# 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 .