\begin{definition}[Definition:Riemannian Inner Product Norm]
Let $\struct {M, g}$ be a Riemannian manifold.
Let $p \in M$ be a point.
Let $T_p M$ be the tangent space of $M$ at $p$.
Let $v \in T_p M$ be a vector.
Then the '''Riemannian inner product norm of $v$''' is:
:$\ds \size {v}_g := \sqrt {g_p \innerprod v v}$
where $g_p$ is the Riemannian metric at $p$.
\end{definition}