Riemannian manifold

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In differential geometry, a Riemannian manifold (or Riemannian space) ${\displaystyle (M,g)}$, so called after the German mathematician Bernhard Riemann, is a real, smooth manifold ${\displaystyle M}$ equipped with a smoothly-varying family ${\displaystyle g}$ of positive-definite inner products ${\displaystyle g_{p}}$ on the tangent spaces ${\displaystyle T_{p}M}$ at each point ${\displaystyle p}$.

The family ${\displaystyle g}$ 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 ${\displaystyle g}$ is smoothly-varying amounts to that for any smooth coordinate chart ${\displaystyle (U,x)}$ on ${\displaystyle M}$, the 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.