Definition:Proper Variation of Admissible Curve

Definition

Let $M$ be a smooth manifold.

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

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

Let $\Gamma : J \times I \to M$ be an variation of admissible curve such that:

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

Suppose:

$\ds \forall s \in I : \paren{\map {\Gamma_s} a = \map \gamma a} \land \paren{\map {\Gamma_s} b = \map \gamma b}$

Then $\Gamma$ is called the proper variation of $\gamma$.

Sources

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