Riemannian manifold

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In differential geometry, a Riemannian manifold (or Riemannian space) (M, g), so called after the German mathematician Bernhard Riemann, is a real, smooth manifold M equipped with a smoothly-varying positive-definite inner product g_p on the tangent space T_pM at each point p.

The family g_p of inner products is called a Riemannian metric (or a Riemannian metric tensor, or just a metric). It is a special case of a metric tensor. Riemannian geometry is the study of Riemannian manifolds.

A Riemannian metric makes it possible to define many geometric notions, including angles, lengths of curves, areas of surfaces, higher-dimensional analogues of area (volumes, etc.), extrinsic curvature of submanifolds, and the intrinsic curvature of the manifold itself.

The requirement that g_p is smoothly-varying amounts to that for any smooth coordinate chart (U, x) on M, the n² functions

${\displaystyle g_{ij}=g\left({\frac {\partial }{\partial x^{i}}},{\frac {\partial }{\partial x^{j}}}\right):U\to \mathbb {R} }$

are smooth functions, i.e., they are infinitely differentiable. The section Riemannian manifolds with continuous metrics handles the case where the ${\displaystyle g_{ij}}$ are merely continuous.