Isometry Invariance of Riemannian Length of Admissible Curve

Theorem

Let $\struct {M, g}$ and $\struct {\tilde M, \tilde g}$ be Riemannian manifolds with or without boundary.

Let $\closedint a b$ be a closed real interval.

Let $\gamma : \closedint a b \to M$ with $t \stackrel \gamma \mapsto \map \gamma t$ be an admissible curve.

Let $\phi : M \to \tilde M$ be an isometry.

Let $\map {L_g} \gamma$ be the Riemannian length of $\gamma$ from $t = a$ to $t = b$.

Then:

$\map {L_g} {\gamma} = \map {L_{\tilde g} } {\phi \circ \gamma}$

where $\circ$ denotes the composition of mappings.

Proof

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Sources

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