First variation formula

Statement

Notation

Let $M$ be a Riemannian manifold and $p,q\in M$. Let $\Omega =\Omega (M;p,q)$ be the path space of $M$, viz the space of piecewise smooth paths from $p$ to $q$.

Let the points of non-smoothness by $0=t_{0}<t_{1}<t_{2}\ldots <t_{k}$. Let $\alpha$ be a variation of $\omega$, and $W$ be its variation vector field. Let $V_{t}=d\omega /dt$ wherever $\omega$ is differentiable. For each $t_{i}$ where $\omega$ is not smooth, let $\Delta _{t_{i}}V=V_{t_{i}^{+}}-V_{t_{i}^{-}}$.

Denote by ${\overline {\alpha }}$ the map sending $u$ to the curve $t\mapsto \alpha (u,t)$.

The formula

${\frac {1}{2}}{\frac {dE({\overline {\alpha }}(u))}{du}}=-\sum _{i=1}^{k-1}<W_{t},\Delta _{t}V>-\int _{0}^{1}<W_{t},A_{t}>dt$