Let $\mathcal{M}$ be a real, compact, orientable manifold and let $X$ be a vector field on $\mathcal{M}$. Consider the functional

$$E(f) = \int_{\mathcal{M}} X_p(f)^2 dV$$

where $X_p(f)$ is the directional (Lie) derivative of $f$ along $X$ at the point $p$ and $dV$ is a volume form on $\mathcal{M}$ -- this functional essentially measures the total amount of change in $f$ along $X$ over all of $\mathcal{M}$ in the $L^2$ sense. Then $\delta E(f)$ is a differential operator whose eigenspectrum

$$\delta E(f) = \lambda f$$

(for $\lambda \in \mathbb{R}$) yields the critical points of $E$ over the set of functions with unit norm. Is there an established name for this operator (or functional) and/or its corresponding eigenspectrum?

The prototype for this operator is Dirichlet's energy

$$\int_{\mathcal{M}} ||\nabla f||^2 dV$$

which has as its (unit-norm) critical points the Laplacian eigenspectrum

$$\nabla^2 f = \lambda f,$$

the main difference being that Dirichlet's energy measures the total *gradient*, i.e., the change in all directions, rather than just the change along a particular direction at each point.

riemannianmanifold. $\endgroup$ – José Figueroa-O'Farrill Jan 26 '10 at 9:41