Definition: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 and:

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

Let $T_p M$ be the tangent space at $p \in M$.

Let $V : M \to T_\gamma M$ be a piecewise smooth vector field along $\gamma$ such that:

$\map V t = \valueat{\dfrac {\partial \map \Gamma{s, t} }{\partial s}}{s \mathop = 0}$

Then $V$ is said to be the 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