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6 votes
0 answers
267 views

Elliptic foliations of the plane

A $1$ dimensional foliation of the plane $\mathbb{R}^2$is called elliptic if it admits a non vanishing smooth tangent vector field $X$ with the following properties: The differential operator ...
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
4 votes
0 answers
148 views

Exponential map for tangent space of space of distributions $\mathscr{P}_2(X)$

In Chapter 8 of the book Gradient Flows In Metric Spaces and in the Space of Probability Measures by Ambrosio et al., the tangent space to the space of distributions on $X$ (let's say $X=\mathbb{R}^d$)...
Juno Kim's user avatar
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
4 votes
1 answer
847 views

Does $P_xP_y+Q_xQ_y=0 \implies$ "non-existence of limit cycle" for $P\partial_x+Q\partial_y$"? (Complex dilatation and limit cycle theory)

Let $X=P\partial_x+Q\partial_y$ be a vector field on the plane $\mathbb{R}^2$. Assume that we have :$$P_xP_y+Q_xQ_y=0$$ Does this imply that the vector field $X$ is a divergence-free vector field ...
Ali Taghavi's user avatar
4 votes
0 answers
111 views

Integrability of Continuous Tangent Subbundles

Are there any field of mathematics, except dynamical systems, where one needs to integrate continuous sub-bundles of the tangent space? More specifically given a smooth manifold of $M$ and a ...
Avicenna's user avatar
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2 votes
0 answers
479 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
1 vote
0 answers
240 views

A Lie algebra associated with a one dimensional foliation

A non vanishing vector field $X$ on a manifold is called "well behaved" if for every non vanishing smooth function $f$ we have $$C(X)\simeq C(fX)$$ This means that the centralizer Lie algebras $C(...
Ali Taghavi's user avatar