Second variation formula: Difference between revisions

Latest revision as of 20:08, 18 May 2008

Statement

Notation

Let $M$ be a Riemannian manifold and $p, q \in M$ . Let $Ω = Ω (M; p, q)$ 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 $α$ be a 2-parameter variation of $ω$ , and $W_{1}$ and $W_{2}$ be its variation vector fields along the two parameters. Let $V_{t} = d ω / d t$ wherever $ω$ is differentiable. For each $t_{i}$ where $ω$ is not smooth, let $Δ_{t_{i}} V = V_{t_{i}^{+}} - V_{t_{i}^{-}}$ .

Denote by $\bar{α}$ the map sending $u \in U$ (here $U$ is the open set in $R^{2}$ over which the variation is defined) to the curve $t \mapsto α (u_{1}, u_{2}, t)$ .

The formula

$\frac{1}{2} \frac{\partial^{2} E (\bar{α} (u^{1}, u^{2}))}{\partial u^{1} \partial u^{2}} (0) = - \sum_{t} < W_{t}, Δ_{t} D W / d t > - \int_{0}^{1} < W_{2}, D^{2} W_{1} / d t^{2} + R (W, V) V > d t$

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

@@ Line 13: / Line 13: @@
 <math>\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</math>
+==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