Variation vector field

Definition

Let $M$ be a differential manifold.

Let $\alpha :(-ϵ,ϵ)\times [0,1]\to M$ be a piecewise smooth variation of a curve $\omega :[0,1]\to M$. The variation vector field of $\alpha$ is defined as the following element of the tangent space $T_{\omega }\mathrm{\Omega }$:

$W(t):=\frac{\partial \alpha (u,t)}{\partial u}{|}_{u=0}$

Facts

Given any element of the tangent space of a piecewise smooth path in the path space, there exists a piecewise smooth variation thereof for which it is the variation vector field. This is obtained by simply exponentiating the vector field.