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4 votes
0 answers
246 views

Dynamical obstruction for a vector field to have a Harmonic divergence

Let $(M,g)$ be an analytic Riemannian manifold and $X$ be an analytic vector field on $M$. Can we always have a volume form $\Omega$ such that $\operatorname{Div}_{\Omega} X$ is a harmonic ...
Ali Taghavi's user avatar
5 votes
1 answer
597 views

A vector field whose flow has constant singular values

$\newcommand{\tr}{\operatorname{tr}}$ $\renewcommand{\div}{\operatorname{div}}$ Let $D \subseteq \mathbb{R}^2$ be the closed unit disk. Given a vector field $X$ on $D$, let $\psi_t$ be its flow. Does ...
Asaf Shachar's user avatar
  • 6,741
2 votes
0 answers
480 views

A Fourier elliptic vector field on a Riemannian manifold

Motivation for this question: Let $X$ be a vector field on a manifold $M$. Obviously the differential operator $D:C^{\infty}(M)\to C^{\infty}(M)$ with $D(f)=X.f$ is not an elliptic opetator when $\...
Ali Taghavi's user avatar
6 votes
2 answers
624 views

On equation $\Delta \circ \partial/\partial X=\partial/\partial X \circ \Delta$ on a Riemannian manifold

Assume that $M$ is a compact Riemannian manifold whose Laplacian is denoted by $\Delta$. Assume that the Euler characteristic of $M$ is zero. Does $M$ admit a non vanishing vector field ...
Ali Taghavi's user avatar
6 votes
0 answers
283 views

A concept weaker than geodesibility of flows which is possibly useful in limit cycle theory

The main objective of this post is to apply the Gauss Bonnet Theorem to count the number of limit cycles of a polynomial vector field as described in this MO post and its linked MO posts But in this ...
Ali Taghavi's user avatar
3 votes
1 answer
156 views

Is the space of harmonic functions invariant under the derivational operator associated with a geodesible flow?

Assume that $V$ is a vector field on a Riemannian manifold $(M,g)$ with natural volume form $\Omega$ arising from $g$. Assume that the solution curves of $V$ are parametrized geodesics of the ...
Ali Taghavi's user avatar
9 votes
1 answer
839 views

Conformal changes of metric and geodesics

Suppose $(M,g)$ is a Riemannian manifold. Let us assume that $X$ denotes a vector field in this manifold and consider the integral curves of this vector field. Does there exist a conformal factor $c$ ...
Ali's user avatar
  • 4,143