Geodesic variation

Definition

Let $M$ be a Riemannian manifold.

A geodesic variation is a variation $\alpha :(-\epsilon ,\epsilon )\times [0,1]\to M$ such that for any $u$, the path $t\mapsto \alpha (u,t)$ is a geodesic.

Facts

A vector field along a geodesic is the variation vector field of a geodesic variation if and only if it is a Jacobi field.