\begin{definition}[Definition:Riemannian Metric]
Let $M$ be a smooth manifold.
Let $p \in M$ be a point in $M$.
Let $T_p M$ be the tangent space of $M$ at $p$ with the inner product $\innerprod \cdot \cdot_p$.
Let $g \in \map {\TT^2} M$ be a smooth covariant 2-tensor field such that for all $p$ its value at $p$ is equal to $\innerprod \cdot \cdot_p$:
:$\forall p \in M : g_p = \innerprod \cdot \cdot_p$
Then $g$ is known as a '''Riemannian metric''' on $M$.
\end{definition}