Geodesic variation

From Diffgeom
Jump to: navigation, search

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.