Editing Differential geometry (section)

== Branches ==
===Riemannian geometry===
{{main|Riemannian geometry}}

Riemannian geometry studies [[Riemannian manifold]]s, [[smooth manifold]]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]]. 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]] of Carl Friedrich Gauss showed that for surfaces, the existence of a local isometry imposes that the [[Gaussian curvature]]s 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 space]]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]].

===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===
{{main|Finsler manifold}}
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 <math>M</math> is a function <math>F:\mathrm{T}M\to[0,\infty)</math> such that:
#<math>F(x,my)=mF(x,y)</math> for all <math>(x,y)</math> in <math>\mathrm{T}M</math> and all <math>m\ge 0</math>,
# <math>F</math> is infinitely differentiable in <math>\mathrm{T}M\setminus\{0\}</math>,
# The vertical Hessian of <math>F^2</math> is positive definite.

===Symplectic geometry===
{{main|Symplectic geometry}}

[[Symplectic geometry]] is the study of [[symplectic manifold]]s. 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:  {{nowrap|1=d''ω'' = 0}}.

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 the [[curvature]] 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.<ref>The area preserving condition (or the twisting condition) cannot be removed. If one tries to extend such a theorem to higher dimensions, one would probably guess that a volume preserving map of a certain type must have fixed points. This is false in dimensions greater than 3.</ref>

===Contact geometry===
{{main|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 {{nowrap|(2''n'' + 1)}}-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]] <math>\alpha</math>, which is unique up to multiplication by a nowhere vanishing function:

: <math> H_p = \ker\alpha_p\subset T_{p}M.</math>

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''<sub>''p''</sub> at each point. If the distribution ''H'' can be defined by a global one-form <math>\alpha</math> then this form is contact if and only if the top-dimensional form

: <math>\alpha\wedge (d\alpha)^n</math>

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.

===Complex and Kähler geometry===

{{See also|Complex geometry}}
''Complex differential geometry'' is the study of [[complex manifolds]].
An [[almost complex manifold]] is a ''real'' manifold <math>M</math>, endowed with a [[tensor]] of type (1, 1), i.e. a [[vector bundle|vector bundle endomorphism]] (called an ''[[almost complex structure]]'')
:<math> J:TM\rightarrow TM </math>, such that <math>J^2=-1. \,</math>

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

An almost complex manifold is called ''complex'' if <math>N_J=0</math>, where <math>N_J</math> is a tensor of type (2, 1) related to <math>J</math>, 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
:<math>g(JX,JY)=g(X,Y). \,</math>

An almost Hermitian structure defines naturally a [[differential form|differential two-form]]
:<math>\omega_{J,g}(X,Y):=g(JX,Y). \,</math>

The following two conditions are equivalent:

# <math> N_J=0\mbox{ and }d\omega=0 \,</math>
# <math>\nabla J=0 \,</math>

where <math>\nabla</math> is the [[Levi-Civita connection]] of <math>g</math>. In this case, <math>(J, g)</math> 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 manifold]]s) 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 manifold]]s.

===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 bundle]]s and [[Exterior derivative|de Rham differential]] of [[Differential form|forms]]. Beside [[Lie algebroid]]s, also [[Courant algebroid]]s 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 field]]s. 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 ===
{{Main article|Gauge theory (mathematics)}}

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 the [[standard model of particle physics]]. Gauge theory is concerned with the study of differential equations for connections on bundles, and the resulting geometric [[moduli space]]s 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]], and so their study is of considerable interest in physics.