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Let $M$ be a compact Riemannian manifold. The norm of vector fields are computed with respect to the metric. Moreover for every distribution $D$, the orthogonal projection on $D$ is den …

I would like to apply the known version of the conjectural formula (11) page 10 of the paper Number theory and dynamical Lefschetz trace formula. Disclaimer: I do not have a complete unde …

Motivation: The aim of this post is to extend the Lie algebra of a foliation to a bigger Lie algebra. We assume that a manifold $M$ is foliated by compat leaves. The Lie algebra of the foliation is th …

Let $P\to M$ be a $G$-principal bundle where $P,M$ are smooth manifolds and $G$ is a Lie group with Lie algebra $\mathfrak{g}$, whose center is denoted by $C(\mathfrak{g})$. Let $\omega$ be the con …

No. 706, Séminaire Bourbaki, Vol. 1988/89, 155–181 That is, two topologically equivalent foliations have the same Godbillon-Vey class. What are some updates on this conjecture? …

Inspired by An algebraic Hamiltonian vector field with a finite number of periodic orbits (2) we ask if there is a 1 dimensional analytic foliation of $\mathbb{R}^4$ which has at least 1 compact l …

Is there a non-vanishing vector field $X$ on $S^3$ which does not admit a transversal $2$-dimensional foliation? if the answer is negative, is there a non-vanishing vector field $X$ on $S^3$ which doe …

Can the Reeb foliation of $S^3$ be realized as foliation associated to stable(or unstable) manifolds of a hyperbolic discrete dynamic on $S^3$?If yes what is a precise formulation for that Hyp …

Inspired by this question about Jacobi equation. conjugate points and limit cycle theory we ask the following question: Is there a geodesible 1 dimensional foliation $\mathcal{F}$ on a manifold $M$, …

Updated 1/25/2023 I just added a related post below: Jacobi fields, Conjugate points and limit cycle theory EDIT: Here is a related post which concern quadratic vector fields rather than Van de …

What is an example of a non vanishing smooth vector field on a manifold $M$ whose corresponding foliation is a Riemannian foliation with respect to no Riemannian metric on $M$

Inspired by this question and the counter example provided in its answer we ask: Is there a polynomial vector field on $\mathbb{R}^2$ such that after complexification of the equation, the cor …

Inspired by this question on homothety vector fields we realize that non homotheticity is some how an intrinsic property of the foliation associated to the vector field. See the comment by Prof. Bryan …

Is there a polynomial vector field $X$ on $\mathbb{R}^2$ such that every complex limit cycle of the corresponding foliation of $\mathbb{C}^2$ must necessarily intersect the real plane $\mathbb{R}^2$. …

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 …