Geodesic variation

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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.


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