Definition:Variation of Admissible Curve
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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 $J \subseteq \R$ be an open real interval containing $0$.
Let $\Gamma : J \times I \to M$ be an admissible family of curves such that:
- $\ds \forall s \in J : \forall t \in I : \tuple {s, t} \stackrel \Gamma \mapsto \map {\Gamma_s} t$
Let $\Gamma$ have the property that:
- $\map {\Gamma_0} t = \map \gamma t$
Then $\Gamma$ is called the 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