In differential geometry, a Riemannian manifold is an object which attempts to capture the intuitive notion of a "curved surface" or "curved space". A Riemannian manifold is a real differentiable manifold in which each tangent space is equipped with an inner product in a manner which varies smoothly from point to point. This allows to define of various notions familiar from multivariable calculus: the length of curves, angles, areas (or volumes), curvature, gradients of functions and divergence of vector fields.
Every open subset of Euclidean space Rn is an n-dimensional Riemannian manifold in a natural manner: as charts for the manifold structure one can use identity maps, and the inner product structure comes from the dot product given on Rn.
If γ : [a, b] → M is a continuously differentiable curve in the Riemannian manifold M, then we define its length L(γ) by
The Nash embedding theorem states that every Riemannian manifold M can be thought of as a submanifold of some Euclidean space Rn, with the notions of "length", "curvature" and "angle" on M coinciding with the ordinary ones in Rn.