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 **R**^{n} 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 **R**^{n}.

If γ : [*a*, *b*] → *M* is a continuously differentiable curve in the Riemannian manifold *M*, then we define its length *L*(γ) by

With this definition of length, every connected Riemannian manifold *M* becomes a metric space in a natural fashion: the distance *d*(*x*, *y*) between the points *x* and *y* of *M* is defined as

*d*(*x*,*y*) = inf{ L(γ) : γ is a continuously differentiable curve joining*x*and*y*}.

The Nash embedding theorem states that every Riemannian manifold *M* can be thought of as a submanifold of some Euclidean space **R**^{n}, with the notions of "length", "curvature" and "angle" on *M* coinciding with the ordinary ones in **R**^{n}.

- see also: Riemannian geometry