Differentiable Manifolds/Riemannian manifolds

Definition (Riemannian metric):

Let $M$ be a ${\mathcal {C}}^{k}$-differentiable manifold. A Riemannian metric $g$ on $M$ is a collection of inner products

$g_{p}:T_{p}M\times T_{p}M\to \mathbb {R}$

such that the components of the matrix of $g_{p}$ are contained within ${\mathcal {C}}^{k}(M)$.

Definition (Riemannian manifold):

A Riemannian manifold is a pair $(M,g)$, where $M$ is a differentiable manifold and $g$ a Riemannian metric on $M$.