Linked Questions

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

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