Definition:Proper Variation Field of Family of Curves
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Definition
Let $M$ be a smooth manifold.
Let $\gamma$ be an admissible curve on $M$.
Let $I = \closedint a b$ is a closed real interval.
Let $J$ is an open real interval.
Let $\Gamma : J \times I \to M$ be the variation of $\gamma$, where $\times$ denotes the cartesian product.
Let $V$ be the variation field of $\Gamma$.
Suppose $\map V a = \map V b = 0$.
Then $V$ is said to be the proper variation field of $\Gamma$.
Sources
- 2018: John M. Lee: Introduction to Riemannian Manifolds (2nd ed.) ... (previous) ... (next): $\S 6$: Geodesics and Distance. Geodesics and Minimizing Curves