Definition:Minimizing Curve on Riemannian Manifold

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