Editing Differential geometry (section)

== 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.