Definition:Minimizing Curve on Riemannian Manifold
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Definition
Let $\struct {M, g}$ be a Riemannian manifold.
Let $L_g$ be the Riemannian length.
Let $S$ be the set of all admissible curves in $M$ with same endpoints $p_i, p_f \in M$.
Let $\gamma_{min} \in S$ be such that:
- $\forall \tilde \gamma \in S : \map {L_g} {\gamma_{min}} \le \map {L_g} {\tilde \gamma}$.
Then $\gamma_{min}$ is called the minimizing curve.
Sources
- 2018: John M. Lee: Introduction to Riemannian Manifolds (2nd ed.) ... (previous) ... (next): $\S 6$: Geodesics and Distance. Geodesics and Minimizing Curves