Definition:Critical Point of Riemannian Length

Definition

Let $\struct {M, g}$ be a Riemannian manifold.

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

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

Let $\gamma : I \to M$ be an admissible curve.

Let $L_g$ be the Riemannian length of some admissible curve.

Let $\Gamma : J \times I \to M$ be the proper variation of $\gamma$ such that:

$\forall s \in J, \forall t \in I : \tuple {s, t} \stackrel {\Gamma}{\mapsto} \map {\Gamma_s} t$

Suppose:

$\forall \Gamma : \ds \dfrac d {d s} \map {L_g} {\Gamma_s} = 0$

Then $\gamma$ is called the critical point of $L_g$.

Sources

• 2018: John M. Lee: Introduction to Riemannian Manifolds (2nd ed.) ... (previous) ... (next): $\S 6$: Geodesics and Distance. Minimizing Curves Are Geodesics