Second variation formula

Statement

Notation

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

Let the points of non-smoothness by $0=t_0<t_1<t_2\dots <t_k$. 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 ${\mathrm{\Delta }}_{t_i}V=V_{t_i^{+}}-V_{t_i^{-}}$.

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

$\frac{1}{2}\frac{{\partial }^{2}E(\overline{\alpha }(u^1,u^2))}{\partial u^1\partial u^2}(0)=-{\sum }_t<W_t,{\mathrm{\Delta }}_{t}DW/dt>-\underset{0}{\overset{1}{\int }}<W_2,D^2{W}_1/d{t}^2+R(W,V)V>dt$

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