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 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 Riemannian metric tensor). Riemannian geometry is the study of Riemannian manifolds.

A common convention is to take g to be smooth, which means that for any smooth coordinate chart (U, x) on M, the n² functions

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

are smooth functions. These functions are commonly designated as ${\displaystyle g_{ij}}$.

With further restrictions on the ${\displaystyle g_{ij}}$, one could also consider Lipschitz Riemannian metrics or measurable Riemannian metrics, among many other possibilities.

A Riemannian metric (tensor) makes it possible to define several geometric notions on a Riemannian manifold, such as angle at an intersection, length of a curve, area of a surface and higher-dimensional analogues (volume, etc.), extrinsic curvature of submanifolds, and intrinsic curvature of the manifold itself.