Since the late 19th century, differential geometry has grown into a field concerned more generally with geometric structures ondifferentiable manifolds. A geometric structure is one which defines some notion of size, distance, shape, volume, or other rigidifying structure. For example, inRiemannian geometry distances and angles are specified, insymplectic geometry volumes may be computed, inconformal geometry only angles are specified, and ingauge theory certainfields 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 ofdifferential equations, otherwise known asgeometric analysis.
The history and development of differential geometry as a subject begins at least as far back asclassical antiquity. It is intimately linked to the development of geometry more generally, of the notion of space and shape, and oftopology, especially the study ofmanifolds. In this section we focus primarily on the history of the application ofinfinitesimal methods to geometry, and later to the ideas oftangent spaces, and eventually the development of the modern formalism of the subject in terms oftensors andtensor fields.
Classical antiquity until the Renaissance (300 BC – 1600 AD)
The study of differential geometry, or at least the study of the geometry of smooth shapes, can be traced back at least toclassical antiquity. In particular, much was known about the geometry of theEarth, aspherical geometry, in the time of theancient Greek mathematicians. Famously,Eratosthenes calculated thecircumference of the Earth around 200 BC, and around 150 ADPtolemy in hisGeography introduced thestereographic projection for the purposes of mapping the shape of the Earth.[1] Implicitly throughout this time principles that form the foundation of differential geometry and calculus were used ingeodesy, although in a much simplified form. Namely, as far back asEuclid'sElements 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 theEarth leads to the conclusion thatgreat 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 suchgeodesic paths by Eratosthenes and others can be considered a rudimentary measure ofarclength 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 ofinfinitesimals to the study of geometry, a precursor to the modern calculus-based study of the subject. InEuclid'sElements the notion oftangency of a line to a circle is discussed, andArchimedes applied themethod of exhaustion to compute the areas of smooth shapes such as thecircle, and the volumes of smooth three-dimensional solids such as the sphere, cones, and cylinders.[1]
There was little development in the theory of differential geometry between antiquity and the beginning of theRenaissance. Before the development of calculus byNewton andLeibniz, the most significant development in the understanding of differential geometry came fromGerardus Mercator's development of theMercator 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 theconformal nature of his projection, as well as the difference betweenpraga, the lines of shortest distance on the Earth, and thedirectio, the straight line paths on his map. Mercator noted that the praga wereoblique curvatur in this projection.[1] This fact reflects the lack of ametric-preserving map of the Earth's surface onto a flat plane, a consequence of the laterTheorema Egregium ofGauss.
The first systematic or rigorous treatment of geometry using the theory of infinitesimals and notions fromcalculus began around the 1600s when calculus was first developed byGottfried Leibniz andIsaac Newton. At this time, the recent work ofRené Descartes introducinganalytic coordinates to geometry allowed geometric shapes of increasing complexity to be described rigorously. In particular around this timePierre de Fermat, Newton, and Leibniz began the study ofplane curves and the investigation of concepts such as points ofinflection and circles ofosculation, which aid in the measurement ofcurvature. Indeed, already in hisfirst paper on the foundations of calculus, Leibniz notes that the infinitesimal condition indicates the existence of an inflection point. Shortly after this time theBernoulli brothers,Jacob andJohann made important early contributions to the use of infinitesimals to study geometry. In lectures by Johann Bernoulli at the time, later collated byL'Hopital intothe first textbook on differential calculus, the tangents to plane curves of various types are computed using the condition, and similarly points of inflection are calculated.[1] At this same time theorthogonality 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 ofcurvature, is written down.
In the wake of the development of analytic geometry and plane curves,Alexis Clairaut began the study ofspace curves at just the age of 16.[2][1] In his book Clairaut introduced the notion of tangent andsubtangent directions to space curves in relation to the directions which lie along asurface on which the space curve lies. Thus Clairaut demonstrated an implicit understanding of thetangent space of a surface and studied this idea using calculus for the first time. Importantly Clairaut introduced the terminology ofcurvature anddouble curvature, essentially the notion ofprincipal curvatures 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.[3] In regards to differential geometry, Euler studied the notion of ageodesic on a surface deriving the first analyticalgeodesic equation, and later introduced the first set of intrinsic coordinate systems on a surface, beginning the theory ofintrinsic geometry upon which modern geometric ideas are based.[1] Around this time Euler's study of mechanics in theMechanica 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'sgeneral relativity, and also to theEuler–Lagrange equations and the first theory of thecalculus of variations, which underpins in modern differential geometry many techniques insymplectic geometry andgeometric analysis. This theory was used byLagrange, a co-developer of the calculus of variations, to derive the first differential equation describing aminimal 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 asEuler's theorem.
Later in the 1700s, the new French school led byGaspard Monge began to make contributions to differential geometry. Monge made important contributions to the theory of plane curves, surfaces, and studiedsurfaces of revolution andenvelopes of plane curves and space curves. Several students of Monge made contributions to this same theory, and for exampleCharles Dupin provided a new interpretation of Euler's theorem in terms of the principle curvatures, which is the modern form of the equation.[1]
Intrinsic geometry and non-Euclidean geometry (1800–1900)
Dubbed the single most important work in the history of differential geometry,[4] in 1827 Gauss produced theDisquisitiones generales circa superficies curvas detailing the general theory of curved surfaces.[5][4][6] 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.[6] In his fundamental paper Gauss introduced theGauss map,Gaussian curvature,first andsecond fundamental forms, proved theTheorema Egregium showing the intrinsic nature of the Gaussian curvature, and studied geodesics, computing the area of ageodesic triangle in various non-Euclidean geometries on surfaces.
At this time Gauss was already of the opinion that the standard paradigm ofEuclidean geometry should be discarded, and was in possession of private manuscripts on non-Euclidean geometry which informed his study of geodesic triangles.[6][7] Around this same timeJános Bolyai and Lobachevsky independently discoveredhyperbolic geometry and thus demonstrated the existence of consistent geometries outside Euclid's paradigm. Concrete models of hyperbolic geometry were produced byEugenio Beltrami later in the 1860s, andFelix Klein coined the term non-Euclidean geometry in 1871, and through theErlangen program put Euclidean and non-Euclidean geometries on the same footing.[8] Implicitly, thespherical geometry of the Earth that had been studied since antiquity was a non-Euclidean geometry, anelliptic geometry.
The development of intrinsic differential geometry in the language of Gauss was spurred on by his student,Bernhard Riemann in hisHabilitationsschrift,On the hypotheses which lie at the foundation of geometry.[9] In this work Riemann introduced the notion of aRiemannian metric and theRiemannian 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 by Riemann, was the development of an idea of Gauss's about the linear element of a surface. At this time Riemann began to introduce the systematic use oflinear algebra andmultilinear algebra into the subject, making great use of the theory ofquadratic 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 atopological space had not been encountered, but he did propose that it might be possible to investigate or measure the properties of the metric ofspacetime 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 theequivalence principle a full 60 years before it appeared in the scientific literature.[6][4]
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 oftensor calculus and Klein's Erlangen program, and progress increased in the field. The notion of groups of transformations was developed bySophus Lie andJean Gaston Darboux, leading to important results in the theory ofLie groups andsymplectic geometry. The notion of differential calculus on curved spaces was studied byElwin Christoffel, who introduced theChristoffel symbols which describe thecovariant derivative in 1868, and by others includingEugenio Beltrami who studied many analytic questions on manifolds.[10] In 1899Luigi Bianchi produced hisLectures on differential geometry which studied differential geometry from Riemann's perspective, and a year laterTullio Levi-Civita andGregorio Ricci-Curbastro produced their textbook systematically developing the theory ofabsolute differential calculus andtensor calculus.[11][4] It was in this language that differential geometry was used by Einstein in the development of general relativity andpseudo-Riemannian geometry.
The subject of modern differential geometry emerged from the early 1900s in response to the foundational contributions of many mathematicians, including importantlythe work ofHenri Poincaré on the foundations oftopology.[12] 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 asHilbert's program. As part of this broader movement, the notion of atopological space was distilled in byFelix Hausdorff in 1914, and by 1942 there were many different notions of manifold of a combinatorial and differential-geometric nature.[12]
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 for a Riemannian metric, and for the Christoffel symbols, both coming fromG inGravitation.Élie Cartan helped reformulate the foundations of the differential geometry of smooth manifolds in terms ofexterior calculus and the theory ofmoving frames, leading in the world of physics toEinstein–Cartan theory.[13][4]
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 ofgauge theory andYang–Mills theory in physics brought bundles and connections into focus, leading to developments ingauge theory. Many analytical results were investigated including the proof of theAtiyah–Singer index theorem. The development ofcomplex geometry was spurred on by parallel results inalgebraic geometry, and results in the geometry and global analysis of complex manifolds were proven byShing-Tung Yau and others. In the latter half of the 20th century new analytic techniques were developed in regards to curvature flows such as theRicci flow, which culminated inGrigori Perelman's proof of thePoincaré conjecture. During this same period primarily due to the influence ofMichael Atiyah, new links betweentheoretical physics and differential geometry were formed. Techniques from the study of theYang–Mills equations andgauge theory were used by mathematicians to develop new invariants of smooth manifolds. Physicists such asEdward Witten, the only physicist to be awarded aFields medal, made new impacts in mathematics by usingtopological quantum field theory andstring theory to make predictions and provide frameworks for new rigorous mathematics, which has resulted for example in the conjecturalmirror symmetry and theSeiberg–Witten invariants.
A distance-preservingdiffeomorphism between Riemannian manifolds is called anisometry. This notion can also be definedlocally, i.e. for small neighborhoods of points. Any two regular curves are locally isometric. However, theTheorema Egregium of Carl Friedrich Gauss showed that for surfaces, the existence of a local isometry imposes that theGaussian curvatures at the corresponding points must be the same. In higher dimensions, theRiemann 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 theRiemannian symmetric spaces, whose curvature is not necessarily constant. These are the closest analogues to the "ordinary" plane and space considered in Euclidean andnon-Euclidean geometry.
Finsler geometry hasFinsler manifolds as the main object of study. This is a differential manifold with aFinsler metric, that is, aBanach 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:
Symplectic geometry is the study ofsymplectic manifolds. Analmost symplectic manifold is a differentiable manifold equipped with a smoothly varyingnon-degenerateskew-symmetricbilinear form on each tangent space, i.e., a nondegenerate 2-formω, called thesymplectic form. A symplectic manifold is an almost symplectic manifold for which the symplectic formω is closed:dω = 0.
A diffeomorphism between two symplectic manifolds which preserves the symplectic form is called asymplectomorphism. 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. Thephase space of a mechanical system is a symplectic manifold and they made an implicit appearance already in the work ofJoseph Louis Lagrange onanalytical mechanics and later inCarl Gustav Jacobi's andWilliam Rowan Hamilton'sformulations of classical mechanics.
By contrast with Riemannian geometry, where thecurvature 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 thePoincaré–Birkhoff theorem, conjectured byHenri Poincaré and then proved byG.D. Birkhoff in 1912. It claims that if an area preserving map of anannulus twists each boundary component in opposite directions, then the map has at least two fixed points.[14]
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. Acontact structure on a(2n + 1)-dimensional manifoldM is given by a smooth hyperplane fieldH in thetangent bundle that is as far as possible from being associated with the level sets of a differentiable function onM (the technical term is "completely nonintegrable tangent hyperplane distribution"). Near each pointp, a hyperplane distribution is determined by a nowhere vanishing1-form, which is unique up to multiplication by a nowhere vanishing function:
A local 1-form onM is acontact form if the restriction of itsexterior derivative toH is a non-degenerate two-form and thus induces a symplectic structure onHp at each point. If the distributionH can be defined by a global one-form then this form is contact if and only if the top-dimensional form
is avolume form onM, 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.
It follows from this definition that an almost complex manifold is even-dimensional.
An almost complex manifold is calledcomplex if, where is a tensor of type (2, 1) related to, called theNijenhuis tensor (or sometimes thetorsion).An almost complex manifold is complex if and only if it admits aholomorphiccoordinate atlas.Analmost Hermitian structure is given by an almost complex structureJ, along with aRiemannian metricg, satisfying the compatibility condition
ALie group is agroup 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-invariantvector fields. Beside the structure theory there is also the wide field ofrepresentation theory.
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 is the study of connections on vector bundles and principal bundles, and arises out of problems inmathematical physics and physicalgauge theories which underpin thestandard model of particle physics. Gauge theory is concerned with the study of differential equations for connections on bundles, and the resulting geometricmoduli spaces of solutions to these equations as well as the invariants that may be derived from them. These equations often arise as theEuler–Lagrange equations describing the equations of motion of certain physical systems inquantum field theory, and so their study is of considerable interest in physics.
The apparatus ofvector bundles,principal bundles, andconnections on bundles plays an extraordinarily important role in modern differential geometry. A smooth manifold always carries a natural vector bundle, thetangent 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 ofparallel transport. An important example is provided byaffine connections. For a surface inR3, 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. InRiemannian geometry, theLevi-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 bespacetime and the bundles and connections are related to various physical fields.
From the beginning and through the middle of the 19th century, differential geometry was studied from theextrinsic point of view: curves and surfaces were considered as lying in a Euclidean space of higher dimension (for example a surface in anambient space of three dimensions). The simplest results are those in the differential geometry of curves and differential geometry of surfaces. Starting with the work ofRiemann, theintrinsic 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'stheorema egregium, to the effect thatGaussian 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 andconnections 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 theNash embedding theorem.) In the formalism ofgeometric calculus both extrinsic and intrinsic geometry of a manifold can be characterized by a singlebivector-valued one-form called theshape operator.[15]
Below are some examples of how differential geometry is applied to other fields of science and mathematics.
Inphysics, differential geometry has many applications, including:
Differential geometry is the language in whichAlbert Einstein'sgeneral 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 ofspacetime. Understanding this curvature is essential for the positioning ofsatellites into orbit around the Earth. Differential geometry is also indispensable in the study ofgravitational lensing andblack holes.
Riemannian geometry and contact geometry have been used to construct the formalism ofgeometrothermodynamics which has found applications in classical equilibriumthermodynamics.
Inchemistry andbiophysics when modelling cell membrane structure under varying pressure.
Inimage processing, differential geometry is used to process and analyse data on non-flat surfaces.[20]
Grigori Perelman's proof of thePoincaré conjecture using the techniques ofRicci flows demonstrated the power of the differential-geometric approach to questions intopology and it highlighted the important role played by its analytic methods.
Ingeodesy, for calculating distances and angles on the mean sea level surface of theEarth, modelled by an ellipsoid of revolution.
Inneuroimaging andbrain-computer interface, symmetric positive definite manifolds are used to model functional, structural, or electrophysiological connectivity matrices.[22]
^1868On the hypotheses which lie at the foundation of geometry, translated byW.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: 652–61.
^abDieudonné, J., 2009. A history of algebraic and differential topology, 1900-1960. Springer Science & Business Media.
^abFré, P.G., 2018. A Conceptual History of Space and Symmetry. Springer, Cham.
^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.
^Marriott, Paul; Salmon, Mark, eds. (2000).Applications of Differential Geometry to Econometrics. Cambridge University Press.ISBN978-0-521-65116-5.
^Manton, Jonathan H. (2005). "On the role of differential geometry in signal processing".Proceedings. (ICASSP '05). IEEE International Conference on Acoustics, Speech, and Signal Processing, 2005. Vol. 5. pp. 1021–1024.doi:10.1109/ICASSP.2005.1416480.ISBN978-0-7803-8874-1.S2CID12265584.
^Bullo, Francesco; Lewis, Andrew (2010).Geometric Control of Mechanical Systems : Modeling, Analysis, and Design for Simple Mechanical Control Systems. Springer-Verlag.ISBN978-1-4419-1968-7.
Elsa Abbena; Simon Salamon; Alfred Gray (2017).Modern Differential Geometry of Curves and Surfaces with Mathematica (3rd ed.). Boca Raton: Chapman and Hall/CRC.ISBN978-1-351-99220-6.OCLC1048919510.
ter Haar Romeny, Bart M. (2003).Front-end vision and multi-scale image analysis : multi-scale computer vision theory and applications, written in Mathematica. Dordrecht: Kluwer Academic.ISBN978-1-4020-1507-6.OCLC52806205.
