Riemannian manifold

Helping orphans the way you would do it

In Riemannian geometry, 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 as the length of curves, angles, areas (or volumes), curvature, gradients of functions and divergence of vector fields.

Every smooth submanifold of Rⁿ has an induced Riemannian metric: the inner product on each tangent space is the restriction of the inner product on Rⁿ. In fact, as it follows from the Nash embedding theorem, all Riemannian manifolds can be realized this way. In particular one could define Riemannian manifold as a metric space which is isometric to a smooth submanifold of Rⁿ with the induced intrinsic metric. This definition might theoretically not be flexible enough, but it is quite useful to build the first geometric intuitions in Riemannian geometry.

Usually a Riemannian manifold is defined as a smooth manifold with a smooth section of positive-definite quadratic forms on the tangent bundle. Then one has to work to show that it can be turned to a metric space:

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

(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 (and even a length 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 are usually "curved", there is still a notion of "straight line" on them: the geodesics. These are curves which locally join their points along shortest pathss.

Riemannian manifold

See also