Differentiable Manifolds/Riemannian manifolds

Definition (Riemannian metric):

Let be a -differentiable manifold. A Riemannian metric on is a collection of inner products

such that the components of the matrix of are contained within .

Definition (Riemannian manifold):

A Riemannian manifold is a pair , where is a differentiable manifold and a Riemannian metric on .

Path length on Riemannian differentiable manifolds