Riemannian manifold

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 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

<math>L(\gamma) = \int_a^b \|\gamma'(t)\|\;dt</math>

(Note that γ'(t) is an element of the tangent space to M at the point γ(t); ||.|| denotes the norm resulting from the given inner product on that tangent space.)

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}.

Even though Riemannian manifolds can be "curved", there is still a notion of "straight line" on them: the geodesics. These are curves which locally join their points with shortest paths.

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.