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The Poincare compactification is a method to carry a polynomial vector field on the plane to an analytic vector field on $S^{2}$ via analytic embedding $$(x,y)\to (\frac{x}{\sqrt{1+x^{2}+y^{2}}},\fra …

Assume that $M$ is a surface in $\mathbb{R}^{3}$. We denote its shape operator by $S$. A vector field $X$ is shape related to $Y$ if $S(X)=Y$. (of course it is not an equivalent relation). Ass …

Edit: According to the comment of Prof. Bryant we revise the question as follows: Assume that $X$ is a smooth vector field on an open manifold $M$, for exmple $\mathbb{R}^2$. Is there a non degenera …

Let $(M, g)$ be a $2$ dimensional Riemannian manifold. Then we consider the Riemannian metric on TM described here. Assume that $X:M\to TM$ is a vector field. For every $p\ …

We consider the following standard partial order relation on $\mathbb{R}^n$: We say $X=(x_1,x_2,\ldots,x_n)\leq (y_1,y_2,\ldots,y_n)=Y$ iff $\sum_{i=1}^k x_i \leq \sum_{i=1}^k y_i,\quad \forall k: 1 …

What is an example of a gradient vector field $X$ on a Riemannian manifold $(M,g)$ which cannot be converted to a divergence free vector field via the following processes: First we remove the sin …

What is an example of a compact manifold which does not admit a diffeomorphism with at least one dense orbit? Moreover, is it true to say that every isometry of $\mathbb{C}P^n$ with the Fubini-Study …

Is there a non constant entire function $\gamma(t)=(x(t),y(t)): \mathbb{C} \to \mathbb{C}^{2}$ which satisfy the following Vander pol dififferential equation? $$\begin{cases}\dot{x}=y-x^{3}\\\dot y= …

Inspired by this question on homothety vector field we ask the following question Let $M$ be a manifold equiped by a volum form $\Omega$. A strongly constant divergence vector field is a vector f …

Inspired by An algebraic Hamiltonian vector field with a finite number of periodic orbits (2) we ask if there is a 1 dimensional analytic foliation of $\mathbb{R}^4$ which has at least 1 compact l …

Let $\mathbb{D}=\{z\in \mathbb{R}^2\mid |z|<1\}$ Is it true to say that every homeomorphism of $\mathbb{D}$ is conjugate to a self homeomrphism of the disk extendable to a homeomorphism of $\bar{\math …

Assume that $$\begin{cases}\dot x=P(x,y)\\\dot y=Q(x,y)\end{cases}$$ is a non vanishing holomorphic vector field on an open subset $U$ of $\mathbb{C}^{2}\simeq \mathbb{R}^{4}$. It defines a two dime …

Motivated by the following RG question we ask a related question as follows: We identify $\mathbb{R}^{n} \otimes \mathbb{R}^{m}$ with $\mathbb{R}^{mn}$. We define $GL(\mathbb{R}^{n})\otimes GL(\math …

Let $M$ be a smooth Riemannian manifold. The Riemannian metric enables us to equip the tangent bundle $TM$ with a symplectic structure $\omega$, which is the pullback of the standard symplectic $2$ …

Inspired by the Lie algebra discussed in this answer, we consider the following Lie agebra associated to a given foliation: Let $\mathcal{F}$ be a nontrivial foliation of a manifol …