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. In other words, a Riemannian manifold is a differentiable manifold in which the tangent space at each point is a finite-dimensional Euclidean space.
This allows one to define various geometric notions on a Riemannian manifold such as angles, lengths of curves, areas (or volumes), curvature, gradients of functions and divergence of vector fields.
Riemannian manifolds should not be confused with Riemann surfaces, manifolds that locally appear like patches of the complex plane.
The terms are named after German mathematician Bernhard Riemann.
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′(t0) in the tangent space TM(t0) at any point t0 ∈ (0, 1), and each such vector has length ‖α′(t0)‖, where ‖·‖ denotes the norm induced by the inner product on TM(t0). The integral of these lengths gives the length of the curve α:
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