Linked Questions

13 votes
1 answer
700 views

Counterexample showing that G-invariant de Rham cohomology different from cohomology of G-invariant sub-complex?

If $G$ is a discrete or a Lie Group acting smoothly on a manifold $M$, we can define the algebra of $G$-invariant de Rham classes, $H(M)^G$, and we can also consider the cohomology of the sub-complex ...
ychemama's user avatar
  • 1,326
7 votes
2 answers
527 views

A non integrable distribution which is totally geodesic

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 ...
Ali Taghavi's user avatar
2 votes
2 answers
365 views

Computer algebra for calculating curvature when the tensor metric is very big

Is there a computer algebra method to compute the curvature of a Riemannian metric on the plane when the metric tensor has long entries $E,F,G$ The computation by hand is very ...
Ali Taghavi's user avatar
12 votes
3 answers
2k views

Limit cycles as closed geodesics (in negatively or positively curved space)

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 ...
Ali Taghavi's user avatar
4 votes
1 answer
353 views

A cubic system with two nested limit cycles with opposite orientations

What is an example of polynomial vector field $$\begin{cases} x'=P(x,y)\\ y'=Q(x,y) \end{cases}$$ such that two closed orbits $C_1,C_2$ of the system surrounds an annular region $R$ such that $...
Ali Taghavi's user avatar
6 votes
2 answers
1k views

The adjoint operators as elliptic operators

Edit: It seems that the link "https://cms.math.ca/Events/Toulouse2004/abs/ss7.html#lt" which contains a talk by Loic Teyssier about homological equations and vanishing cycles is temporally ...
Ali Taghavi's user avatar
2 votes
1 answer
179 views

Riemannian metric adapted to singular $1$-dimensional foliation

Is there a real analytic vector field $X$, locally defined around $0\in \mathbb{R}^{2n}$, with the following properties: 1) The origin is an isolated singularity for $X$ and its linear ...
Ali Taghavi's user avatar
4 votes
1 answer
710 views

A certain generalization of the Poincare Bendixson theorem

Assume that we have a $n-1$ dimensional integrable distribution $D $ on $\mathbb{R}^n \setminus \{0\}$ which generates a foliation $\mathcal{F}$. We fix an orientation for $D$.(For $n=2$ we ...
Ali Taghavi's user avatar
3 votes
1 answer
193 views

An explicit formula for a flat metric compatible to certain polynomial vector field with center

Let $X$ be the following vector field on the plane: $$\begin{cases} x'=y\\ y'=-x-x^3\end{cases}\;\;\;\;\;(X)$$ The vector field $ (X)$ has a non isochronous center at the origin.The ...
Ali Taghavi's user avatar
13 votes
0 answers
705 views

Hilbert 16th problem and dynamical Lefschetz trace formula

I would like to apply the known version of the conjectural formula (11) page !0 of the paper Number theory and dynamical Lefschetz trace formula. Disclaimer: I do not have a complete ...
Ali Taghavi's user avatar
5 votes
1 answer
201 views

The diversity of Riemannian metrics adapted to a given (1 dimensional) foliation, A Krein Millman view point

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 $d\alpha=0,\;\alpha(X)=1$...
Ali Taghavi's user avatar
7 votes
0 answers
516 views

Limit cycles as closed geodesics(2)

Hilbert 16th problem asks for a uniform upper bound $H(n)$ for the number of limit cycles of a polynomial vector field of degree $n$ on the plane. Here is an updated proof of the ...
Ali Taghavi's user avatar
8 votes
0 answers
497 views

A cohomology associated to a (not necessarily integrable) distribution (Hilbert 16th problem and dynamical Lefschetz trace formula 2)

Let $(M,D)$ be a pair consisting of a manifold $M$ and a distribution $D$ on $M$. The de Rham complex $\Omega^*(M)$ has the following subcomplex $$\Omega^*(M,D)=\{ \alpha \in \Omega^*(M)\mid \alpha_{|...
Ali Taghavi's user avatar
4 votes
0 answers
486 views

Limit cycles of quadratic systems and closed geodesics(Finitness of $H(2)$)

This question is inspired by this answer to the question Finding a 1-form adapted to a smooth flow. Assume that $V$ is a polynomial vector field of degree $2$ as follows:$$\begin{cases} x'=P(...
Ali Taghavi's user avatar
2 votes
1 answer
463 views

A curvature description for center condition for quadratic vector field

We consider the quadratic vector field $V$ $$\begin{cases} x'=P(x,y)\\ y'=Q(x,y) \end {cases}\;\;\;\;(V)$$ where $P,Q \in \mathbb{R}[x,y]$ are polynomials of degree $2$ with $P(0,0)=Q(0,0)=...
Ali Taghavi's user avatar
3 votes
1 answer
187 views

Is there a connection $\nabla$ for which this particular non geodesible vector field $X$ satisfy $\nabla_X X=0$?

Let $X$ be the following vector field on $\mathbb{R}^2\setminus \{0\}$ \begin{align} x' &= x\,(1-x^2-y^2)(x^2+y^2-3) - y\,(2-x^2-y^2)\\ y' &= y\,(1-x^2-y^2)(x^2+y^2-3) + x\,(2-x^2-y^2). \...
Ali Taghavi's user avatar
4 votes
1 answer
145 views

Is a linear vector field a geodesible vector field?

I have already asked this question in MSE; I repeat it here at MO. Assume that $A\in M_n(\mathbb{R})$ is a non-singular matrix. Is the flow of linear vector field $X'=AX$ a geodesible flow on $\...
Ali Taghavi's user avatar
5 votes
0 answers
214 views

Singular foliations of $\mathbb{C}P^2$ that are compatible to Fubini-Study metric

Is there a complete classification of quadratic polynomial vector fields on $\mathbb{C}^2$ whose corresponding singular foliation of $\mathbb{C}P^2$ satisfies the property quoted below? The regular ...
Ali Taghavi's user avatar
1 vote
1 answer
161 views

The configuration of the zero locus of certain polynomial

What is a complete description for the configuration of zero locus of the algebraic curve $C$ defined by $$yP(x,y)-xQ(x,y)=0$$ where $P,Q \in \mathbb{R}[x,y]$ are arbitrary polynomials of ...
Ali Taghavi's user avatar
3 votes
0 answers
161 views

Flat Riemannian metrics adapted to quadratic vector fields with center

Assume that $P(x,y),Q(x,y)\in \mathbb{R}[x,y]$ are two polynomials of degree $2$ with $P(0,0)=Q(0,0)=0.$ Suppose that the vector field $$\begin{cases} x'=P(x,y)\\ y'=Q(x,y) \end{cases}$$ has a center ...
Ali Taghavi's user avatar
3 votes
0 answers
161 views

Is a non vanishing holomorphic vector field necessarily a geodesible vector field?

Motivated by the "The obvious Fact" part of this answer,, we ask the following question: First we recall a definition, which is used in the above link: Definition: A non vanishing vector ...
Ali Taghavi's user avatar
4 votes
0 answers
120 views

Are all linear vector fields geodesible vector fields?

I had already asked this question in MSE then I ask here at MO. Assume that $A\in M_n(\mathbb{R})$ is a non singular matrix. Is the flow of the linear vector field $X'=AX$ a geodesible flow on $\...
Ali Taghavi's user avatar
3 votes
0 answers
131 views

Is there a non geodesible vector field $P\partial_x+Q\partial_y$ which satisfies $P_xP_y+Q_xQ_y=0$

Inspired by the following two posts Finding a 1-form adapted to a smooth flow Does $P_xP_y+Q_xQ_y=0 \implies$ "non-existence of limit cycle" for $P\partial_x+Q\partial_y$"? (Complex ...
Ali Taghavi's user avatar
3 votes
0 answers
71 views

The diversity of Riemannian metrics adapted to a given foliation، A Krein Millman view point(2)

Inspired by this answer to the linked question we add a more bounded conditions to this post. This question is asked seperately because the previous one had a complete answer so we did not revise ...
Ali Taghavi's user avatar
1 vote
0 answers
55 views

Which planar smooth foliations are not smooth equivalent to a foliation arising from level sets of a harmonic function?

Is there an smooth foliation of the plane which is not smoothly equivalent to a foliation $dH=0$ where H is a harmonic function without critical values? If the answer is negative then we conclude ...
Ali Taghavi's user avatar