Definition:Speed of Smooth Curve

Definition

Let $\struct {M, g}$ be a Riemannian manifold.

Let $I$ be a closed real interval.

Let $\gamma : \R \to M$ be a smooth curve.

Then for any $t \in I$ the speed of $\gamma$ is $\size {\map {\gamma'} t}_g$ where $\size {\, \cdot \,}_g$ is the Riemannian inner product norm.

This article, or a section of it, needs explaining. In particular: $\gamma'$ is clearly a derivative, but in the context of a Riemann manifold it probably needs to be explained -- needs a link to what it actually is You can help $\mathsf{Pr} \infty \mathsf{fWiki}$ by explaining it. To discuss this page in more detail, feel free to use the talk page. When this work has been completed, you may remove this instance of {{Explain}} from the code.

Sources

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