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Riemannian geometry

: [Elliptic geometry] * is also sometimes called "Riemannian geometry". *

** Riemannian geometry ** is the branch of [differential geometry] that studies [Riemannian manifold] s, [smooth manifolds] with a * Riemannian metric * , i.e. with an [inner product] on the [tangent space] at each point which varies [smoothly] from point to point. This gives in particular local notions of [angle] , [length of curves] , [surface area] , and [volume] . From those some other global quantities can be derived by [integrating] local contributions.

Riemannian geometry originated with the vision of [Bernhard Riemann] expressed in his inaugurational lecture ** (English: On the hypotheses on which geometry is based). It is a very broad and abstract generalization of the ** [differential geometry of surfaces] ** in ** R ** 3. Development of Riemannian geometry resulted in synthesis of diverse results concerning the geometry of surfaces and the behavior of ** [geodesic] ** s on them, with techniques that can be applied to the study of ** [differentiable manifold] ** s of higher dimensions. It enabled ** [Einstein] ** 's ** [general relativity theory] ** , made profound impact on ** [group theory] ** and ** [representation theory] ** , as well as ** [analysis] ** , and spurred the development of ** [algebraic] ** and ** [differential topology] ** . **

** Introduction **

Riemannian geometry was first put forward in generality by [Bernhard Riemann] in the nineteenth century. It deals with a broad range of geometries whose metric properties vary from point to point, as well as two standard types of [Non-Euclidean geometry] , [spherical geometry] and [hyperbolic geometry] , as well as [Euclidean geometry] itself.

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