Editing Riemannian manifold (section)

== History ==

[[File:Georg_Friedrich_Bernhard_Riemann.jpeg|right|thumb|Riemannian manifolds were first conceptualized by their namesake, German mathematician [[Bernhard Riemann]].]]

In 1827, [[Carl Friedrich Gauss]] discovered that the [[Gaussian curvature]] of a surface embedded in 3-dimensional space only depends on local measurements made within the surface (the [[first fundamental form]]).{{sfn|do Carmo|1992|pp=35–36}} This result is known as the [[Theorema Egregium]] ("remarkable theorem" in Latin).

A map that preserves the local measurements of a surface is called a [[local isometry]]. A property of a surface is called an intrinsic property if it is preserved by local isometries and it is called an extrinsic property if it is not. In this language, the Theorema Egregium says that the Gaussian curvature is an intrinsic property of surfaces.

Riemannian manifolds and their curvature were first introduced non-rigorously by [[Bernhard Riemann]] in 1854.{{sfn|do Carmo|1992|p=37}} However, they would not be formalized until much later. In fact, the more primitive concept of a [[smooth manifold]] was first explicitly defined only in 1913 in a book by [[Hermann Weyl]].{{sfn|do Carmo|1992|p=37}}

[[Élie Cartan]] introduced the [[Cartan connection]], one of the first concepts of a [[Connection (vector bundle)|connection]]. [[Tullio Levi-Civita|Levi-Civita]] defined the [[Levi-Civita connection]], a special connection on a Riemannian manifold. 

[[Albert Einstein]] used the theory of [[pseudo-Riemannian manifold]]s (a generalization of Riemannian manifolds) to develop [[general relativity]]. Specifically, the [[Einstein field equations]] are constraints on the curvature of [[spacetime]], which is a 4-dimensional pseudo-Riemannian manifold.