Differential geometry and topology - AbsoluteAstronomy.com

Differential geometry is a mathematical

Mathematics

Mathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...

discipline that uses the techniques of differential

Differential calculus

In mathematics, differential calculus is a subfield of calculus concerned with the study of the rates at which quantities change. It is one of the two traditional divisions of calculus, the other being integral calculus....

and integral calculus

Calculus

Calculus is a branch of mathematics focused on limits, functions, derivatives, integrals, and infinite series. This subject constitutes a major part of modern mathematics education. It has two major branches, differential calculus and integral calculus, which are related by the fundamental theorem...

, as well as linear

Linear algebra

Linear algebra is a branch of mathematics that studies vector spaces, also called linear spaces, along with linear functions that input one vector and output another. Such functions are called linear maps and can be represented by matrices if a basis is given. Thus matrix theory is often...

and multilinear algebra

Multilinear algebra

In mathematics, multilinear algebra extends the methods of linear algebra. Just as linear algebra is built on the concept of a vector and develops the theory of vector spaces, multilinear algebra builds on the concepts of p-vectors and multivectors with Grassmann algebra.-Origin:In a vector space...

, to study problems in geometry

Geometry

Geometry arose as the field of knowledge dealing with spatial relationships. Geometry was one of the two fields of pre-modern mathematics, the other being the study of numbers ....

. The theory of plane and space curves

Differential geometry of curves

Differential geometry of curves is the branch of geometry that dealswith smooth curves in the plane and in the Euclidean space by methods of differential and integral calculus....

and of surfaces

Differential geometry of surfaces

In mathematics, the differential geometry of surfaces deals with smooth surfaces with various additional structures, most often, a Riemannian metric....

in the three-dimensional Euclidean space

Euclidean space

In mathematics, Euclidean space is the Euclidean plane and three-dimensional space of Euclidean geometry, as well as the generalizations of these notions to higher dimensions...

formed the basis for its initial development in the eighteenth and nineteenth centuries. Since the late nineteenth century, differential geometry has grown into a field concerned more generally with geometric structures on differentiable manifold

Differentiable manifold

A differentiable manifold is a type of manifold that is locally similar enough to a linear space to allow one to do calculus. Any manifold can be described by a collection of charts, also known as an atlas. One may then apply ideas from calculus while working within the individual charts, since...

s. It is closely related to differential topology

Differential topology

In mathematics, differential topology is the field dealing with differentiable functions on differentiable manifolds. It is closely related to differential geometry and together they make up the geometric theory of differentiable manifolds.- Description :...

, and to the geometric aspects of the theory of differential equation

Differential equation

A differential equation is a mathematical equation for an unknown function of one or several variables that relates the values of the function itself and its derivatives of various orders...

s. Grigori Perelman

Grigori Perelman

Grigori Yakovlevich Perelman is a Russian mathematician who has made landmark contributions to Riemannian geometry and geometric topology.In 1992, Perelman proved the soul conjecture. In 2002, he proved Thurston's geometrization conjecture...

's proof of the Poincaré conjecture

Poincaré conjecture

In mathematics, the Poincaré conjecture is a theorem about the characterization of the three-dimensional sphere , which is the hypersphere that bounds the unit ball in four-dimensional space...

using the techniques of Ricci flow

Ricci flow

In differential geometry, the Ricci flow is an intrinsic geometric flow. It is a process that deforms the metric of a Riemannian manifold in a way formally analogous to the diffusion of heat, smoothing out irregularities in the metric....

demonstrated the power of the differential-geometric approach to questions in topology

Topology

Topology is a major area of mathematics concerned with properties that are preserved under continuous deformations of objects, such as deformations that involve stretching, but no tearing or gluing...

and highlighted the important role played by the analytic methods. Differential geometry of surfaces

Differential geometry of surfaces

In mathematics, the differential geometry of surfaces deals with smooth surfaces with various additional structures, most often, a Riemannian metric....

captures many of the key ideas and techniques characteristic of the field.

Riemannian geometry

Riemannian geometry studies Riemannian manifold

Riemannian manifold

In Riemannian geometry and the differential geometry of surfaces, a Riemannian manifold or Riemannian space is a real differentiable manifold M in which each tangent space is equipped with an inner product g, a Riemannian metric, which varies smoothly from point to point...

s, smooth manifolds with a Riemannian metric. This is a concept of distance expressed by means of a smooth

Smooth function

In mathematical analysis, a differentiability class is a classification of functions according to the properties of their derivatives. Higher order differentiability classes correspond to the existence of more derivatives. Functions that have derivatives of all orders are called smooth.Most of...

positive definite symmetric bilinear form

Symmetric bilinear form

A symmetric bilinear form is a bilinear form on a vector space that is symmetric. Symmetric bilinear forms are of great importance in the study of orthogonal polarity and quadrics....

defined on the tangent space at each point. Riemannian geometry generalizes Euclidean geometry

Euclidean geometry

Euclidean geometry is a mathematical system attributed to the Alexandrian Greek mathematician Euclid, which he described in his textbook on geometry: the Elements. Euclid's method consists in assuming a small set of intuitively appealing axioms, and deducing many other propositions from these...

to spaces that are not necessarily flat, although they still resemble the Euclidean space

Euclidean space

In mathematics, Euclidean space is the Euclidean plane and three-dimensional space of Euclidean geometry, as well as the generalizations of these notions to higher dimensions...

at each point "infinitesimally", i.e. in the first order of approximation. Various concepts based on length, such as the arc length

Arc length

Determining the length of an irregular arc segment is also called rectification of a curve. Historically, many methods were used for specific curves...

of curve

Curve

In mathematics, a curve is, generally speaking, an object similar to a line but which is not required to be straight...

s, area

Area

Area is a quantity that expresses the extent of a two-dimensional surface or shape in the plane. Area can be understood as the amount of material with a given thickness that would be necessary to fashion a model of the shape, or the amount of paint necessary to cover the surface with a single coat...

of plane regions, and volume

Volume

Volume is the quantity of three-dimensional space enclosed by some closed boundary, for example, the space that a substance or shape occupies or contains....

of solids all admit natural analogues in Riemannian geometry. The notion of a directional derivative

Directional derivative

In mathematics, the directional derivative of a multivariate differentiable function along a given vector V at a given point P intuitively represents the instantaneous rate of change of the function, moving through P in the direction of V...

of a function from multivariable calculus

Multivariable calculus

Multivariable calculus is the extension of calculus in one variable to calculus in more than one variable: the differentiated and integrated functions involve multiple variables, rather than just one....

is extended in Riemannian geometry to the notion of a covariant derivative

Covariant derivative

In mathematics, the covariant derivative is a way of specifying a derivative along tangent vectors of a manifold. Alternatively, the covariant derivative is a way of introducing and working with a connection on a manifold by means of a differential operator, to be contrasted with the approach given...

of a tensor

Tensor

Tensors are geometric objects that describe linear relations between vectors, scalars, and other tensors. Elementary examples include the dot product, the cross product, and linear maps. Vectors and scalars themselves are also tensors. A tensor can be represented as a multi-dimensional array of...

. Many concepts and techniques of analysis and differential equations have been generalized to the setting of Riemannian manifolds.

A distance-preserving diffeomorphism

Diffeomorphism

In mathematics, a diffeomorphism is an isomorphism in the category of smooth manifolds. It is an invertible function that maps one differentiable manifold to another, such that both the function and its inverse are smooth.- Definition :...

between Riemannian manifolds is called an isometry

Isometry

In mathematics, an isometry is a distance-preserving map between metric spaces. Geometric figures which can be related by an isometry are called congruent.Isometries are often used in constructions where one space is embedded in another space...

. This notion can also be defined locally, i.e. for small neighborhoods of points. Any two regular curves are locally isometric. However, Theorema Egregium

Theorema Egregium

Gauss's Theorema Egregium is a foundational result in differential geometry proved by Carl Friedrich Gauss that concerns the curvature of surfaces...

of Gauss

Carl Friedrich Gauss

Johann Carl Friedrich Gauss was a German mathematician and scientist who contributed significantly to many fields, including number theory, statistics, analysis, differential geometry, geodesy, geophysics, electrostatics, astronomy and optics.Sometimes referred to as the Princeps mathematicorum...

showed that already for surfaces, the existence of a local isometry imposes strong compatibility conditions on their metrics: the Gaussian curvature

Gaussian curvature

In differential geometry, the Gaussian curvature or Gauss curvature of a point on a surface is the product of the principal curvatures, κ1 and κ2, of the given point. It is an intrinsic measure of curvature, i.e., its value depends only on how distances are measured on the surface, not on the way...

s at the corresponding points must be the same. In higher dimensions, the Riemann curvature tensor

Riemann curvature tensor

In the mathematical field of differential geometry, the Riemann curvature tensor, or Riemann–Christoffel tensor after Bernhard Riemann and Elwin Bruno Christoffel, is the most standard way to express curvature of Riemannian manifolds...

is an important pointwise invariant associated to a Riemannian manifold that measures how close it is to being flat. An important class of Riemannian manifolds is the Riemannian symmetric space

Riemannian symmetric space

In differential geometry, representation theory and harmonic analysis, a symmetric space is a smooth manifold whose group of symmetries contains an inversion symmetry about every point. There are two ways to formulate the inversion symmetry, via Riemannian geometry or via Lie theory...

s, whose curvature is not necessarily constant. These are the closest analogues to the "ordinary" plane and space considered in Euclidean and non-Euclidean geometry

Non-Euclidean geometry

Non-Euclidean geometry is the term used to refer to two specific geometries which are, loosely speaking, obtained by negating the Euclidean parallel postulate, namely hyperbolic and elliptic geometry. This is one term which, for historical reasons, has a meaning in mathematics which is much...

Pseudo-Riemannian geometry

Pseudo-Riemannian manifold

In differential geometry, a pseudo-Riemannian manifold is a generalization of a Riemannian manifold. It is one of many mathematical objects named after Bernhard Riemann. The key difference between a Riemannian manifold and a pseudo-Riemannian manifold is that on a pseudo-Riemannian manifold the...

generalizes Riemannian geometry to the case in which the metric tensor

Metric tensor

In the mathematical field of differential geometry, a metric tensor is a type of function defined on a manifold which takes as input a pair of tangent vectors v and w and produces a real number g in a way that generalizes many of the familiar properties of the dot product of vectors in Euclidean...

need not be positive-definite

Definite bilinear form

In mathematics, a definite bilinear form is a bilinear form B over some vector space V such that the associated quadratic formQ=B \,...

.
A special case of this is a Lorentzian manifold, which is the mathematical basis of Einstein's general relativity theory of gravity

General relativity

General relativity or the general theory of relativity is the geometric theory of gravitation published by Albert Einstein in 1916. It is the current description of gravitation in modern physics...

Finsler geometry

Finsler geometry has the Finsler manifold as the main object of study. This is a differential manifold with a Finsler metric, i.e. a Banach norm defined on each tangent space. A Finsler metric is a much more general structure than a Riemannian metric. A Finsler structure on a manifold M is a function F : TM → [0,∞) such that:

F(x, my) = |m|F(x,y) for all x, y in TM,
F is infinitely differentiable in TM − {0},
The vertical Hessian of F² is positive definite.

Symplectic geometry

Symplectic geometry is the study of symplectic manifold

Symplectic manifold

In mathematics, a symplectic manifold is a smooth manifold, M, equipped with a closed nondegenerate differential 2-form, ω, called the symplectic form. The study of symplectic manifolds is called symplectic geometry or symplectic topology...

s. An almost symplectic manifold is a differentiable manifold equipped with a smoothly varying

Smooth function

non-degenerate skew-symmetric

Skew-symmetric matrix

In mathematics, and in particular linear algebra, a skew-symmetric matrix is a square matrix A whose transpose is also its negative; that is, it satisfies the equation If the entry in the and is aij, i.e...

bilinear form on each tangent space, i.e., a nondegenerate 2-form

Differential form

In the mathematical fields of differential geometry and tensor calculus, differential forms are an approach to multivariable calculus that is independent of coordinates. Differential forms provide a better definition for integrands in calculus...

ω, called the symplectic form. A symplectic manifold is an almost symplectic manifold for which the symplectic form ω is closed: dω = 0.

A diffeomorphism

Diffeomorphism

between two symplectic manifolds which preserves the symplectic form is called a symplectomorphism

Symplectomorphism

In mathematics, a symplectomorphism is an isomorphism in the category of symplectic manifolds.-Formal definition:A diffeomorphism between two symplectic manifolds f: \rightarrow is called symplectomorphism, iff^*\omega'=\omega,...

. 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

Surface

In mathematics, specifically in topology, a surface is a two-dimensional topological manifold. The most familiar examples are those that arise as the boundaries of solid objects in ordinary three-dimensional Euclidean space R3 — for example, the surface of a ball...

endowed with an area form and a symplectomorphism is an area-preserving diffeomorphism. The phase space

Phase space

In mathematics and physics, a phase space, introduced by Willard Gibbs in 1901, is a space in which all possible states of a system are represented, with each possible state of the system corresponding to one unique point in the phase space...

of a mechanical system is a symplectic manifold and they made an implicit appearance already in the work of Lagrange

Joseph Louis Lagrange

Joseph-Louis Lagrange , born Giuseppe Lodovico Lagrangia, was a mathematician and astronomer, who was born in Turin, Piedmont, lived part of his life in Prussia and part in France, making significant contributions to all fields of analysis, to number theory, and to classical and celestial mechanics...

on analytical mechanics

Analytical mechanics

Analytical mechanics is a term used for a refined, mathematical form of classical mechanics, constructed from the 18th century onwards as a formulation of the subject as founded by Isaac Newton. Often the term vectorial mechanics is applied to the form based on Newton's work, to contrast it with...

and later in Jacobi's and Hamilton's

William Rowan Hamilton

Sir William Rowan Hamilton was an Irish physicist, astronomer, and mathematician, who made important contributions to classical mechanics, optics, and algebra. His studies of mechanical and optical systems led him to discover new mathematical concepts and techniques...

formulation of classical mechanics

Hamiltonian mechanics

Hamiltonian mechanics is a reformulation of classical mechanics that was introduced in 1833 by Irish mathematician William Rowan Hamilton.It arose from Lagrangian mechanics, a previous reformulation of classical mechanics introduced by Joseph Louis Lagrange in 1788, but can be formulated without...

.

By contrast with Riemannian geometry, where the curvature

Curvature

In mathematics, curvature refers to any of a number of loosely related concepts in different areas of geometry. Intuitively, curvature is the amount by which a geometric object deviates from being flat, or straight in the case of a line, but this is defined in different ways depending on the context...

provides a local invariant of Riemannian manifolds, Darboux's theorem

Darboux's theorem

Darboux's theorem is a theorem in the mathematical field of differential geometry and more specifically differential forms, partially generalizing the Frobenius integration theorem. It is a foundational result in several fields, the chief among them being symplectic geometry...

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

Henri Poincaré

Jules Henri Poincaré was a French mathematician, theoretical physicist, engineer, and a philosopher of science...

and proved by George Birkhoff

George David Birkhoff

-External links:* − from National Academies Press, by Oswald Veblen....

in 1912. It claims that if an area preserving map of an annulus

Annulus (mathematics)

In mathematics, an annulus is a ring-shaped geometric figure, or more generally, a term used to name a ring-shaped object. Or, it is the area between two concentric circles...

twists each boundary component in opposite directions, then the map has at least two fixed points.

Contact geometry

In mathematics, contact geometry is the study of a geometric structure on smooth manifolds given by a hyperplane distribution in the tangent bundle and specified by a one-form, both of which satisfy a 'maximum non-degeneracy' condition called 'complete non-integrability'...

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 (2n+1)-dimensional manifold M is given by a smooth hyperplane field H in the tangent bundle

Tangent bundle

In differential geometry, the tangent bundle of a differentiable manifold M is the disjoint unionThe disjoint union assures that for any two points x1 and x2 of manifold M the tangent spaces T1 and T2 have no common vector...

that is as far as possible from being associated with the level sets of a differentiable function on M (the technical term is "completely nonintegrable tangent hyperplane distribution"). Near each point p, a hyperplane distribution is determined by a nowhere vanishing 1-form

Differential form

, which is unique up to multiplication by a nowhere vanishing function:

A local 1-form on M is a contact form if the restriction of its exterior derivative

Exterior derivative

In differential geometry, the exterior derivative extends the concept of the differential of a function, which is a 1-form, to differential forms of higher degree. Its current form was invented by Élie Cartan....

to H is a non-degenerate 2-form and thus induces a symplectic structure on H_p at each point. If the distribution H can be defined by a global 1-form

then this form is contact if and only if the top-dimensional form

is a volume form

Volume form

In mathematics, a volume form on a differentiable manifold is a nowhere-vanishing differential form of top degree. Thus on a manifold M of dimension n, a volume form is an n-form, a section of the line bundle Ωn = Λn, that is nowhere equal to zero. A manifold has a volume...

on M, i.e. does not vanish anywhere. A contact analogue of the Darboux theorem holds: all contact structures on an odd-dimensional manifold are locally isomorphic and can be brought to a certain local normal form by a suitable choice of the coordinate system.

Complex and Kähler geometry

Complex differential geometry is the study of complex manifolds.
An almost complex manifold

Almost complex manifold

In mathematics, an almost complex manifold is a smooth manifold equipped with smooth linear complex structure on each tangent space. The existence of this structure is a necessary, but not sufficient, condition for a manifold to be a complex manifold. That is, every complex manifold is an almost...

is a real manifold

, endowed with a tensor

Tensor

of type

Tensor

(1,1), i.e. a vector bundle endomorphism

Vector bundle

In mathematics, a vector bundle is a topological construction that makes precise the idea of a family of vector spaces parameterized by another space X : to every point x of the space X we associate a vector space V in such a way that these vector spaces fit together...

(called an almost complex structure)

, such that

.

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

An almost complex manifold is called complex if

, where

is a tensor of type (2,1) related to

, called the Nijenhuis tensor (or sometimes the torsion).
An almost complex manifold is complex if and only if it admits a holomorphic coordinate atlas

Atlas (topology)

In mathematics, particularly topology, one describesa manifold using an atlas. An atlas consists of individualcharts that, roughly speaking, describe individual regionsof the manifold. If the manifold is the surface of the Earth,...

.
An almost Hermitian structure
Hermitian manifold
In mathematics, a Hermitian manifold is the complex analog of a Riemannian manifold. Specifically, a Hermitian manifold is a complex manifold with a smoothly varying Hermitian inner product on each tangent space...

is given by an almost complex structure J, along with a riemannian metric g, satisfying the compatibility condition

.

An almost hermitian structure
defines naturally a differential 2-form

Differential form

.

The following two conditions are equivalent:

where

is the Levi-Civita connection

Levi-Civita connection

In Riemannian geometry, the Levi-Civita connection is a specific connection on the tangent bundle of a manifold. More specifically, it is the torsion-free metric connection, i.e., the torsion-free connection on the tangent bundle preserving a given Riemannian metric.The fundamental theorem of...

. In this case,

is called a Kähler structure, and a Kähler manifold is a manifold endowed with a Kähler structure. In particular, a Kähler manifold is both a complex and a symplectic manifold

Symplectic manifold

. A large class of Kähler manifolds (the class of Hodge manifolds) is given by all the smooth complex projective varieties

Algebraic geometry

Algebraic geometry is a branch of mathematics which combines techniques of abstract algebra, especially commutative algebra, with the language and the problems of geometry. It occupies a central place in modern mathematics and has multiple conceptual connections with such diverse fields as complex...

CR geometry

CR geometry is the study of the intrinsic geometry of boundaries of domains in complex manifold

Complex manifold

In differential geometry, a complex manifold is a manifold with an atlas of charts to the open unit disk in Cn, such that the transition maps are holomorphic....

Differential topology

is the study of (global) geometric invariants without a metric or symplectic form. It starts from the natural operations such as Lie derivative

Lie derivative

In mathematics, the Lie derivative , named after Sophus Lie by Władysław Ślebodziński, evaluates the change of a vector field or more generally a tensor field, along the flow of another vector field...

of natural vector bundle

Vector bundle

s and de Rham differential

Exterior derivative

of forms

Differential form

. Beside Lie algebroid

Lie algebroid

In mathematics, Lie algebroids serve the same role in the theory of Lie groupoids that Lie algebras serve in the theory of Lie groups: reducing global problems to infinitesimal ones...

s, also Courant algebroid

Courant algebroid

In a field of mathematics known as differential geometry, a Courant algebroid is a combination of a Lie algebroid and a quadratic Lie algebra. It was originally introduced in 1990 by Theodore James Courant in his dissertation at UC Berkeley where he first called them Dirac Manifolds, and then were...

s start playing a more important role.

Lie groups

A Lie group

Lie group

In mathematics, a Lie group is a group which is also a differentiable manifold, with the property that the group operations are compatible with the smooth structure...

is a group

Group (mathematics)

In mathematics, a group is an algebraic structure consisting of a set together with an operation that combines any two of its elements to form a third element. To qualify as a group, the set and the operation must satisfy a few conditions called group axioms, namely closure, associativity, identity...

in the category of smooth manifolds. I.e. beside the algebraic properties this enjoys also differential geometric properties. The most obvious construction is that of a Lie algebra which is the tangent space at the unit endowed with the Lie bracket between left-invariant vector field

Vector field

In vector calculus, a vector field is an assignmentof a vector to each point in a subset of Euclidean space. A vector field in the plane for instance can be visualized as an arrow, with a given magnitude and direction, attached to each point in the plane...

s. Beside the structure theory there is also the wide field of representation theory

Representation of a Lie group

In mathematics and theoretical physics, the idea of a representation of a Lie group plays an important role in the study of continuous symmetry. A great deal is known about such representations, a basic tool in their study being the use of the corresponding 'infinitesimal' representations of Lie...

Bundles and connections

The apparatus of vector bundles

Vector bundle

, principal bundles

Principal bundle

In mathematics, a principal bundle is a mathematical object which formalizes some of the essential features of the Cartesian product X × G of a space X with a group G...

, and connections

Connection (mathematics)

In geometry, the notion of a connection makes precise the idea of transporting data along a curve or family of curves in a parallel and consistent manner. There are a variety of kinds of connections in modern geometry, depending on what sort of data one wants to transport...

on bundles plays an extraordinarily important role in modern differential geometry. A smooth manifold always carries a natural vector bundle, the tangent bundle

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

Parallel transport

In geometry, parallel transport is a way of transporting geometrical data along smooth curves in a manifold. If the manifold is equipped with an affine connection , then this connection allows one to transport vectors of the manifold along curves so that they stay parallel with respect to the...

. An important example is provided by affine connection

Affine connection

In the branch of mathematics called differential geometry, an affine connection is a geometrical object on a smooth manifold which connects nearby tangent spaces, and so permits tangent vector fields to be differentiated as if they were functions on the manifold with values in a fixed vector space...

s. For a surface

Surface

in R³, tangent planes at different points can be identified using a natural path-wise parallelism induced by the ambient Euclidean space, which has a well-known standard definition of metric and parallelism. In Riemannian geometry

Riemannian geometry

Riemannian geometry is the branch of differential geometry that studies Riemannian manifolds, smooth manifolds with a Riemannian metric, i.e. with an inner product on the tangent space at each point which varies smoothly from point to point. This gives, in particular, local notions of angle, length...

, the Levi-Civita connection

Levi-Civita connection

serves a similar purpose. (The Levi-Civita connection defines path-wise parallelism in terms of a given arbitrary Riemannian metric on a manifold.) 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 the space-time continuum

Spacetime

In physics, spacetime is any mathematical model that combines space and time into a single continuum. Spacetime is usually interpreted with space as being three-dimensional and time playing the role of a fourth dimension that is of a different sort from the spatial dimensions...

and the bundles and connections are related to various physical fields.

Intrinsic versus extrinsic

Initially and up to the middle of the nineteenth century, differential geometry was studied from the extrinsic point of view: curve

Curve

In mathematics, a curve is, generally speaking, an object similar to a line but which is not required to be straight...

s and surface

Surface

s were considered as lying in a Euclidean space

Euclidean space

In mathematics, Euclidean space is the Euclidean plane and three-dimensional space of Euclidean geometry, as well as the generalizations of these notions to higher dimensions...

of higher dimension (for example a surface in an ambient space

Ambient space

An ambient space, ambient configuration space, or electroambient space, is the space surrounding an object.-Mathematics:In mathematics, especially in geometry and topology, an ambient space is the space surrounding a mathematical object...

of three dimensions). The simplest results are those in the differential geometry of curves

Differential geometry of curves

Differential geometry of curves is the branch of geometry that dealswith smooth curves in the plane and in the Euclidean space by methods of differential and integral calculus....

and differential geometry of surfaces

Differential geometry of surfaces

In mathematics, the differential geometry of surfaces deals with smooth surfaces with various additional structures, most often, a Riemannian metric....

. Starting with the work of Riemann

Bernhard Riemann

Georg Friedrich Bernhard Riemann was an influential German mathematician who made lasting contributions to analysis and differential geometry, some of them enabling the later development of general relativity....

, the intrinsic point of view was developed, in which one cannot speak of moving 'outside' the geometric object because it is considered as given in a free-standing way. The fundamental result here is Gauss's theorema egregium

Theorema Egregium

Gauss's Theorema Egregium is a foundational result in differential geometry proved by Carl Friedrich Gauss that concerns the curvature of surfaces...

, to the effect that Gaussian curvature

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 (what would be 'outside' it?). With the intrinsic point of view it is harder to define the central concept of curvature

Curvature

and other structures such as connections

Connection (mathematics)

, so there is a price to pay.

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

Nash embedding theorem

The Nash embedding theorems , named after John Forbes Nash, state that every Riemannian manifold can be isometrically embedded into some Euclidean space. Isometric means preserving the length of every path...

Applications of differential geometry

Below are some examples of how differential geometry is applied to other fields of science and mathematics.

In physics
Physics
Physics is a natural science that involves the study of matter and its motion through spacetime, along with related concepts such as energy and force. More broadly, it is the general analysis of nature, conducted in order to understand how the universe behaves.Physics is one of the oldest academic...

, three uses will be mentioned:
- Differential geometry is the language in which Einstein's general theory of relativity is expressed. According to the theory, the universe is a smooth manifold equipped with a pseudo-Riemannian metric, which describes the curvature
  Curvature
  In mathematics, curvature refers to any of a number of loosely related concepts in different areas of geometry. Intuitively, curvature is the amount by which a geometric object deviates from being flat, or straight in the case of a line, but this is defined in different ways depending on the context...
  
  of space-time. 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 useful in the study of electromagnetism
  Electromagnetism
  Electromagnetism is one of the four fundamental interactions in nature. The other three are the strong interaction, the weak interaction and gravitation...
  
  .
- Differential geometry has applications to both Lagrangian mechanics
  Lagrangian mechanics
  Lagrangian mechanics is a re-formulation of classical mechanics that combines conservation of momentum with conservation of energy. It was introduced by the Italian-French mathematician Joseph-Louis Lagrange in 1788....
  
  and Hamiltonian mechanics
  Hamiltonian mechanics
  Hamiltonian mechanics is a reformulation of classical mechanics that was introduced in 1833 by Irish mathematician William Rowan Hamilton.It arose from Lagrangian mechanics, a previous reformulation of classical mechanics introduced by Joseph Louis Lagrange in 1788, but can be formulated without...
  
  . Symplectic manifold
  Symplectic manifold
  In mathematics, a symplectic manifold is a smooth manifold, M, equipped with a closed nondegenerate differential 2-form, ω, called the symplectic form. The study of symplectic manifolds is called symplectic geometry or symplectic topology...
  
  s in particular can be used to study Hamiltonian system
  Hamiltonian system
  In physics and classical mechanics, a Hamiltonian system is a physical system in which forces are momentum invariant. Hamiltonian systems are studied in Hamiltonian mechanics....
  
  s.
In economics
Economics
Economics is the social science that analyzes the production, distribution, and consumption of goods and services. The term economics comes from the Ancient Greek from + , hence "rules of the house"...

, differential geometry has applications to the field of econometrics
Econometrics
Econometrics has been defined as "the application of mathematics and statistical methods to economic data" and described as the branch of economics "that aims to give empirical content to economic relations." More precisely, it is "the quantitative analysis of actual economic phenomena based on...

.
Geometric modeling
Geometric modeling
Geometric modeling is a branch of applied mathematics and computational geometry that studies methods and algorithms for the mathematical description of shapes....

(including computer graphics
Computer graphics
Computer graphics are graphics created using computers and, more generally, the representation and manipulation of image data by a computer with help from specialized software and hardware....

) and computer-aided geometric design draw on ideas from differential geometry.
In engineering
Engineering
Engineering is the discipline, art, skill and profession of acquiring and applying scientific, mathematical, economic, social, and practical knowledge, in order to design and build structures, machines, devices, systems, materials and processes that safely realize improvements to the lives of...

, differential geometry can be applied to solve problems in digital signal processing
Digital signal processing
Digital signal processing is concerned with the representation of discrete time signals by a sequence of numbers or symbols and the processing of these signals. Digital signal processing and analog signal processing are subfields of signal processing...

.
In probability
Probability
Probability is ordinarily used to describe an attitude of mind towards some proposition of whose truth we arenot certain. The proposition of interest is usually of the form "Will a specific event occur?" The attitude of mind is of the form "How certain are we that the event will occur?" The...

, statistics
Statistics
Statistics is the study of the collection, organization, analysis, and interpretation of data. It deals with all aspects of this, including the planning of data collection in terms of the design of surveys and experiments....

, and information theory
Information theory
Information theory is a branch of applied mathematics and electrical engineering involving the quantification of information. Information theory was developed by Claude E. Shannon to find fundamental limits on signal processing operations such as compressing data and on reliably storing and...

, one can interpret various structures as Riemannian manifolds, which yields the field of information geometry
Information geometry
Information geometry is a branch of mathematics that applies the techniques of differential geometry to the field of probability theory. It derives its name from the fact that the Fisher information is used as the Riemannian metric when considering the geometry of probability distribution families...

, particularly via the Fisher information metric
Fisher information metric
In information geometry, the Fisher information metric is a particular Riemannian metric which can be defined on a smooth statistical manifold, i.e., a smooth manifold whose points are probability measures defined on a common probability space....

.
In structural geology
Structural geology
Structural geology is the study of the three-dimensional distribution of rock units with respect to their deformational histories. The primary goal of structural geology is to use measurements of present-day rock geometries to uncover information about the history of deformation in the rocks, and...

, differential geometry is used to analyze and describe geologic structures.
In computer vision
Computer vision
Computer vision is a field that includes methods for acquiring, processing, analysing, and understanding images and, in general, high-dimensional data from the real world in order to produce numerical or symbolic information, e.g., in the forms of decisions...

, differential geometry is used to analyze shapes.
In image processing
Image processing
In electrical engineering and computer science, image processing is any form of signal processing for which the input is an image, such as a photograph or video frame; the output of image processing may be either an image or, a set of characteristics or parameters related to the image...

, differential geometry is used to process and analyse data on non-flat surfaces.

External links

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Riemannian geometry

Pseudo-Riemannian geometry

Finsler geometry

Symplectic geometry

Contact geometry

Complex and Kähler geometry

CR geometry

Differential topology

Lie groups

Bundles and connections

Intrinsic versus extrinsic

Applications of differential geometry

See also

Further reading

External links