Assume that $\mathcal{F}$ is a foliation of a $3$ dimensional compact Riemannian manifold $M$ which is invariant under flow of a transversal non vanishing vector field $X$. Let $\Delta_{\mathcal{F}}$ be the Laplace operator on each leaf which comes by restriction of Riemannian metric of $M$ to leaves.

Let $A=\{f\in C^{\infty}(M)\mid \Delta_{\mathcal{F}}f=X.f\}$

Is $A$ invariant under the derivation $g\mapsto X.g$? Under which condition the later is the case? What is an example of this situation, invariance of $A$, with extra condition that the codimension of the derivation operator $(g\mapsto X.g)|A$ is a finite number?

This question is some how motivated by the following two posts(An indirect motivation not a direct motivation):

Elliptic operators corresponds to non vanishing vector fields

Hilbert 16th problem and dynamical Lefschetz trace formula

  • $\begingroup$ The title recalls me " The Horrible and Terrifying Deeds and Words of the Very Renowned Pantagruel King of the Dipsodes, Son of the Great Giant Gargantua" (see en.wikipedia.org/wiki/Gargantua_and_Pantagruel). $\endgroup$ – user64494 Jun 13 at 18:29
  • $\begingroup$ @user64494 what is the reason the title of this post recalls you the title of that novel/ $\endgroup$ – Ali Taghavi Jun 13 at 21:00

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