Editing Differential geometry (section)

{{short description|Branch of mathematics dealing with functions and geometric structures on differentiable manifolds}}
[[File:Hyperbolic triangle.svg|thumb|235px|right|A triangle immersed in a saddle-shape plane (a [[hyperbolic paraboloid]]), as well as two diverging [[Hyperbolic geometry#Non-intersecting / parallel lines|ultraparallel lines]]]]
{{General geometry |branches}}

'''Differential geometry''' is a [[Mathematics|mathematical]] discipline that studies the [[geometry]] of smooth shapes and smooth spaces, otherwise known as [[smooth manifold]]s. It uses the techniques of [[Calculus|single variable calculus]], [[vector calculus]], [[linear algebra]] and [[multilinear algebra]]. The field has its origins in the study of [[spherical geometry]] as far back as [[classical antiquity|antiquity]]. It also relates to [[astronomy]], the [[geodesy]] of the [[Earth]], and later  the study of [[hyperbolic geometry]] by [[Nikolai Lobachevsky|Lobachevsky]]. The simplest examples of smooth spaces are the [[Differential geometry of curves|plane and space curves]] and [[Differential geometry of surfaces|surfaces]] in the three-dimensional [[Euclidean space]], and the study of these shapes formed the basis for development of modern differential geometry during the 18th and 19th centuries.

Since the late 19th century, differential geometry has grown into a field concerned more generally with geometric structures on [[differentiable manifold]]s. A geometric structure is one which defines some notion of size, distance, shape, volume, or other rigidifying structure. For example, in [[Riemannian geometry]] distances and angles are specified, in [[symplectic geometry]] volumes may be computed, in [[conformal geometry]] only angles are specified, and in [[gauge theory (mathematics)|gauge theory]] certain [[tensor field|fields]] are given over the space. Differential geometry is closely related to, and is sometimes taken to include, [[differential topology]], which concerns itself with properties of differentiable manifolds that 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]]s, otherwise known as [[geometric analysis]].

Differential geometry finds applications throughout mathematics and the [[natural science]]s. Most prominently the language of differential geometry was used by [[Albert Einstein]] in his [[theory of general relativity]], and subsequently by [[physicists]] in the development of [[quantum field theory]] and the [[standard model of particle physics]]. Outside of physics, differential geometry finds applications in [[chemistry]], [[economics]], [[engineering]], [[control theory]], [[computer graphics]] and [[computer vision]], and recently in [[machine learning]].