Variation vector field

From Diffgeom
Jump to: navigation, search


Let M be a differential manifold.

Let \alpha: (-\epsilon,\epsilon) \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\Omega:

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


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.