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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 …

Consider the following two foliations of torus: 1)The Kronecker foliation with slope $\sqrt{2}$ 2)The Kronecker foliation with slope $\pi$ As I learn from the literature, these two foliations are … But intuitively it is difficult to imagine that these two foliations are different. because in both foliations all leaves are dense! …

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 …

Is it possible to have a 2 dimensional foliation of $\mathbb{R}^{3}-\{0\}$ such that each leaf is homeomorphic to the torus? what algebraic topological obstruction exist? Another question: is there …

Is there a non integrable $2$ dimensional distribution $D$ of a $3$ dimensional Riemannian manifold such that the distribution is totally geodesic in the following sense: Every geodesic whose tang …

A $n-1$ dimensional submanifold $N\subset \mathbb{R}^n$ is called a convex submanifold if for every $x\in N$ ,ther is a neighborhood $W$ of $x$ in $N$ such that $W$ entirly lies at one side of $T_x N …

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 …

Assume that we have a codimension one foliation of $\mathbb{R}^{n}-\{0\}$ with compact leaves. Is it true to say that the foliation is stable at origin:That is: for every neighborhood $V$ of $0 …

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? …

Is there a $1$-dimensional foliation of $S^3$ which is not a geodesible foliation? Is there a $1$-dimensional foliation of $S^2\times S^1$ which is not a geodesible foliation? If the answer is affirm …

Let $X$ be a Kronecker vector field on the two dimensional torus $\mathbb{T}^2$. Let $K$ be the space of all 1- forms $\alpha$ of class $C^1$ on $\mathbb{T}^2$ which satisfy $d\alpha=0,\;\alpha(X)=1 …

Is there a non vanishing real analytic vector field $X$ on $S^3$ such that $X$ has an attractor periodic orbit(An asymptotically stable periodic orbit) ? What about the smooth case? …

This question is motivated by the finitness of limit cycles for polynomial vector fields on $\mathbb{R}^2$ Assume that $X,Y$ are two independent polynomial vector fields on $\mathbb{R}^{3}$ such tha …

I would appreciate if you consider the following two questions on $1$ dimensional foliations whose leaves are geodesic. 1)Assume that $M$ is a Riemannian manifold which is either an open manifold …

We consider the standard symplectic structure $\omega=\sum dx_i\wedge dy_i$ on $\mathbb{R}^{2n}$. To every codimension $1$ submanifold $M\subset \mathbb{R}^{2n}$ we associate a vector bundle …