Second 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 2-parameter variation of $\omega$, and $W_{1}$ and $W_{2}$ be its variation vector fields along the two parameters. 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 $u\in U$ (here $U$ is the open set in $\mathbb {R} ^{2}$ over which the variation is defined) to the curve $t\mapsto \alpha (u_{1},u_{2},t)$.

The formula

${\frac {1}{2}}{\frac {\partial ^{2}E({\overline {\alpha }}(u^{1},u^{2}))}{\partial u^{1}\partial u^{2}}}(0)=-\sum _{t}<W_{t},\Delta _{t}DW/dt>-\int _{0}^{1}<W_{2},D^{2}W_{1}/dt^{2}+R(W,V)V>dt$

Facts

Here are two important and apparently remarkable things about the second variation formula:

• The dependence of the right side on the 2-parameter variation is only through the variation vector field
• Although the left side is symmetric, the right side does not a priori appear symmetric