Linked Questions
25 questions linked to/from Finding a 1-form adapted to a smooth flow
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 ...
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 ...
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 ...
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 ...
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 $...
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 ...
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 ...
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 ...
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 ...
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 ...
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 $d\alpha=0,\;\alpha(X)=1$...
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 ...
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 $$\Omega^*(M,D)=\{ \alpha \in \Omega^*(M)\mid \alpha_{|...
4
votes
0
answers
486
views
Limit cycles of quadratic systems and closed geodesics(Finitness of $H(2)$)
This question is inspired by this answer to the question Finding a 1-form adapted to a smooth flow.
Assume that $V$ is a polynomial vector field of degree $2$ as follows:$$\begin{cases} x'=P(...
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)=...
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).
\...
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 $\...
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 ...
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 ...
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 ...
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 ...
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 $\...
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 ...
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 ...
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 ...