# First variation formula

From Diffgeom

## Statement

### Notation

Let be a Riemannian manifold and . Let be the path space of , viz the space of piecewise smooth paths from to .

Let the points of non-smoothness by . Let be a variation of , and be its variation vector field. Let wherever is differentiable. For each where is not smooth, let .

Denote by the map sending to the curve .

### The formula