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Riemannian manifold

In [Riemannian geometry] , a Riemannian manifold or Riemannian space ( M , g ) is a real [differentiable manifold] M in which each [tangent space] is equipped with an [inner product] g , a Riemannian metric , in a manner which varies smoothly from point to point. The metric g is a [positive definite] [symmetric tensor] : a [metric tensor] . This allows one to define various notions such as [angle] s, lengths of [curve] s, [area] s (or [volume] s), [curvature] , [gradient] s of functions and [divergence] of [vector field] s. In other words, a Riemannian manifold is a differentiable manifold in which the [tangent space] at each point is a finite-dimensional [Euclidean space] . The terms are named after German mathematician [Bernhard Riemann] .

Overview
The [tangent bundle] of a [smooth manifold] M assigns to each fixed point of M a vector space called the [tangent space] , and each tangent space can be equipped with an inner product. If such a collection of inner products on the tangent bundle of a manifold varies smoothly as one traverses the manifold, then concepts that were defined only pointwise at each tangent space can be extended to yield analogous notions over finite regions of the manifold. For example, a smooth curve α( t ): [0, 1] → M has tangent vector α′( t 0) in the tangent space T M ( t 0) at any point t 0 ∈ (0, 1), and each such vector has length α′( t 0) , where · denotes the [norm] induced by the inner product on T M ( t 0). The [integral] of these lengths gives the length of the curve α:

:L(\alpha) = \int_0^1{\ \alpha'(t)\ \, \mathrm{d}t}.

Smoothness of α( t ) for t in [0, 1] guarantees that the integral L (α) exists and the length of this curve is defined.

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