Length of Velocity of Smooth Curve is Constant iff Covariant Derivative of Velocity along Curve is Orthogonal to Velocity

Theorem

Let $\struct {M, g}$ be a Riemannian or pseudo-Riemannian manifold with or without boundary.

Let $\nabla$ be a metric connection on $M$.

Let $I \subseteq \R$ be an open real interval.

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

Let $\gamma'$ be the velocity of $\gamma$.

Suppose $D_t$ is the covariant derivative along $\gamma$.

Let $\size {\, \cdot \,}$ be the Riemannian or pseudo-Riemannian inner product norm.

Then $\size {\map {\gamma'} t}$ is constant if and only if $D_t \map {\gamma'} t$ is orthogonal to $\map {\gamma'} t$ for all $t \in I$.

Proof

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Sources

• 2018: John M. Lee: Introduction to Riemannian Manifolds (2nd ed.) ... (previous) ... (next): $\S 5$: The Levi-Civita Connection. Connections on Abstract Riemannian Manifolds