Editing Differential geometry

{{short description|Branch of mathematics dealing with functions and geometric structures on differentiable manifolds}}
[[File:Hyperbolic triangle.svg|thumb|235px|right|A triangle immersed in a saddle-shape plane (a [[hyperbolic paraboloid]]), as well as two diverging [[Hyperbolic geometry#Non-intersecting / parallel lines|ultraparallel lines]]]]
{{General geometry |branches}}

'''Differential geometry''' is a [[Mathematics|mathematical]] discipline that studies the [[geometry]] of smooth shapes and smooth spaces, otherwise known as [[smooth manifold]]s. It uses the techniques of [[Calculus|single variable calculus]], [[vector calculus]], [[linear algebra]] and [[multilinear algebra]]. The field has its origins in the study of [[spherical geometry]] as far back as [[classical antiquity|antiquity]]. It also relates to [[astronomy]], the [[geodesy]] of the [[Earth]], and later  the study of [[hyperbolic geometry]] by [[Nikolai Lobachevsky|Lobachevsky]]. The simplest examples of smooth spaces are the [[Differential geometry of curves|plane and space curves]] and [[Differential geometry of surfaces|surfaces]] in the three-dimensional [[Euclidean space]], 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]]s. 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 (mathematics)|gauge theory]] certain [[tensor field|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 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 equation]]s, otherwise known as [[geometric analysis]].

Differential geometry finds applications throughout mathematics and the [[natural science]]s. Most prominently the language of differential geometry was used by [[Albert Einstein]] 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]], [[economics]], [[engineering]], [[control theory]], [[computer graphics]] and [[computer vision]], and recently in [[machine learning]].

== History and development ==

The history and development of differential geometry as a subject begins at least as far back as [[classical antiquity]]. It is intimately linked to the development of geometry more generally, of the notion of space and shape, and of [[topology]], especially the study of [[manifold]]s. In this section we focus primarily on the history of the application of [[infinitesimal]] methods to geometry, and later to the ideas of [[tangent space]]s, and eventually the development of the modern formalism of the subject in terms of [[tensor]]s and [[tensor field]]s.

=== Classical antiquity until the Renaissance (300 BC{{Snd}}1600 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]]. In particular, much was known about the geometry of the [[Earth]], a [[spherical geometry]], in the time of the [[ancient Greek]] mathematicians. Famously, [[Eratosthenes]] calculated the [[circumference]] of the Earth around 200 BC, and around 150 AD [[Ptolemy]] in his ''[[Geography (Ptolemy)|Geography]]'' introduced the [[stereographic projection]] for the purposes of mapping the shape of the Earth.<ref name="struik1">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.</ref> 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]]'s ''[[Euclid's Elements|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]] 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 [[infinitesimal]]s to the study of geometry, a precursor to the modern calculus-based study of the subject. In [[Euclid]]'s ''[[Euclid's Elements|Elements]]'' the notion of [[tangency]] of a line to a circle is discussed, and [[Archimedes]] applied the [[method of exhaustion]] to compute the areas of smooth shapes such as the [[circle]], and the volumes of smooth three-dimensional solids such as the sphere, cones, and cylinders.<ref name="struik1" />

There was little development in the theory of differential geometry between antiquity and the beginning of the [[Renaissance]]. Before the development of calculus by [[Isaac Newton|Newton]] and [[Leibniz]], the most significant development in the understanding of differential geometry came from [[Gerardus Mercator]]'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 map projection|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.<ref name="struik1" /> This fact reflects the lack of a [[isometry|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) ===
[[File:Osculating circle.svg|thumb|right|An osculating circle of plane curve]]
The first systematic or rigorous treatment of geometry using the theory of infinitesimals and notions from [[calculus]] began around the 1600s when calculus was first developed by [[Gottfried Leibniz]] and [[Isaac Newton]]. At this time, the recent work of [[René Descartes]] introducing [[analytic geometry|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 curve]]s and the investigation of concepts such as points of [[inflection point|inflection]] and circles of [[osculating circle|osculation]], which aid in the measurement of [[curvature]]. Indeed, already in his [[Nova Methodus pro Maximis et Minimis|first paper]] on the foundations of calculus, Leibniz notes that the infinitesimal condition <math>d^2 y = 0</math> indicates the existence of an inflection point. Shortly after this time the [[Bernoulli family|Bernoulli brothers]], [[Jacob Bernoulli|Jacob]] and [[Johann Bernoulli|Johann]] made important early contributions to the use of infinitesimals to study geometry. In lectures by Johann Bernoulli at the time, later collated by [[Guillaume de l'Hôpital|L'Hopital]] into [[Analyse des Infiniment Petits pour l'Intelligence des Lignes Courbes|the first textbook on differential calculus]], the tangents to plane curves of various types are computed using the condition <math>dy=0</math>, and similarly points of inflection are calculated.<ref name="struik1" /> 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]]s at just the age of 16.<ref>Clairaut, A.C., 1731. Recherches sur les courbes à double courbure. Nyon.</ref><ref name="struik1" /> 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]] 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]]s later studied by Gauss and others.

Around this same time, [[Leonhard Euler]], originally a student of Johann Bernoulli, provided many significant contributions not just to the development of geometry, but to mathematics more broadly.<ref>{{MacTutor|id=Euler|title=Leonhard Euler}}</ref> In regards to differential geometry, Euler studied the notion of a [[geodesic]] 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.<ref name="struik1" /> 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]], and also to the [[Euler–Lagrange equations]] and the first theory of the [[calculus of variations]], 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 (differential geometry)|Euler's theorem]].

Later in the 1700s, the new French school led by [[Gaspard Monge]] began to make contributions to differential geometry. Monge made important contributions to the theory of plane curves, surfaces, and studied [[surfaces of revolution]] and [[envelope (mathematics)|envelopes]] 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.<ref name="struik1" />

=== 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]] 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,<ref name="spivak2">Spivak, M., 1975. A comprehensive introduction to differential geometry (Vol. 2). Publish or Perish, Incorporated.</ref> in 1827 Gauss produced the ''Disquisitiones generales circa superficies curvas'' detailing the general theory of curved surfaces.<ref name="Gauss">Gauss, C.F., 1828. Disquisitiones generales circa superficies curvas (Vol. 1). Typis Dieterichianis.</ref><ref name="spivak2" /><ref name="struik2">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</ref> 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.<ref name="struik2" /> In his fundamental paper Gauss introduced the [[Gauss map]], [[Gaussian curvature]], [[first fundamental form|first]] and [[second fundamental form]]s, 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]] should be discarded, and was in possession of private manuscripts on non-Euclidean geometry which informed his study of geodesic triangles.<ref name="struik2" /><ref>{{MacTutor|id=Non-Euclidean_Geometry|title=Non-Euclidean Geometry|class=HistTopics}}</ref> Around this same time [[János Bolyai]] 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.<ref>{{aut|[[John Milnor|Milnor, John W.]]}}, (1982) ''[http://projecteuclid.org/euclid.bams/1183548588 Hyperbolic geometry: The first 150 years]'', Bull. Amer. Math. Soc. (N.S.) Volume 6, Number 1, pp. 9–24.</ref> 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''.<ref>1868 ''On the hypotheses which lie at the foundation of geometry'', translated by [[William Kingdon Clifford|W.K.Clifford]], Nature 8 1873 183 – reprinted in Clifford's Collected Mathematical Papers, London 1882 (MacMillan); New York 1968 (Chelsea) http://www.emis.de/classics/Riemann/. Also in Ewald, William B., ed., 1996 "From Kant to Hilbert: A Source Book in the Foundations of Mathematics", 2 vols. Oxford Uni. Press: 652&ndash;61.</ref> 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 <math>ds^2</math> by Riemann, was the development of an idea of Gauss's about the linear element <math>ds</math> of a surface. At this time Riemann began to introduce the systematic use of [[linear algebra]] and [[multilinear algebra]] into the subject, making great use of the theory of [[quadratic form]]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]] 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.<ref name="struik2" /><ref name="spivak2" />

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]] and [[Jean Gaston Darboux]], 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]], 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.<ref>{{cite journal |last=Christoffel |first=E.B. |year=1869 |title=Ueber die Transformation der homogenen Differentialausdrücke zweiten Grades |url=http://resolver.sub.uni-goettingen.de/purl?PPN243919689_0070 |journal=Journal für die Reine und Angewandte Mathematik |volume=70}}</ref> 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]] and [[Gregorio Ricci-Curbastro]] produced their textbook systematically developing the theory of [[absolute differential calculus]] and [[tensor calculus]].<ref>{{cite journal |last1=Ricci |first1=Gregorio |last2=Levi-Civita |first2=Tullio |author-link2=Tullio Levi-Civita |title=Méthodes de calcul différentiel absolu et leurs applications |trans-title=Methods of the absolute differential calculus and their applications |journal=[[Mathematische Annalen]] |date=March 1900 |volume=54 |issue=1–2 |pages=125–201 |doi=10.1007/BF01454201 |url=http://gdz.sub.uni-goettingen.de/dms/resolveppn/?PPN=GDZPPN002258102 |publisher=Springer |s2cid=120009332 |language=fr}}</ref><ref name="spivak2" /> 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 [[Analysis Situs (paper)|the work]] of [[Henri Poincaré]] on the foundations of [[topology]].<ref name="dieudonne">Dieudonné, J., 2009. A history of algebraic and differential topology, 1900-1960. Springer Science & Business Media.</ref> 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]] 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.<ref name="dieudonne" />

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 <math>g</math> for a Riemannian metric, and <math>\Gamma</math> 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]].<ref name="fre">Fré, P.G., 2018. A Conceptual History of Space and Symmetry. Springer, Cham.</ref><ref name="spivak2" />

Following this early development, many mathematicians contributed to the development of the modern theory, including [[Jean-Louis Koszul]] who introduced [[connection (vector bundle)|connections on vector bundles]], [[Shiing-Shen Chern]] who introduced [[characteristic class]]es to the subject and began the study of [[complex manifold]]s, [[W. V. D. Hodge|Sir William Vallance Douglas Hodge]] and [[Georges de Rham]] who expanded understanding of [[differential forms]], [[Charles Ehresmann]] who introduced the theory of fibre bundles and [[Ehresmann connection]]s, and others.<ref name="fre" /><ref name="spivak2" /> Of particular importance was [[Hermann Weyl]] who made important contributions to the foundations of general relativity, introduced the [[Weyl tensor]] providing insight into [[conformal geometry]], and first defined the notion of a [[gauge (mathematics)|gauge]] leading to the development of [[gauge theory]] in physics and [[gauge theory (mathematics)|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 physics brought bundles and connections into focus, leading to developments in [[gauge theory (mathematics)|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]], which culminated in [[Grigori Perelman]]'s proof of the [[Poincaré conjecture]]. 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 (string theory)|mirror symmetry]] and the [[Seiberg–Witten invariant]]s.

== Branches ==
===Riemannian geometry===
{{main|Riemannian geometry}}

Riemannian geometry studies [[Riemannian manifold]]s, [[smooth manifold]]s with a ''Riemannian metric''. This is a concept of distance expressed by means of a [[Smooth function|smooth]] [[positive definite bilinear form|positive definite]] [[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]] of curves, [[area]] of plane regions, and [[volume]] of solids all possess natural analogues in Riemannian geometry. The notion of a [[directional derivative]] of a function from [[multivariable calculus]] 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]]. 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 curvature]]s 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 space]]s, 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 manifold|Pseudo-Riemannian geometry]] generalizes Riemannian geometry to the case in which the [[metric tensor]] need not be [[Definite bilinear form|positive-definite]]. 
A special case of this is a [[Lorentzian manifold]], which is the mathematical basis of Einstein's [[General relativity|general relativity theory of gravity]].

===Finsler geometry===
{{main|Finsler manifold}}
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 <math>M</math> is a function <math>F:\mathrm{T}M\to[0,\infty)</math> such that:
#<math>F(x,my)=mF(x,y)</math> for all <math>(x,y)</math> in <math>\mathrm{T}M</math> and all <math>m\ge 0</math>,
# <math>F</math> is infinitely differentiable in <math>\mathrm{T}M\setminus\{0\}</math>,
# The vertical Hessian of <math>F^2</math> is positive definite.

===Symplectic geometry===
{{main|Symplectic geometry}}

[[Symplectic geometry]] is the study of [[symplectic manifold]]s. An '''almost symplectic manifold''' is a differentiable manifold equipped with a smoothly varying [[non-degenerate]] [[skew-symmetric matrix|skew-symmetric]] [[bilinear form]] on each tangent space, i.e., a nondegenerate 2-[[Differential form|form]] ''ω'', called the ''symplectic form''. A symplectic manifold is an almost symplectic manifold for which the symplectic form ''ω'' is closed:  {{nowrap|1=d''ω'' = 0}}.

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]]'s [[Hamiltonian mechanics|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 (mathematics)|annulus]] twists each boundary component in opposite directions, then the map has at least two fixed points.<ref>The area preserving condition (or the twisting condition) cannot be removed. If one tries to extend such a theorem to higher dimensions, one would probably guess that a volume preserving map of a certain type must have fixed points. This is false in dimensions greater than 3.</ref>

===Contact geometry===
{{main|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 {{nowrap|(2''n'' + 1)}}-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 [[Differential form|1-form]] <math>\alpha</math>, which is unique up to multiplication by a nowhere vanishing function:

: <math> H_p = \ker\alpha_p\subset T_{p}M.</math>

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''<sub>''p''</sub> at each point. If the distribution ''H'' can be defined by a global one-form <math>\alpha</math> then this form is contact if and only if the top-dimensional form

: <math>\alpha\wedge (d\alpha)^n</math>

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===

{{See also|Complex geometry}}
''Complex differential geometry'' is the study of [[complex manifolds]].
An [[almost complex manifold]] is a ''real'' manifold <math>M</math>, endowed with a [[tensor]] of type (1, 1), i.e. a [[vector bundle|vector bundle endomorphism]] (called an ''[[almost complex structure]]'')
:<math> J:TM\rightarrow TM </math>, such that <math>J^2=-1. \,</math>

It follows from this definition that an almost complex manifold is even-dimensional.

An almost complex manifold is called ''complex'' if <math>N_J=0</math>, where <math>N_J</math> is a tensor of type (2, 1) related to <math>J</math>, called the [[Nijenhuis tensor]] (or sometimes the ''torsion'').
An almost complex manifold is complex if and only if it admits a [[Holomorphic function|holomorphic]] [[Atlas (topology)|coordinate atlas]].
An ''[[Hermitian manifold|almost Hermitian structure]]'' is given by an almost complex structure ''J'', along with a [[Riemannian metric]] ''g'', satisfying the compatibility condition
:<math>g(JX,JY)=g(X,Y). \,</math>

An almost Hermitian structure defines naturally a [[differential form|differential two-form]]
:<math>\omega_{J,g}(X,Y):=g(JX,Y). \,</math>

The following two conditions are equivalent:

# <math> N_J=0\mbox{ and }d\omega=0 \,</math>
# <math>\nabla J=0 \,</math>

where <math>\nabla</math> is the [[Levi-Civita connection]] of <math>g</math>. In this case, <math>(J, g)</math> is called a ''[[Kähler manifold|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 manifold]]s) is given by all the smooth [[algebraic geometry|complex projective varieties]].

===CR geometry===
[[CR structure|CR geometry]] is the study of the intrinsic geometry of boundaries of domains in [[complex manifold]]s.

===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 bundle]]s and [[Exterior derivative|de Rham differential]] of [[Differential form|forms]]. Beside [[Lie algebroid]]s, also [[Courant algebroid]]s start playing a more important role.

=== Lie groups ===
A [[Lie group]] is a [[Group (mathematics)|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]]s. Beside the structure theory there is also the wide field of [[representation of a Lie group|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 ===
{{Main article|Gauge theory (mathematics)}}

Gauge theory is the study of connections on vector bundles and principal bundles, and arises out of problems in [[mathematical physics]] and physical [[gauge theory|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 space]]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 [[quantum field theory]], and so their study is of considerable interest in physics.

== Bundles and connections ==

The apparatus of [[vector bundle]]s, [[principal bundle]]s, and [[connection (mathematics)|connection]]s 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 connection]]s. For a surface in '''R'''<sup>3</sup>, 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 [[Bernhard Riemann|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 [[connection (mathematics)|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]].) 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]].<ref>{{cite book |first=David |last=Hestenes |chapter=The Shape of Differential Geometry in Geometric Calculus |chapter-url=https://davidhestenes.net/geocalc/pdf/Shape%20in%20GC-2012.pdf |title=Guide to Geometric Algebra in Practice |editor-first=L. |editor-last=Dorst |editor2-first=J. |editor2-last=Lasenby|editor2-link=Joan Lasenby |publisher=Springer Verlag |year=2011 |pages=393–410 }}</ref>

== Applications ==
{{Spacetime|cTopic=Introduction}}
Below are some examples of how differential geometry is applied to other fields of science and mathematics.
*In [[physics]], differential geometry has many applications, including:
**Differential geometry is the language in which [[Albert Einstein]]'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]].
**Differential geometry has applications to both [[Lagrangian mechanics]] and [[Hamiltonian mechanics]]. Symplectic manifolds in particular can be used to study [[Hamiltonian system]]s.
**Riemannian geometry and contact geometry have been used to construct the formalism of [[geometrothermodynamics]] which has found applications in classical equilibrium [[thermodynamics]].
*In [[chemistry]] and [[biophysics]] when modelling cell membrane structure under varying pressure.
*In [[economics]], differential geometry has applications to the field of [[econometrics]].<ref>{{cite book |editor-first=Paul |editor-last=Marriott |editor2-first=Mark |editor2-last=Salmon |title=Applications of Differential Geometry to Econometrics |publisher=Cambridge University Press |year=2000 |isbn=978-0-521-65116-5 }}</ref>
*[[Geometric modeling]] (including [[computer graphics]]) and [[computer-aided geometric design]] draw on ideas from differential geometry.
*In [[engineering]], differential geometry can be applied to solve problems in [[digital signal processing]].<ref>{{cite book |first=Jonathan H. |last=Manton |chapter=On the role of differential geometry in signal processing |year=2005 |doi=10.1109/ICASSP.2005.1416480|title=Proceedings. (ICASSP '05). IEEE International Conference on Acoustics, Speech, and Signal Processing, 2005 |volume=5 |pages=1021–1024 |isbn=978-0-7803-8874-1 |s2cid=12265584 }}</ref>
*In [[control theory]], differential geometry can be used to analyze nonlinear controllers, particularly [[geometric control]]<ref>{{cite book |first1=Francesco |last1=Bullo |first2=Andrew |last2=Lewis |title=Geometric Control of Mechanical Systems : Modeling, Analysis, and Design for Simple Mechanical Control Systems |publisher=Springer-Verlag |year=2010 |isbn=978-1-4419-1968-7 }}</ref>
* In [[probability]], [[statistics]], and [[information theory]], one can interpret various structures as Riemannian manifolds, which yields the field of [[information geometry]], particularly via the [[Fisher information metric]].
*In [[structural geology]], differential geometry is used to analyze and describe geologic structures.
*In [[computer vision]], differential geometry is used to analyze shapes.<ref>{{cite thesis |type=Ph.D. |date=May 2008 |first=Mario |last=Micheli |title=The Differential Geometry of Landmark Shape Manifolds: Metrics, Geodesics, and Curvature |url=https://www.math.ucla.edu/~micheli/PUBLICATIONS/micheli_phd.pdf |archive-date=June 4, 2011 |archive-url=https://web.archive.org/web/20110604092900/http://www.math.ucla.edu/~micheli/PUBLICATIONS/micheli_phd.pdf }}</ref>
*In [[image processing]], differential geometry is used to process and analyse data on non-flat surfaces.<ref>{{cite thesis |type=Ph.D. |first=Anand A. |last=Joshi |title=Geometric Methods for Image Processing and Signal Analysis |date=August 2008 |url=http://users.loni.ucla.edu/~ajoshi/final_thesis.pdf |archive-url=https://web.archive.org/web/20110720072929/http://users.loni.ucla.edu/~ajoshi/final_thesis.pdf |archive-date=2011-07-20 |url-status=live }}</ref>
*[[Grigori Perelman]]'s proof of the [[Poincaré conjecture]] using the techniques of [[Ricci flow]]s demonstrated the power of the differential-geometric approach to questions in [[topology]] and it highlighted the important role played by its analytic methods.
* In [[wireless|wireless communications]], [[Grassmannian| Grassmannian manifolds]] are used for [[beamforming]] techniques in [[MIMO|multiple antenna]] systems.<ref>{{cite journal |first1=David J. |last1=Love |first2=Robert W. Jr. |last2=Heath |title=Grassmannian Beamforming for Multiple-Input Multiple-Output Wireless Systems |journal=IEEE Transactions on Information Theory |volume=49 |issue=10 |date=October 2003 |pages=2735–2747 |doi=10.1109/TIT.2003.817466 |url=http://users.ece.utexas.edu/~rheath/papers/2002/grassbeam/paper.pdf |archive-url=https://wayback.archive-it.org/all/20081002134712/http://users.ece.utexas.edu/~rheath/papers/2002/grassbeam/paper.pdf |url-status=dead |archive-date=2008-10-02 |citeseerx=10.1.1.106.4187 }}</ref>
* In [[geodesy]], for calculating distances and angles on the mean sea level surface of the [[Earth]], modelled by an ellipsoid of revolution.
* In [[neuroimaging]] and [[brain-computer interface]], symmetric positive definite manifolds are used to model functional, structural, or electrophysiological connectivity matrices.<ref>{{cite arXiv
|last1  = Ju
|first1 = Ce
|last2  = Kobler 
|first2 = Reinmar
|last3  = Collas 
|first3 = Antoine
|last4  = Kawanabe 
|first4 = Motoaki
|last5  = Guan 
|first5 = Cuntai
|last6  = Thirion 
|first6 = Bertrand
|date  = 26 Apr 2025
|title = SPD Learning for Covariance-Based Neuroimaging Analysis: Perspectives, Methods, and Challenges
|arxiv = 2504.18882
|class =cs.LG}}</ref>

==See also==
{{div col}}
* [[Affine differential geometry]]
* [[Analysis on fractals]]
* [[Basic introduction to the mathematics of curved spacetime]]
* [[Discrete differential geometry]]
* [[Gauss]]
* [[Glossary of differential geometry and topology]]
* [[List of publications in mathematics#Differential geometry|Important publications in differential geometry]]
* [[List of publications in mathematics#Differential topology|Important publications in differential topology]]
* [[Integral geometry]]
* [[List of differential geometry topics]]
* [[Noncommutative geometry]]
* [[Projective differential geometry]]
* [[Synthetic differential geometry]]
* [[Systolic geometry]]
* [[Gauge theory (mathematics)]]
{{div col end}}

== References ==
{{reflist|30em}}

== Further reading ==
*{{cite book|author=Ethan D. Bloch|title=A First Course in Geometric Topology and Differential Geometry|url=https://books.google.com/books?id=unwpBAAAQBAJ&q=%22differential+geometry%22|date=27 June 2011|publisher=Springer Science & Business Media|isbn=978-0-8176-8122-7|location=Boston|oclc=811474509}}
*{{cite book |last=Burke, William L.|url=http://worldcat.org/oclc/53249854|title=Applied differential geometry|date=1997|publisher=Cambridge University Press|isbn=0-521-26929-6|oclc=53249854}}
*{{cite book |first = Manfredo Perdigão|last = do Carmo|author-link=Manfredo do Carmo | isbn = 978-0-13-212589-5|url=https://www.worldcat.org/oclc/1529515|title=Differential geometry of curves and surfaces|date=1976|publisher=Prentice-Hall|location=Englewood Cliffs, N.J.|oclc=1529515}}
*{{cite book |first=Theodore |last=Frankel |author-link=Theodore Frankel |isbn = 978-0-521-53927-2|url=https://www.worldcat.org/oclc/51855212|title=The geometry of physics : an introduction|date=2004|publisher=Cambridge University Press|edition=2nd |location=New York|oclc=51855212}}
*{{cite book|author1=Elsa Abbena|author2=Simon Salamon|author3=Alfred Gray|url=https://www.worldcat.org/oclc/1048919510|title=Modern Differential Geometry of Curves and Surfaces with Mathematica.|date=2017|publisher=Chapman and Hall/CRC|isbn=978-1-351-99220-6|edition=3rd |location=Boca Raton|oclc=1048919510}}
*{{cite book |title = Differential Geometry |first = Erwin |last = Kreyszig | isbn = 978-0-486-66721-8 | year = 1991|url=https://www.worldcat.org/oclc/23384584|publisher=Dover Publications|location=New York|oclc=23384584}}
*{{cite book |title = Differential Geometry: Curves – Surfaces – Manifolds |first=Wolfgang |last=Kühnel |edition = 2nd |year = 2002 |isbn = 978-0-8218-3988-1|url=https://www.worldcat.org/oclc/61500086|publisher=American Mathematical Society|location=Providence, R.I.|oclc=61500086}}
*{{cite book |first = John |last = McCleary | year = 1994|url=http://worldcat.org/oclc/915912917|title=Geometry from a differentiable viewpoint|publisher=Cambridge University Press|isbn=0-521-13311-4|oclc=915912917}}
*{{cite book |title = A Comprehensive Introduction to Differential Geometry (5 Volumes) |edition = 3rd |first=Michael |last=Spivak | author-link=Michael Spivak |url=http://worldcat.org/oclc/179192286|date=1999|publisher=Publish or Perish|isbn=0-914098-72-1|oclc=179192286}}
*{{cite book | first = Bart M. |last = ter Haar Romeny |isbn = 978-1-4020-1507-6|url=https://www.worldcat.org/oclc/52806205|title=Front-end vision and multi-scale image analysis : multi-scale computer vision theory and applications, written in Mathematica|date=2003|publisher=Kluwer Academic|location=Dordrecht|oclc=52806205}}

== External links ==
{{Sister project links| wikt=no | commons=Category:Differential geometry | b=no | n=no | q=Differential geometry | s=no | v=no | voy=no | species=no | d=no}}

* {{Springer |title=Differential geometry |id=p/d032170}}
*[http://math.stanford.edu/~conrad/diffgeomPage/ B. Conrad. Differential Geometry handouts, Stanford University]
*[http://www.maths.adelaide.edu.au/michael.murray/teaching_old.html Michael Murray's online differential geometry course, 1996] {{Webarchive|url=https://web.archive.org/web/20130801003701/http://www.maths.adelaide.edu.au/michael.murray/teaching_old.html |date=2013-08-01 }}
*[http://VirtualMathMuseum.org/Surface/a/bk/curves_surfaces_palais.pdf A Modern Course on Curves and Surfaces, Richard S Palais, 2003] {{Webarchive|url=https://web.archive.org/web/20190409063941/http://virtualmathmuseum.org/ |date=2019-04-09 }}
*[http://VirtualMathMuseum.org/ Richard Palais's 3DXM Surfaces Gallery] {{Webarchive|url=https://web.archive.org/web/20190409063941/http://virtualmathmuseum.org/ |date=2019-04-09 }}
*[http://www.cs.elte.hu/geometry/csikos/dif/dif.html Balázs Csikós's Notes on Differential Geometry] {{Webarchive|url=https://web.archive.org/web/20090605054825/http://www.cs.elte.hu/geometry/csikos/dif/dif.html |date=2009-06-05 }}
*[http://www.wisdom.weizmann.ac.il/~yakov/scanlib/hicks.pdf N. J. Hicks, Notes on Differential Geometry, Van Nostrand.]
*[http://ocw.mit.edu/courses/mathematics/18-950-differential-geometry-fall-2008/ MIT OpenCourseWare: Differential Geometry, Fall 2008]

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[[Category:Differential geometry| ]]
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