Differential geometry

Differential geometry is a discipline that uses the techniques of, ,  and  to study problems in. The and  in the three-dimensional  formed the basis for development of 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 the geometric structures on s. Differential geometry is closely related to and the geometric aspects of the theory of s. The  captures many of the key ideas and techniques endemic to this field.

History of development
Differential geometry arose and developed as a result of and in connection to the mathematical analysis of curves and surfaces. Mathematical analysis of curves and surfaces had been developed to answer some of the nagging and unanswered questions that appeared in, like the reasons for relationships between complex shapes and curves, series and analytic functions. These unanswered questions indicated greater, hidden relationships.

The general idea of natural equations for obtaining curves from local curvature appears to have been first considered by in 1736, and many examples with fairly simple behavior were studied in the 1800s.

When curves, surfaces enclosed by curves, and points on curves were found to be quantitatively, and generally, related by mathematical forms, the formal study of the nature of curves and surfaces became a field of study in its own right, with 's paper in 1795, and especially, with 's publication of his article, titled 'Disquisitiones Generales Circa Superficies Curvas', in Commentationes Societatis Regiae Scientiarum Gottingesis Recentiores in 1827.

Initially applied to the Euclidean space, further explorations led to non-Euclidean space, and metric and topological spaces.

Riemannian geometry
Riemannian geometry studies s, s with a Riemannian metric. This is a concept of distance expressed by means of a   defined on the tangent space at each point. Riemannian geometry generalizes to spaces that are not necessarily flat, although they still resemble the  at each point infinitesimally, i.e. in the. Various concepts based on length, such as the of s,  of plane regions, and  of solids all possess natural analogues in Riemannian geometry. The notion of a of a function from  is extended in Riemannian geometry to the notion of a  of a. Many concepts and techniques of analysis and differential equations have been generalized to the setting of Riemannian manifolds.

A distance-preserving between Riemannian manifolds is called an. This notion can also be defined locally, i.e. for small neighborhoods of points. Any two regular curves are locally isometric. However, the of  showed that for surfaces, the existence of a local isometry imposes strong compatibility conditions on their metrics: the s at the corresponding points must be the same. In higher dimensions, the 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 s, whose curvature is not necessarily constant. These are the closest analogues to the "ordinary" plane and space considered in Euclidean and.

Pseudo-Riemannian geometry
generalizes Riemannian geometry to the case in which the need not be. A special case of this is a, which is the mathematical basis of Einstein's.

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 defined on each tangent space. Riemannian manifolds are special cases of the more general Finsler manifolds. A Finsler structure on a manifold $M$ is a function $F : TM → [0, ∞)$ such that:
 * 1) $F(x, my) = m F(x, y)$ for all $(x, y)$ in $TM$ and all $m≥0$,
 * 2) $F$ is infinitely differentiable in $TM ∖ {0}$,
 * 3) The vertical Hessian of $F^{2}$ is positive definite.

Symplectic geometry
is the study of s. An almost symplectic manifold is a differentiable manifold equipped with a    on each tangent space, i.e., a nondegenerate 2- ω, called the symplectic form. A symplectic manifold is an almost symplectic manifold for which the symplectic form ω is closed: dω = 0.

A between two symplectic manifolds which preserves the symplectic form is called a. 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 endowed with an area form and a symplectomorphism is an area-preserving diffeomorphism. The of a mechanical system is a symplectic manifold and they made an implicit appearance already in the work of  on  and later in 's and 's.

By contrast with Riemannian geometry, where the provides a local invariant of Riemannian manifolds,  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, conjectured by and then proved by  in 1912. It claims that if an area preserving map of an twists each boundary component in opposite directions, then the map has at least two fixed points.

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 (2n + 1)-dimensional manifold M is given by a smooth hyperplane field H in the 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 $$\alpha$$, which is unique up to multiplication by a nowhere vanishing function:


 * $$ H_p = \ker\alpha_p\subset T_{p}M.$$

A local 1-form on M is a contact form if the restriction of its to H is a non-degenerate two-form and thus induces a symplectic structure on Hp at each point. If the distribution H can be defined by a global one-form $$\alpha$$ then this form is contact if and only if the top-dimensional form


 * $$\alpha\wedge (d\alpha)^n$$

is a 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. An is a real manifold $$M$$, endowed with a  of  (1, 1), i.e. a  (called an )
 * $$ J:TM\rightarrow TM $$, such that $$J^2=-1. \,$$

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 (or sometimes the torsion). An almost complex manifold is complex if and only if it admits a. An  is given by an almost complex structure J, along with a g, satisfying the compatibility condition
 * $$g(JX,JY)=g(X,Y) \,$$.

An almost Hermitian structure defines naturally a
 * $$\omega_{J,g}(X,Y):=g(JX,Y) \,$$.

The following two conditions are equivalent:


 * 1) $$ N_J=0\mbox{ and }d\omega=0 \,$$
 * 2) $$\nabla J=0 \,$$

where $$\nabla$$ is the of $$g$$. In this case, $$(J, g)$$ is called a , and a Kähler manifold is a manifold endowed with a Kähler structure. In particular, a Kähler manifold is both a complex and a. A large class of Kähler manifolds (the class of s) is given by all the smooth.

CR geometry
is the study of the intrinsic geometry of boundaries of domains in s.

Differential topology
is the study of global geometric invariants without a metric or symplectic form.

Differential topology starts from the natural operations such as of natural s and  of. Beside s, also s start playing a more important role.

Lie groups
A is a  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 s. Beside the structure theory there is also the wide field of.

Bundles and connections
The apparatus of s, s, and s on bundles plays an extraordinarily important role in modern differential geometry. A smooth manifold always carries a natural vector bundle, the. 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. An important example is provided by s. For a 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, the 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 and the bundles and connections are related to various physical fields.

Intrinsic versus extrinsic
From the beginning and through the middle of the 18th century, differential geometry was studied from the extrinsic point of view: s and s were considered as lying in a of higher dimension (for example a surface in an  of three dimensions). The simplest results are those in the and. Starting with the work of, 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, to the effect that 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" of the universe?). However, there is a price to pay in technical complexity: the intrinsic definitions of and  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 .) In the formalism of both extrinsic and intrinsic geometry of a manifold can be characterized by a single bivector-valued one-form called the.

Applications
Below are some examples of how differential geometry is applied to other fields of science and mathematics.
 * In, differential geometry has many applications, including:
 * Differential geometry is the language in which 's is expressed. According to the theory, the universe is a smooth manifold equipped with a pseudo-Riemannian metric, which describes the  of . Understanding this curvature is essential for the positioning of  into orbit around the earth. Differential geometry is also indispensable in the study of  and.
 * are used in the study of.
 * Differential geometry has applications to both and . s in particular can be used to study s.
 * Riemannian geometry and contact geometry have been used to construct the formalism of which has found applications in classical equilibrium.
 * In and  when modelling cell membrane structure under varying pressure.
 * In, differential geometry has applications to the field of.
 * (including ) and draw on ideas from differential geometry.
 * In, differential geometry can be applied to solve problems in.
 * In, differential geometry can be used to analyze nonlinear controllers, particularly
 * In, , and , one can interpret various structures as Riemannian manifolds, which yields the field of , particularly via the.
 * In, differential geometry is used to analyze and describe geologic structures.
 * In, differential geometry is used to analyze shapes.
 * In, differential geometry is used to process and analyse data on non-flat surfaces.
 * 's proof of the using the techniques of s demonstrated the power of the differential-geometric approach to questions in  and it highlighted the important role played by its analytic methods.
 * In, are used for  techniques in  systems.