Linked Questions
25 questions linked to/from Finding a 1-form adapted to a smooth flow
3
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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
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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
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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
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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
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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
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161
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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
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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
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71
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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
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0
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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 ...