# Linked Questions

20 questions linked to/from Finding a 1-form adapted to a smooth flow

**11**

votes

**1**answer

506 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 ...

**7**

votes

**2**answers

414 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 ...

**2**

votes

**2**answers

253 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 ...

**10**

votes

**3**answers

2k views

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

EDIT: Here is a related post which concern quadratic vector fields rather than Van der Pol equation. In this linked post we see that the convexity of limit cycle play a crucial role. On ...

**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 ...

**3**

votes

**1**answer

278 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 $...

**4**

votes

**1**answer

592 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 ...

**2**

votes

**1**answer

163 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 ...

**3**

votes

**1**answer

174 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 ...

**13**

votes

**0**answers

606 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 ...

**7**

votes

**0**answers

483 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 ...

**8**

votes

**0**answers

452 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_{|...

**4**

votes

**0**answers

401 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(...

**2**

votes

**1**answer

419 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)=...

**3**

votes

**1**answer

175 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).
\...