Editing Bernhard Riemann (section)

==  Riemannian geometry ==
{{General geometry |geometers by name}}

Riemann's published works opened up research areas combining analysis with geometry. These would subsequently become major parts of the theories of [[Riemannian geometry]], [[algebraic geometry]], and [[complex manifold]] theory. The theory of [[Riemann surface]]s was elaborated by [[Felix Klein]] and particularly [[Adolf Hurwitz]]. This area of mathematics is part of the foundation of [[topology]] and is still being applied in novel ways to [[mathematical physics]].

In 1853, [[Carl Friedrich Gauss|Gauss]] asked Riemann, his student, to prepare a ''[[Habilitationsschrift]]'' on the foundations of geometry. Over many months, Riemann developed his theory of [[higher dimensions]] and delivered his lecture at Göttingen on 10 June 1854, entitled ''Ueber die Hypothesen, welche der Geometrie zu Grunde liegen''.<ref>[https://www.deutschestextarchiv.de/book/view/riemann_hypothesen_1867?p=7 Riemann, Bernhard: Ueber die Hypothesen, welche der Geometrie zu Grunde liegen. In: Abhandlungen der Königlichen Gesellschaft der Wissenschaften zu Göttingen 13 (1868), S. 133-150.]</ref><ref>[http://www.emis.de/classics/Riemann/WKCGeom.pdf ''On the Hypotheses which lie at the Bases of Geometry''. Bernhard Riemann. Translated by William Kingdon Clifford [Nature, Vol. VIII. Nos. 183, 184, pp. 14–17, 36, 37.<nowiki>]</nowiki>]</ref><ref>{{Cite book |last=Riemann |first=Bernhard |title=On the Hypotheses Which Lie at the Bases of Geometry |last2=Jost |first2=Jürgen |date=2016 |publisher=Springer International Publishing : Imprint: Birkhäuser |isbn=978-3-319-26042-6 |edition=1st ed. 2016 |series=Classic Texts in the Sciences |location=Cham}}</ref> It was not published until twelve years later in 1868 by Dedekind, two years after his death. Its early reception appears to have been slow, but it is now recognized as one of the most important works in geometry.

The subject founded by this work is [[Riemannian geometry]]. Riemann found the correct way to extend into ''n'' dimensions the [[differential geometry]] of surfaces, which Gauss himself proved in his ''[[theorema egregium]]''. The fundamental objects are called the [[Riemannian metric]] and the [[Riemann curvature tensor]]. For the surface (two-dimensional) case, the curvature at each point can be reduced to a number (scalar), with the surfaces of constant positive or negative curvature being models of the [[non-Euclidean geometry|non-Euclidean geometries]].

The Riemann metric is a collection of numbers at every point in space (i.e., a [[tensor]]) which allows measurements of speed in any trajectory, whose integral gives the distance between the trajectory's endpoints. For example, Riemann found that in four spatial dimensions, one needs ten numbers at each point to describe distances and curvatures on a [[manifold]], no matter how distorted it is.