Definition:Proper Variation Field of Family of Curves

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