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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 $E$. 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. 1 # Is there a name for this differential operator and/or its corresponding spectrum? 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\$. 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 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.