Definition:Riemannian Length of Admissible Curve
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Definition
Let $\struct {M, g}$ be a Riemannian manifold.
Let $\closedint a b$ be a closed real interval.
Let $\gamma : \closedint a b \to M$ be an admissible curve.
Then the (Riemannian) length of $\gamma$, denoted by $\map {L_g} \gamma$, is defined by:
- $\ds \map {L_g} \gamma := \int_a^b \size {\map \gamma t}_g \rd t$
where $\size {\map \gamma t}_g$ is the Riemannian inner product norm.
Also see
Sources
- 2018: John M. Lee: Introduction to Riemannian Manifolds (2nd ed.) ... (previous) ... (next): $\S 2$: Riemannian Metrics. Lengths and Distances