Differential geometry is a

_{''p''} at each point. If the distribution ''H'' can be defined by a global one-form $\backslash alpha$ then this form is contact if and only if the top-dimensional form
: $\backslash alpha\backslash wedge\; (d\backslash alpha)^n$
is a volume form on ''M'', i.e. does not vanish anywhere. A contact analogue of the Darboux theorem holds: all contact structures on an odd-dimensional manifold are locally isomorphic and can be brought to a certain local normal form by a suitable choice of the coordinate system.

^{3}, 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

B. Conrad. Differential Geometry handouts, Stanford University

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

Richard Palais's 3DXM Surfaces Gallery

N. J. Hicks, Notes on Differential Geometry, Van Nostrand.

MIT OpenCourseWare: Differential Geometry, Fall 2008

{{DEFAULTSORT:Differential Geometry Differential geometry, Geometry processing

mathematical
Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and their changes (cal ...

discipline that studies the geometry
Geometry (from the grc, γεωμετρία; ' "earth", ' "measurement") is, with , one of the oldest branches of . It is concerned with properties of space that are related with distance, shape, size, and relative position of figures. A mat ...

of smooth shapes and smooth spaces, otherwise known as smooth manifold
In mathematics, a differentiable manifold (also differential manifold) is a type of manifold
The real projective plane is a two-dimensional manifold that cannot be realized in three dimensions without self-intersection, shown here as Boy's s ...

s, using the techniques of differential calculus
In mathematics
Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). I ...

, integral calculus
In mathematics
Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities a ...

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

and multilinear algebra
In mathematics
Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). I ...

. The field has its origins in the study of spherical geometry
Image:Triangles (spherical geometry).jpg, 300px, The sum of the angles of a spherical triangle is not equal to 180°. A sphere is a curved surface, but locally the laws of the flat (planar) Euclidean geometry are good approximations. In a small tr ...

as far back as antiquity
Antiquity or Antiquities may refer to
Historical objects or periods Artifacts
* Antiquities, objects or artifacts surviving from ancient cultures
Eras
Any period before the European Middle Ages
In the history of Europe, the Middle Ages ...

, as it relates to astronomy
Astronomy (from el, ἀστρονομία, literally meaning the science that studies the laws of the stars) is a natural science that studies astronomical object, celestial objects and celestial event, phenomena. It uses mathematics, phys ...

and the geodesy
Geodesy ( ) is the Earth science of accurately measuring and understanding Earth's geometric shape, orientation in space, and gravitational field. The field also incorporates studies of how these properties change over time and equivalent measu ...

of the Earth
Earth is the third planet from the Sun and the only astronomical object known to harbour and support life. 29.2% of Earth's surface is land consisting of continents and islands. The remaining 70.8% is Water distribution on Earth, covered wi ...

, and later in the study of hyperbolic geometry
In mathematics
Mathematics (from Greek: ) includes the study of such topics as numbers ( and ), formulas and related structures (), shapes and spaces in which they are contained (), and quantities and their changes ( and ). There is no g ...

by Lobachevsky
Nikolai Ivanovich Lobachevsky ( rus, Никола́й Ива́нович Лобаче́вский, p=nʲikɐˈlaj ɪˈvanəvʲɪtɕ ləbɐˈtɕɛfskʲɪj, a=Ru-Nikolai_Ivanovich_Lobachevsky.ogg; – ) was a Russia
Russia (russian: link=no, ...

. The simplest examples of smooth spaces are the plane and space curves and surfaces
Water droplet lying on a damask. Surface tension">damask.html" ;"title="Water droplet lying on a damask">Water droplet lying on a damask. Surface tension is high enough to prevent floating below the textile.
A surface, as the term is most gener ...

in the three-dimensional Euclidean space
Euclidean space is the fundamental space of classical geometry. Originally, it was the three-dimensional space of Euclidean geometry, but in modern mathematics there are Euclidean spaces of any nonnegative integer dimension (mathematics), dimens ...

, and the study of these shapes formed the basis for development of modern differential geometry during the 18th century and the 19th century.
Since the late 19th century, differential geometry has grown into a field concerned more generally with geometric structures on differentiable manifold
In mathematics, a differentiable manifold (also differential manifold) is a type of manifold
The real projective plane is a two-dimensional manifold that cannot be realized in three dimensions without self-intersection, shown here as Boy's surfa ...

s. A geometric structure is one which defines some notion of size, distance, shape, volume, or other rigidifying structure. For example, in Riemannian geometry#REDIRECT Riemannian geometry
Riemannian geometry is the branch of differential geometry
Differential geometry is a Mathematics, mathematical discipline that uses the techniques of differential calculus, integral calculus, linear algebra and m ...

distances and angles are specified, in symplectic geometry
Symplectic geometry is a branch of differential geometry
Differential geometry is a Mathematics, mathematical discipline that uses the techniques of differential calculus, integral calculus, linear algebra and multilinear algebra to study prob ...

volumes may be computed, in conformal geometry
In mathematics
Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It h ...

only angles are specified, and in gauge theory
In physics, a gauge theory is a type of Field theory (physics), field theory in which the Lagrangian (field theory), Lagrangian (and hence the dynamics of the system itself) does not change (is Invariant (physics), invariant) under local symmetry, ...

certain fields
File:A NASA Delta IV Heavy rocket launches the Parker Solar Probe (29097299447).jpg, FIELDS heads into space in August 2018 as part of the ''Parker Solar Probe''
FIELDS is a science instrument on the ''Parker Solar Probe'' (PSP), designed to mea ...

are given over the space. Differential geometry is closely related to, and is sometimes taken to include, differential topology
In mathematics
Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ...

, which concerns itself with properties of differentiable manifolds which do not rely on any additional geometric structure (see that article for more discussion on the distinction between the two subjects). Differential geometry is also related to the geometric aspects of the theory of differential equation
In mathematics, a differential equation is an equation
In mathematics
Mathematics (from Greek: ) includes the study of such topics as numbers ( and ), formulas and related structures (), shapes and spaces in which they are contained ( ...

s, otherwise known as geometric analysis
Geometric analysis is a mathematics, mathematical discipline where tools from differential equations, especially elliptic partial differential equations are used to establish new results in differential geometry and differential topology. The use o ...

.
Differential geometry finds applications throughout mathematics and the natural science
Natural science is a branch
A branch ( or , ) or tree branch (sometimes referred to in botany
Botany, also called , plant biology or phytology, is the science of plant life and a branch of biology. A botanist, plant scientist or ph ...

s. Most prominently the language of differential geometry was used by Albert Einstein
Albert Einstein ( ; ; 14 March 1879 – 18 April 1955) was a German-born theoretical physicist, widely acknowledged to be one of the greatest physicists of all time. Einstein is known for developing the theory of relativity
The theo ...

in his theory of general relativity
General relativity, also known as the general theory of relativity, is the differential geometry, geometric scientific theory, theory of gravitation published by Albert Einstein in 1915 and is the current description of gravitation in modern ph ...

, and subsequently by physicists
A physicist is a scientist
A scientist is a person who conducts scientific research
The scientific method is an Empirical evidence, empirical method of acquiring knowledge that has characterized the development of science since at lea ...

in the development of quantum field theory
In theoretical physics, quantum field theory (QFT) is a theoretical framework that combines classical field theory, special relativity and quantum mechanics. QFT is used in particle physics to construct physical models of subatomic particles and ...

and the standard model of particle physics
The Standard Model of particle physics
Particle physics (also known as high energy physics) is a branch of physics
Physics (from grc, φυσική (ἐπιστήμη), physikḗ (epistḗmē), knowledge of nature, from ''phýsis ...

. Outside of physics, differential geometry finds applications in chemistry
Chemistry is the scientific
Science () is a systematic enterprise that builds and organizes knowledge
Knowledge is a familiarity or awareness, of someone or something, such as facts
A fact is an occurrence in the real world. T ...

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

, engineering
Engineering is the use of scientific principles to design and build machines, structures, and other items, including bridges, tunnels, roads, vehicles, and buildings. The discipline of engineering encompasses a broad range of more speciali ...

, control theory
Control theory deals with the control of dynamical system
In mathematics
Mathematics (from Greek: ) includes the study of such topics as numbers ( and ), formulas and related structures (), shapes and spaces in which they are contai ...

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

and computer vision
Computer vision is an interdisciplinary scientific field that deals with how computer
A computer is a machine that can be programmed to carry out sequences of arithmetic or logical operations automatically. Modern computers can perform ge ...

, and recently in machine learning
Machine learning (ML) is the study of computer algorithms that can improve automatically through experience and by the use of data. It is seen as a part of artificial intelligence. Machine learning algorithms build a model based on sample data ...

.
History and development

The history and development of differential geometry as a subject begins at least as far back asclassical antiquity
Classical antiquity (also the classical era, classical period or classical age) is the period of cultural history
History (from Greek#REDIRECT Greek
Greek may refer to:
Greece
Anything of, from, or related to Greece
Greece ( el, ...

, and is intimately linked to the development of geometry more generally, of the notion of space and shape, and of topology
In mathematics
Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities ...

. For more details about the history of the concept of a manifold see and the history of manifolds and varieties. In this section we focus primarily on the history of the application of infinitesimal
In mathematics, infinitesimals or infinitesimal numbers are quantities that are closer to zero than any standard real number, but are not zero. They do not exist in the standard real number system, but do exist in many other number systems, such a ...

methods to geometry, and later to the ideas of tangent space
In , the tangent space of a generalizes to higher dimensions the notion of tangent planes to surfaces in three dimensions and tangent lines to curves in two dimensions. In the context of physics the tangent space to a manifold at a point can ...

s, and eventually the development of the modern formalism of the subject in terms of tensor
In mathematics
Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities a ...

s and tensor field
In mathematics
Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and t ...

s.
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
Classical antiquity (also the classical era, classical period or classical age) is the period of cultural history
History (from Greek#REDIRECT Greek
Greek may refer to:
Greece
Anything of, from, or related to Greece
Greece ( el, ...

. In particular, much was known about the geometry of the Earth
Earth is the third planet from the Sun and the only astronomical object known to harbour and support life. 29.2% of Earth's surface is land consisting of continents and islands. The remaining 70.8% is Water distribution on Earth, covered wi ...

, a spherical geometry
Image:Triangles (spherical geometry).jpg, 300px, The sum of the angles of a spherical triangle is not equal to 180°. A sphere is a curved surface, but locally the laws of the flat (planar) Euclidean geometry are good approximations. In a small tr ...

, in the time of the ancient Greek
Ancient Greek includes the forms of the Greek language
Greek ( el, label=Modern Greek
Modern Greek (, , or , ''Kiní Neoellinikí Glóssa''), generally referred to by speakers simply as Greek (, ), refers collectively to the diale ...

mathematicians. Famously, Eratosthenes
Eratosthenes of Cyrene (; grc-gre, Ἐρατοσθένης ; – ) was a Greek polymath
A polymath ( el, πολυμαθής, , "having learned much"; la, homo universalis, "universal human") is an individual whose knowledge spans a ...

calculated the circumference
In geometry
Geometry (from the grc, γεωμετρία; ' "earth", ' "measurement") is, with , one of the oldest branches of . It is concerned with properties of space that are related with distance, shape, size, and relative position ...

of the Earth around 200 BC, and around 150 AD Ptolemy
Claudius Ptolemy (; grc-koi, Κλαύδιος Πτολεμαῖος, , ; la, Claudius Ptolemaeus; AD) was a mathematician
A mathematician is someone who uses an extensive knowledge of mathematics
Mathematics (from Greek: ) includes ...

in his ''Geography
Geography (from Ancient Greek, Greek: , ''geographia'', literally "earth description") is a field of science devoted to the study of the lands, features, inhabitants, and phenomena of the Earth and Solar System, planets. The first person t ...

'' introduced the stereographic projection
In geometry
Geometry (from the grc, γεωμετρία; ''wikt:γῆ, geo-'' "earth", ''wikt:μέτρον, -metron'' "measurement") is, with arithmetic, one of the oldest branches of mathematics. It is concerned with properties of space t ...

for the purposes of mapping the shape of the Earth.Struik, D. J. “Outline of a History of Differential Geometry: I.” Isis, vol. 19, no. 1, 1933, pp. 92–120. JSTOR, www.jstor.org/stable/225188. Implicitly throughout this time principles that form the foundation of differential geometry and calculus were used in geodesy
Geodesy ( ) is the Earth science of accurately measuring and understanding Earth's geometric shape, orientation in space, and gravitational field. The field also incorporates studies of how these properties change over time and equivalent measu ...

, although in a much simplified form. Namely, as far back as Euclid
Euclid (; grc-gre, Εὐκλείδης
Euclid (; grc, Εὐκλείδης – ''Eukleídēs'', ; fl. 300 BC), sometimes called Euclid of Alexandria to distinguish him from Euclid of Megara, was a Greek mathematician, often referre ...

's '' Elements'' it was understood that a straight line could be defined by its property of providing the shortest distance between two points, and applying this same principle to the surface of the Earth
Earth is the third planet from the Sun and the only astronomical object known to harbour and support life. 29.2% of Earth's surface is land consisting of continents and islands. The remaining 70.8% is Water distribution on Earth, covered wi ...

leads to the conclusion that great circles
Great may refer to: Descriptions or measurements
* Great, a relative measurement in physical space, see Size
File:Comparison of planets and stars (sheet by sheet) (Oct 2014 update).png, A size comparison illustration comparing the sizes of vario ...

, which are only locally similar to straight lines in a flat plane, provide the shortest path between two points on the Earth's surface. Indeed the measurements of distance along such geodesic
In geometry
Geometry (from the grc, γεωμετρία; ' "earth", ' "measurement") is, with , one of the oldest branches of . It is concerned with properties of space that are related with distance, shape, size, and relative position o ...

paths by Eratosthenes and others can be considered a rudimentary measure of arclength
Arc length is the distance between two points along a section of a curve
In mathematics, a curve (also called a curved line in older texts) is an object similar to a line (geometry), line, but that does not have to be Linearity, straight.
In ...

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
In mathematics, infinitesimals or infinitesimal numbers are quantities that are closer to zero than any standard real number, but are not zero. They do not exist in the standard real number system, but do exist in many other number systems, such a ...

s to the study of geometry, a precursor to the modern calculus-based study of the subject. In Euclid
Euclid (; grc-gre, Εὐκλείδης
Euclid (; grc, Εὐκλείδης – ''Eukleídēs'', ; fl. 300 BC), sometimes called Euclid of Alexandria to distinguish him from Euclid of Megara, was a Greek mathematician, often referre ...

's '' Elements'' the notion of tangency
In geometry, the tangent line (or simply tangent) to a plane curve at a given Point (geometry), point is the straight line that "just touches" the curve at that point. Leibniz defined it as the line through a pair of infinitesimal, infinitely cl ...

of a line to a circle is discussed, and Archimedes
Archimedes of Syracuse (; grc, ; ; ) was a Greek#REDIRECT Greek
Greek may refer to:
Greece
Anything of, from, or related to Greece
Greece ( el, Ελλάδα, , ), officially the Hellenic Republic, is a country located in Southeast Eu ...

applied the method of exhaustion
The method of exhaustion (; ) is a method of finding the area
Area is the quantity
Quantity is a property that can exist as a multitude or magnitude, which illustrate discontinuity and continuity. Quantities can be compared in terms of ...

to compute the areas of smooth shapes such as the circle
A circle is a shape
A shape or figure is the form of an object or its external boundary, outline, or external surface
File:Water droplet lying on a damask.jpg, Water droplet lying on a damask. Surface tension is high enough to preven ...

, and the volumes of smooth three-dimensional solids such as the sphere, cones, and cylinders.
There was little development in the theory of differential geometry between antiquity and the beginning of the Renaissance
The Renaissance ( , ) , from , with the same meanings. is a period
Period may refer to:
Common uses
* Era, a length or span of time
* Full stop (or period), a punctuation mark
Arts, entertainment, and media
* Period (music), a concept in ...

. Before the development of calculus by Newton
Newton most commonly refers to:
* Isaac Newton (1642–1726/1727), English scientist
* Newton (unit), SI unit of force named after Isaac Newton
Newton may also refer to:
Arts and entertainment
* Newton (film), ''Newton'' (film), a 2017 Indian fil ...

and Leibniz
Gottfried Wilhelm (von) Leibniz ; see inscription of the engraving depicted in the "#1666–1676, 1666–1676" section. ( – 14 November 1716) was a German polymath active as a mathematician, philosopher, scientist, and diplomat. He is a promin ...

, the most significant development in the understanding of differential geometry came from Gerardus Mercator
Gerardus Mercator (; 5 March 1512 – 2 December 1594) was a 16th-century geographer
A geographer is a physical scientist, social scientist or humanist whose area of study is geography
Geography (from Greek: , ''geographia'', lite ...

's development of the Mercator projection
The Mercator projection () is a cylindrical map projection presented by Flemish
Flemish (''Vlaams'') is a Low Franconian dialect cluster of the Dutch language. It is sometimes referred to as Flemish Dutch (), Belgian Dutch ( ), or Souther ...

as a way of mapping the Earth. Mercator had an understanding of the advantages and pitfalls of his map design, and in particular was aware of the conformal nature of his projection, as well as the difference between ''praga'', the lines of shortest distance on the Earth, and the ''directio'', the straight line paths on his map. Mercator noted that the praga were ''oblique curvatur'' in this projection. This fact reflects the lack of a metric-preserving map of the Earth's surface onto a flat plane, a consequence of the later Theorema Egregium
without distortion. The Mercator projection, shown here, preserves angles but fails to preserve area.
Gauss's ''Theorema Egregium'' (Latin for "Remarkable Theorem") is a major result of differential geometry
Differential geometry is a Mathe ...

of Gauss
Johann Carl Friedrich Gauss (; german: Gauß ; la, Carolus Fridericus Gauss; 30 April 177723 February 1855) was a German mathematician
This is a List of German mathematician
A mathematician is someone who uses an extensive knowledge of m ...

.
After calculus (1600 - 1800)

The first systematic or rigorous treatment of geometry using the theory of infinitesimals and notions fromcalculus
Calculus, originally called infinitesimal calculus or "the calculus of infinitesimals", is the mathematics, mathematical study of continuous change, in the same way that geometry is the study of shape and algebra is the study of generalizations ...

began around the 1600s when calculus was first developed by Gottfried Leibniz
Gottfried Wilhelm (von) Leibniz ; see inscription of the engraving depicted in the " 1666–1676" section. ( – 14 November 1716) was a German polymath
A polymath ( el, πολυμαθής, , "having learned much"; la, homo universalis, " ...

and Isaac Newton
Sir Isaac Newton (25 December 1642 – 20 March 1726/27) was an English mathematician
A mathematician is someone who uses an extensive knowledge of mathematics
Mathematics (from Greek: ) includes the study of such topics a ...

. At this time, the recent work of René Descartes
René Descartes ( or ; ; Latinized
Latinisation or Latinization can refer to:
* Latinisation of names, the practice of rendering a non-Latin name in a Latin style
* Latinisation in the Soviet Union, the campaign in the USSR during the 1920s ...

introducing analytic coordinates to geometry allowed geometric shapes of increasing complexity to be described rigorously. In particular around this time Pierre de Fermat
Pierre de Fermat (; between 31 October and 6 December 1607 – 12 January 1665) was a French mathematician
A mathematician is someone who uses an extensive knowledge of mathematics
Mathematics (from Greek: ) includes the study of suc ...

, Newton, and Leibniz began the study of plane curve
In mathematics, a plane curve is a curve
In mathematics, a curve (also called a curved line in older texts) is an object similar to a line (geometry), line, but that does not have to be Linearity, straight.
Intuitively, a curve may be thought o ...

s and the investigation of concepts such as points of inflection
In linguistic morphology
Morphology, from the Greek and meaning "study of shape", may refer to:
Disciplines
* Morphology (archaeology), study of the shapes or forms of artifacts
* Morphology (astronomy), study of the shape of astronomical ob ...

and circles of osculation
Image:Osculating circle.svg, A curve ''C'' containing a point ''P'' where the Radius of curvature (mathematics), radius of curvature equals ''r'', together with the tangent line and the osculating circle touching ''C'' at ''P''
In differential geom ...

, which aid in the measurement of curvature
In mathematics
Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities an ...

. Indeed already in his first paper on the foundations of calculus, Leibniz notes that the infinitesimal condition $d^2\; y\; =\; 0$ indicates the existence of an inflection point. Shortly after this time the Bernoulli brothers, Jacob
Jacob (; ; ar, يَعْقُوب, Yaʿqūb; gr, Ἰακώβ, Iakṓb), later given the name Israel
Israel (; he, יִשְׂרָאֵל, translit=Yīsrāʾēl; ar, إِسْرَائِيل, translit=ʾIsrāʾīl), officially the State ...

and Johann
Johann, typically a male given name
A given name (also known as a first name or forename) is the part of a personal name
A personal name, or full name, in onomastic
Onomastics or onomatology is the study of the etymology, history, an ...

made important early contributions to the use of infinitesimals to study geometry. In lectures by Johann Bernoulli at the time, later collated by into the first textbook on differential calculus, the tangents to plane curves of various types are computed using the condition $dy=0$, and similarly points of inflection are calculated. At this same time the orthogonality
In mathematics
Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). ...

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
In mathematics
Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities an ...

, is written down.
In the wake of the development of analytic geometry and plane curves, Alexis Clairaut
Alexis Claude Clairaut (; 13 May 1713 – 17 May 1765) was a French mathematician, astronomer, and geophysicist. He was a prominent Newtonian whose work helped to establish the validity of the principles and results that Sir Isaac Newton
...

began the study of space curve
In mathematics
Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). I ...

s at just the age of 16. In his book Clairaut introduced the notion of tangent and directions to space curves in relation to the directions which lie along a surface
File:Water droplet lying on a damask.jpg, Water droplet lying on a damask. Surface tension is high enough to prevent floating below the textile.
A surface, as the term is most generally used, is the outermost or uppermost layer of a physical obje ...

on which the space curve lies. Thus Clairaut demonstrated an implicit understanding of the tangent space
In , the tangent space of a generalizes to higher dimensions the notion of tangent planes to surfaces in three dimensions and tangent lines to curves in two dimensions. In the context of physics the tangent space to a manifold at a point can ...

of a surface and studied this idea using calculus for the first time. Importantly Clairaut introduced the terminology of ''curvature'' and ''double curvature'', essentially the notion of principal curvature
In differential geometry
Differential geometry is a Mathematics, mathematical discipline that uses the techniques of differential calculus, integral calculus, linear algebra and multilinear algebra to study problems in geometry. The Differentia ...

s later studied by Gauss and others.
Around this same time, Leonhard Euler
Leonhard Euler ( ; ; 15 April 170718 September 1783) was a Swiss mathematician
A mathematician is someone who uses an extensive knowledge of mathematics
Mathematics (from Greek: ) includes the study of such topics as numbers ( and ) ...

, originally a student of Johann Bernoulli, provided many significant contributions not just to the development of geometry, but to mathematics more broadly. In regards to differential geometry, Euler studied the notion of a geodesic
In geometry
Geometry (from the grc, γεωμετρία; ' "earth", ' "measurement") is, with , one of the oldest branches of . It is concerned with properties of space that are related with distance, shape, size, and relative position o ...

on a surface deriving the first analytical geodesic equation
In geometry, a geodesic () is commonly a curve representing in some sense the shortest path (Arc (geometry), arc) between two points in a Differential geometry of surfaces, surface, or more generally in a Riemannian manifold. The term also has me ...

, and later introduced the first set of intrinsic coordinate systems on a surface, beginning the theory of ''intrinsic geometry'' upon which modern geometric ideas are based. Around this time Euler's study of mechanics in the Mechanica lead to the realization that a mass traveling along a surface not under the effect of any force would traverse a geodesic path, an early precursor to the important foundational ideas of Einstein's general relativity
General relativity, also known as the general theory of relativity, is the geometric
Geometry (from the grc, γεωμετρία; '' geo-'' "earth", '' -metron'' "measurement") is, with arithmetic, one of the oldest branches of mathema ...

, and also to the Euler–Lagrange equations and the first theory of the calculus of variations
The calculus of variations is a field of mathematical analysis that uses variations, which are small changes in Function (mathematics), functions
and functional (mathematics), functionals, to find maxima and minima of functionals: Map (mathematic ...

, which underpins in modern differential geometry many techniques in symplectic geometry
Symplectic geometry is a branch of differential geometry
Differential geometry is a Mathematics, mathematical discipline that uses the techniques of differential calculus, integral calculus, linear algebra and multilinear algebra to study prob ...

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

. This theory was used by Lagrange
Joseph-Louis Lagrange (born Giuseppe Luigi LagrangiaRoman_Forum.html" ;"title="Curia Julia in the Roman Forum">Curia Julia in the Roman Forum
A senate is a deliberative assembly, often the upper house or Debating chamber, chamber of a bicame ...

, a co-developer of the calculus of variations, to derive the first differential equation describing a minimal surface
180px, A helicoid minimal surface formed by a soap film on a helical frame
In mathematics
Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), ...

in terms of the Euler–Lagrange equation. In 1760 Euler proved a theorem expressing the curvature of a space curve on a surface in terms of the principal curvatures, known as Euler's theorem
In number theory, Euler's theorem (also known as the Fermat–Euler theorem or Euler's totient theorem) states that if ''n'' and ''a'' are coprime
In number theory, two integer
An integer (from the Latin wikt:integer#Latin, ''integer'' mea ...

.
Later in the 1700s, the new French school lead 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.
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 onhyperbolic geometry
In mathematics
Mathematics (from Greek: ) includes the study of such topics as numbers ( and ), formulas and related structures (), shapes and spaces in which they are contained (), and quantities and their changes ( and ). There is no g ...

and non-Euclidean geometry and throughout the same period the development of projective geometry.
Dubbed the single most important work in the history of differential geometry,Spivak, M., 1975. A comprehensive introduction to differential geometry (Vol. 2). Publish or Perish, Incorporated. in 1827 Gauss produced the ''Disquisitiones generales circa superficies curvas'' detailing the general theory of curved surfaces.Gauss, C.F., 1828. Disquisitiones generales circa superficies curvas (Vol. 1). Typis Dieterichianis.Struik, D.J. “Outline of a History of Differential Geometry (II).” Isis, vol. 20, no. 1, 1933, pp. 161–191. JSTOR, www.jstor.org/stable/224886 In this work and his subsequent papers and unpublished notes on the theory of surfaces, Gauss has been dubbed the inventor of non-Euclidean geometry and the inventor of intrinsic differential geometry. In his fundamental paper Gauss introduced the Gauss map, Gaussian curvature, first fundamental form, first and second fundamental forms, proved the Theorema Egregium
without distortion. The Mercator projection, shown here, preserves angles but fails to preserve area.
Gauss's ''Theorema Egregium'' (Latin for "Remarkable Theorem") is a major result of differential geometry
Differential geometry is a Mathe ...

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. Around this same time János Bolyai and Lobachevsky independently discovered hyperbolic geometry
In mathematics
Mathematics (from Greek: ) includes the study of such topics as numbers ( and ), formulas and related structures (), shapes and spaces in which they are contained (), and quantities and their changes ( and ). There is no g ...

and thus demonstrated the existence of consistent geometries outside Euclid's paradigm. Concrete models of hyperbolic geometry were produced by Eugenio Beltrami later in the 1860s, and Felix Klein coined the term non-Euclidean geometry in 1871, and through the Erlangen program put Euclidean and non-Euclidean geometries on the same footing. Implicitly, the spherical geometry
Image:Triangles (spherical geometry).jpg, 300px, The sum of the angles of a spherical triangle is not equal to 180°. A sphere is a curved surface, but locally the laws of the flat (planar) Euclidean geometry are good approximations. In a small tr ...

of the Earth that had been studied since antiquity was a non-Euclidean geometry, an elliptic geometry.
The development of intrinsic differential geometry in the language of Gauss was spurred on by his student, Bernhard Riemann in his Habilitationsschrift, ''On the hypotheses which lie at the foundation of geometry''. In this work Riemann introduced the notion of a Riemannian metric and the Riemannian curvature tensor for the first time, and began the systematic study of differential geometry in higher dimensions. This intrinsic point of view in terms of the Riemannian metric, denoted by $ds^2$ by Riemann, was the development of an idea of Gauss' about the linear element $ds$ of a surface. At this time Riemann began to introduce the systematic use of linear algebra
Linear algebra is the branch of mathematics concerning linear equations such as:
:a_1x_1+\cdots +a_nx_n=b,
linear maps such as:
:(x_1, \ldots, x_n) \mapsto a_1x_1+\cdots +a_nx_n,
and their representations in vector spaces and through matrix (mat ...

and multilinear algebra
In mathematics
Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). I ...

into the subject, making great use of the theory of quadratic forms in his investigation of metrics and curvature. At this time Riemann did not yet develop the modern notion of a manifold, as even the notion of a topological space had not been encountered, but he did propose that it might be possible to investigate or measure the properties of the metric of spacetime through the analysis of masses within spacetime, linking with the earlier observation of Euler that masses under the effect of no forces would travel along geodesics on surfaces, and predicting Einstein's fundamental observation of the equivalence principle a full 60 years before it appeared in the scientific literature.
In the wake of Riemann's new description, the focus of techniques used to study differential geometry shifted from the ad hoc and extrinsic methods of the study of curves and surfaces to a more systematic approach in terms of tensor calculus and Klein's Erlangen program, and progress increased in the field. The notion of groups of transformations was developed by Sophus Lie and Jean Gaston Darboux, leading to important results in the theory of Lie groups and symplectic geometry
Symplectic geometry is a branch of differential geometry
Differential geometry is a Mathematics, mathematical discipline that uses the techniques of differential calculus, integral calculus, linear algebra and multilinear algebra to study prob ...

. 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. 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. 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 out of 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 oftopology
In mathematics
Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities ...

.Dieudonné, J., 2009. A history of algebraic and differential topology, 1900-1960. Springer Science & Business Media. At the start of the 1900s there was a major movement within mathematics to formalise the foundational aspects of the subject to avoid crises of rigour and accuracy, known as Hilbert's program. As part of this broader movement, the notion of a topological space was distilled in by Felix Hausdorff in 1914, and by 1942 there were many different notions of manifold of a combinatorial and differential-geometric nature.
Interest in the subject was also focused by the emergence of Einstein's theory of general relativity and the importance of the Einstein Field equations. Einstein's theory popularised the tensor calculus of Ricci and Levi-Civita and introduced the notation $g$ for a Riemannian metric, and $\backslash Gamma$ for the Christoffel symbols, both coming from ''G'' in ''Gravitation''. Élie Cartan helped reformulate the foundations of the differential geometry of smooth manifolds in terms of exterior calculus and the theory of moving frames, leading in the world of physics to Einstein–Cartan theory.Fré, P.G., 2018. A Conceptual History of Space and Symmetry. Springer, Cham.
Following this early development, many mathematicians contributed to the development of the modern theory, including Jean-Louis Koszul who introduced connection (vector bundle), connections on vector bundles, Shiing-Shen Chern who introduced characteristic classes to the subject and began the study of complex manifolds, 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 fibre bundles and Ehresmann connections, and others. Of particular important was Hermann Weyl who made important contributions to the foundations of general relativity, introduced the Weyl tensor providing insight into conformal geometry
In mathematics
Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It h ...

, 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
In physics, a gauge theory is a type of Field theory (physics), field theory in which the Lagrangian (field theory), Lagrangian (and hence the dynamics of the system itself) does not change (is Invariant (physics), invariant) under local symmetry, ...

. 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 invariants.
Branches

Riemannian geometry

Riemannian geometry studies Riemannian manifolds,smooth manifold
In mathematics, a differentiable manifold (also differential manifold) is a type of manifold
The real projective plane is a two-dimensional manifold that cannot be realized in three dimensions without self-intersection, shown here as Boy's s ...

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
In mathematics
Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities a ...

. 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
without distortion. The Mercator projection, shown here, preserves angles but fails to preserve area.
Gauss's ''Theorema Egregium'' (Latin for "Remarkable Theorem") is a major result of differential geometry
Differential geometry is a Mathe ...

of Carl Friedrich Gauss showed that for surfaces, the existence of a local isometry imposes that the Gaussian curvatures at the corresponding points must be the same. In higher dimensions, the Riemann curvature tensor is an important pointwise invariant associated with a Riemannian manifold that measures how close it is to being flat. An important class of Riemannian manifolds is the Riemannian symmetric spaces, whose curvature is not necessarily constant. These are the closest analogues to the "ordinary" plane and space considered in Euclidean and non-Euclidean geometry.
Pseudo-Riemannian geometry

pseudo-Riemannian 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

Finsler geometry has ''Finsler manifolds'' as the main object of study. This is a differential manifold with a ''Finsler metric'', that is, a Banach norm defined on each tangent space. Riemannian manifolds are special cases of the more general Finsler manifolds. A Finsler structure on a manifold is a function such that: # for all in and all , # is infinitely differentiable in , # The vertical Hessian of is positive definite.Symplectic geometry

Symplectic geometry is the study of symplectic manifolds. An almost symplectic manifold is a differentiable manifold equipped with a smoothly varying non-degenerate 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: . 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 thecurvature
In mathematics
Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities an ...

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.
Contact geometry

Contact geometry deals with certain manifolds of odd dimension. It is close to symplectic geometry and like the latter, it originated in questions of classical mechanics. A ''contact structure'' on a -dimensional manifold ''M'' is given by a smooth hyperplane field ''H'' in the tangent bundle 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 $\backslash alpha$, which is unique up to multiplication by a nowhere vanishing function: : $H\_p\; =\; \backslash ker\backslash alpha\_p\backslash subset\; T\_M.$ A local 1-form on ''M'' is a ''contact form'' if the restriction of its exterior derivative to ''H'' is a non-degenerate two-form and thus induces a symplectic structure on ''H''Complex and Kähler geometry

''Complex differential geometry'' is the study of complex manifolds. An almost complex manifold is a ''real'' manifold $M$, endowed with atensor
In mathematics
Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities a ...

of type (1, 1), i.e. a vector bundle, vector bundle endomorphism (called an ''almost complex structure'')
:$J:TM\backslash rightarrow\; TM$, such that $J^2=-1.\; \backslash ,$
It follows from this definition that an almost complex manifold is even-dimensional.
An almost complex manifold is called ''complex'' if $N\_J=0$, where $N\_J$ is a tensor of type (2, 1) related to $J$, called the Nijenhuis tensor (or sometimes the ''torsion'').
An almost complex manifold is complex if and only if it admits a Holomorphic 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
:$g(JX,JY)=g(X,Y)\; \backslash ,$.
An almost Hermitian structure defines naturally a differential form, differential two-form
:$\backslash omega\_(X,Y):=g(JX,Y)\; \backslash ,$.
The following two conditions are equivalent:
# $N\_J=0\backslash mboxd\backslash omega=0\; \backslash ,$
# $\backslash nabla\; J=0\; \backslash ,$
where $\backslash nabla$ is the Levi-Civita connection of $g$. In this case, $(J,\; g)$ 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 manifolds) 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 manifolds.Conformal geometry

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

Differential topology is the study of global geometric invariants without a metric or symplectic form. Differential topology starts from the natural operations such as Lie derivative of natural vector bundles and Exterior derivative, de Rham differential of Differential form, forms. Beside Lie algebroids, also Courant algebroids 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 fields. 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

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 thestandard model of particle physics
The Standard Model of particle physics
Particle physics (also known as high energy physics) is a branch of physics
Physics (from grc, φυσική (ἐπιστήμη), physikḗ (epistḗmē), knowledge of nature, from ''phýsis ...

. Gauge theory is concerned with the study of differential equations for connections on bundles, and the resulting geometric moduli spaces of solutions to these equations as well as the invariants that may be derived from them. These equations often arise as the Euler–Lagrange equations describing the equations of motion of certain physical systems in quantum field theory
In theoretical physics, quantum field theory (QFT) is a theoretical framework that combines classical field theory, special relativity and quantum mechanics. QFT is used in particle physics to construct physical models of subatomic particles and ...

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

The apparatus of vector bundles, principal bundles, and connection (mathematics), connections on bundles plays an extraordinarily important role in modern differential geometry. A smooth manifold always carries a natural vector bundle, the tangent bundle. Loosely speaking, this structure by itself is sufficient only for developing analysis on the manifold, while doing geometry requires, in addition, some way to relate the tangent spaces at different points, i.e. a notion of parallel transport. An important example is provided by affine connections. For a surface in RRiemannian geometry#REDIRECT Riemannian geometry
Riemannian geometry is the branch of differential geometry
Differential geometry is a Mathematics, mathematical discipline that uses the techniques of differential calculus, integral calculus, linear algebra and m ...

, 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.Applications

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 whichAlbert Einstein
Albert Einstein ( ; ; 14 March 1879 – 18 April 1955) was a German-born theoretical physicist, widely acknowledged to be one of the greatest physicists of all time. Einstein is known for developing the theory of relativity
The theo ...

'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 systems.
**Riemannian geometry and contact geometry have been used to construct the formalism of geometrothermodynamics which has found applications in classical equilibrium thermodynamics.
*In chemistry
Chemistry is the scientific
Science () is a systematic enterprise that builds and organizes knowledge
Knowledge is a familiarity or awareness, of someone or something, such as facts
A fact is an occurrence in the real world. T ...

and biophysics when modelling cell membrane structure under varying pressure.
*In economics
Economics () is a social science that studies the Production (economics), production, distribution (economics), distribution, and Consumption (economics), consumption of goods and services.
Economics focuses on the behaviour and interact ...

, differential geometry has applications to the field of econometrics.
*Geometric modeling (including computer graphics
Computer graphics deals with generating images with the aid of computers. Today, computer graphics is a core technology in digital photography, film, video games, cell phone and computer displays, and many specialized applications. A great dea ...

) and computer-aided geometric design draw on ideas from differential geometry.
*In engineering
Engineering is the use of scientific principles to design and build machines, structures, and other items, including bridges, tunnels, roads, vehicles, and buildings. The discipline of engineering encompasses a broad range of more speciali ...

, differential geometry can be applied to solve problems in digital signal processing.
*In control theory
Control theory deals with the control of dynamical system
In mathematics
Mathematics (from Greek: ) includes the study of such topics as numbers ( and ), formulas and related structures (), shapes and spaces in which they are contai ...

, differential geometry can be used to analyze nonlinear controllers, particularly geometric control
* 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
Computer vision is an interdisciplinary scientific field that deals with how computer
A computer is a machine that can be programmed to carry out sequences of arithmetic or logical operations automatically. Modern computers can perform ge ...

, differential geometry is used to analyze shapes.
*In image processing, differential geometry is used to process and analyse data on non-flat surfaces.
*Grigori Perelman's proof of the Poincaré conjecture using the techniques of Ricci flows demonstrated the power of the differential-geometric approach to questions in topology
In mathematics
Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities ...

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.
See also

* Abstract differential geometry * Affine differential geometry * Analysis on fractals * Basic introduction to the mathematics of curved spacetime * Discrete differential geometry *Gauss
Johann Carl Friedrich Gauss (; german: Gauß ; la, Carolus Fridericus Gauss; 30 April 177723 February 1855) was a German mathematician
This is a List of German mathematician
A mathematician is someone who uses an extensive knowledge of m ...

* 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)
References

Further reading

* * * * * * * * * *External links

*B. Conrad. Differential Geometry handouts, Stanford University

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

Richard Palais's 3DXM Surfaces Gallery

N. J. Hicks, Notes on Differential Geometry, Van Nostrand.

MIT OpenCourseWare: Differential Geometry, Fall 2008

{{DEFAULTSORT:Differential Geometry Differential geometry, Geometry processing