate. This is a generalization of a Riemannian manifold in which th ...
generalizes Riemannian geometry to the case in which the metric tensor need not be positive-definite. A special case of this is a Lorentzian manifold, 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 defined on each tangent space. Riemannian manifolds are special cases of the more general Finsler manifolds. A Finsler structure on a manifold is a function such that: # for all in and all , # is infinitely differentiable in , # The vertical Hessian of is positive definite.


Symplectic geometry

Symplectic geometry is the study of symplectic manifolds. An almost symplectic manifold is a differentiable manifold equipped with a smoothly varying
non-degenerate In mathematics, specifically linear algebra, a degenerate bilinear form on a vector space ''V'' is a bilinear form such that the map from ''V'' to ''V''∗ (the dual space of ''V'' ) given by is not an isomorphism. An equivalent definit ...
skew-symmetric bilinear form on each tangent space, i.e., a nondegenerate 2-
form Form is the shape, visual appearance, or configuration of an object. In a wider sense, the form is the way something happens. Form also refers to: *Form (document), a document (printed or electronic) with spaces in which to write or enter data * ...
''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. 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 of a mechanical system is a symplectic manifold and they made an implicit appearance already in the work of Joseph Louis Lagrange on analytical mechanics and later in Carl Gustav Jacobi's and
William Rowan Hamilton Sir William Rowan Hamilton Doctor of Law, LL.D, Doctor of Civil Law, DCL, Royal Irish Academy, MRIA, Royal Astronomical Society#Fellow, FRAS (3/4 August 1805 – 2 September 1865) was an Irish mathematician, astronomer, and physicist. He was the ...
's formulations of classical mechanics. By contrast with Riemannian geometry, where the curvature 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 Henri Poincaré and then proved by G.D. Birkhoff in 1912. It claims that if an area preserving map of an
annulus Annulus (or anulus) or annular indicates a ring- or donut-shaped area or structure. It may refer to: Human anatomy * ''Anulus fibrosus disci intervertebralis'', spinal structure * Annulus of Zinn, a.k.a. annular tendon or ''anulus tendineus com ...
twists each boundary component in opposite directions, then the map has at least two fixed points.


Contact geometry

Contact geometry In mathematics, contact geometry is the study of a geometric structure on smooth manifolds given by a hyperplane distribution in the tangent bundle satisfying a condition called 'complete non-integrability'. Equivalently, such a distributio ...
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 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 on a differentiable manifold is a smooth section of the cotangent bundle. Equivalently, a one-form on a manifold M is a smooth mapping of the total space of the tangent bundle of M to \R whose restriction to ...
\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 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 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 com ...
is a ''real'' manifold M, endowed with a tensor 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 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 complex ...
(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 Riemannian metric ''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 of 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 symplectic manifold. A large class of Kähler manifolds (the class of Hodge manifolds) is given by all the smooth complex projective varieties.


CR geometry

CR geometry In mathematics, a CR manifold, or Cauchy–Riemann manifold, is a differentiable manifold together with a geometric structure modeled on that of a real hypersurface in a complex vector space, or more generally modeled on an edge of a wedge. Forma ...
is the study of the intrinsic geometry of boundaries of domains in complex manifolds.


Conformal geometry

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


Differential topology

Differential topology 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 bundles and de Rham differential of forms. Beside Lie algebroids, also Courant algebroids start playing a more important role.


Lie groups

A Lie group 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 fields. Beside the structure theory there is also the wide field of representation theory.


Geometric analysis

Geometric analysis 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 refers to the development of 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 developm ...
and physical gauge theories 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 spaces 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 quantum field theory, and so their study is of considerable interest in physics.


Bundles and connections

The apparatus of vector bundles, principal bundles, and connections on bundles plays an extraordinarily important role in modern differential geometry. A smooth manifold always carries a natural vector bundle, the tangent bundle. 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. 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 Riemannian geometry, the Levi-Civita connection 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 spacetime 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 Gaussian curvature 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 embedded into some Euclidean space. Isometric means preserving the length of every path. For instan ...
.) In the formalism of geometric calculus 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 natural science that studies matter, its 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 which ...
, differential geometry has many applications, including: **Differential geometry is the language in which
Albert Einstein Albert Einstein ( ; ; 14 March 1879 – 18 April 1955) was a German-born theoretical physicist, widely acknowledged to be one of the greatest and most influential physicists of all time. Einstein is best known for developing the theor ...
'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 spacetime. Understanding this curvature is essential for the positioning of satellites into orbit around the earth. Differential geometry is also indispensable in the study of gravitational lensing and black holes. ** Differential forms are used in the study of
electromagnetism In physics, electromagnetism is an interaction that occurs between particles with electric charge. It is the second-strongest of the four fundamental interactions, after the strong force, and it is the dominant force in the interactions o ...
. **Differential geometry has applications to both Lagrangian mechanics and Hamiltonian mechanics. 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, and temperature, and their relation to energy, entropy, and the physical properties of matter and radiation. The behavior of these quantities is governed by the four laws ...
. *In
chemistry Chemistry is the scientific study of the properties and behavior of matter. It is a natural science that covers the elements that make up matter to the compounds made of atoms, molecules and ions: their composition, structure, proper ...
and biophysics when modelling cell membrane structure under varying pressure. *In
economics Economics () is the social science that studies the production, distribution, and consumption of goods and services. Economics focuses on the behaviour and interactions of economic agents and how economies work. Microeconomics anal ...
, differential geometry has applications to the field of
econometrics Econometrics is the 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. ...
. *
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-dimensio ...
(including
computer graphics Computer graphics deals with generating images with the aid of computers. Today, computer graphics is a core technology in digital photography, film, video games, cell phone and computer displays, and many specialized applications. A great de ...
) and computer-aided geometric design draw on ideas from differential geometry. *In
engineering Engineering is the use of scientific principles to design and build machines, structures, and other items, including bridges, tunnels, roads, vehicles, and buildings. The discipline of engineering encompasses a broad range of more speciali ...
, differential geometry can be applied to solve problems in digital signal processing. *In
control theory Control theory is a field of 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 application of system inputs to drive ...
, differential geometry can be used to analyze nonlinear controllers, particularly geometric control * In
probability Probability is the branch of mathematics concerning numerical descriptions of how likely an event is to occur, or how likely it is that a proposition is true. The probability of an event is a number between 0 and 1, where, roughly speaking, ...
,
statistics Statistics (from German: '' Statistik'', "description of a state, a country") is the discipline that concerns the collection, organization, analysis, interpretation, and presentation of data. In applying statistics to a scientific, indust ...
, and information theory, 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 pro ...
, 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 measures defined on a common probability spa ...
. *In structural geology, differential geometry is used to analyze and describe geologic structures. *In computer vision, differential geometry is used to analyze shapes. *In image processing, differential geometry is used to process and analyse data on non-flat surfaces. * Grigori Perelman'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 ...
using the techniques of
Ricci flow In the mathematical fields of differential geometry and geometric analysis, the Ricci flow ( , ), sometimes also referred to as Hamilton's Ricci flow, is a certain partial differential equation for a Riemannian metric. It is often said to be an ...
s demonstrated the power of the differential-geometric approach to questions in
topology In mathematics, topology (from the Greek words , and ) is concerned with the properties of a geometric object that are preserved under continuous deformations, such as stretching, twisting, crumpling, and bending; that is, without closing ...
and it highlighted the important role played by its analytic methods. * In wireless communications, Grassmannian manifolds are used for beamforming techniques in multiple antenna systems.


See also

* Abstract differential geometry * Affine differential geometry * Analysis on fractals * Basic introduction to the mathematics of curved spacetime *
Discrete differential geometry Discrete differential geometry is the study of discrete counterparts of notions in differential geometry. Instead of smooth curves and surfaces, there are polygons, meshes, and simplicial complexes. It is used in the study of computer graphics, ...
* Gauss * Glossary of differential geometry and topology * Important publications in differential geometry * Important publications in differential topology * Integral geometry * List of differential geometry topics * Noncommutative geometry * Projective differential geometry *
Synthetic differential geometry In mathematics, synthetic differential geometry is a formalization of the theory of differential geometry in the language of topos theory. There are several insights that allow for such a reformulation. The first is that most of the analytic ...
*
Systolic geometry In mathematics, systolic geometry is the study of systolic invariants of manifolds and polyhedra, as initially conceived by Charles Loewner and developed by Mikhail Gromov, Michael Freedman, Peter Sarnak, Mikhail Katz, Larry Guth, and o ...
* Gauge theory (mathematics)


References


Further reading

* * * * * * * * * *


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