Definition:Riemannian Inner Product Norm

Definition

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 $p$.

Sources

• 2018: John M. Lee: Introduction to Riemannian Manifolds (2nd ed.): $\S 2$: Riemannian Metrics. Definitions