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Let $f: G= \mbox{GL}(n,\mathbb{R}) \to \mathbb{R}$ be the determinant function. Then $\mbox{Hess} (f)$ is a two linear map on $M_{n}(\mathbb{R})\simeq T_{e}(G)$ where $e$ is the neutral element of $G$ …

Assume that $(M,g)$ is a Riemannian manifold. Is there any relation between the sequence of eigenvalues of Laplace operator acting on the space of smooth functions and the sequence of eigenvalues of …

Title edited I thank მამუკა ჯიბლაძე and Corbennick for their suggestion on the title of this question. I changed the title based on the suggestion of Corbennick. What is an example of a m …

EDIT: According to some comments on this post I revise the title to remove the misunderestanding. Assume that $M$ is a Riemannian manifold of dimension $n$. The natural Laplace operator associated t …

The principal curvatures of a surface is denoted by $\kappa_{1}, \kappa_{2}$. Let $P(x,y)$ be a polynomial with real coefficients. Assume that $P(\kappa_{1}, \kappa_{2})$ is an intrinsically inv …

If the vector field is geodesible then such $1$ - form exists. A geodesible flow on a manifold $M$ is a one dimensional foliation associated with a non vanishing vector field such that …

In this question, we are interested in the number of limit cycles which appears in the following perturbational system: \begin{equation}\cases{ x'=y -x^{2}+\epsilon P(x,y) \\ y'=-x+\epsilon Q(x,y) } …

I would like to apply the known version of the conjectural formula (11) page 10 of the paper Number theory and dynamical Lefschetz trace formula. Disclaimer: I do not have a complete unde …

Motivated by a geometric proof of the Fundamental Theorem of Algebra we ask: Is there a geometric proof for the Gauss-Lucas theorem? Since we are working on a half plane, can one imagine a possible p …

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 de …

What is an example of a nontrivial principal bundle whose fibre space $G$, total space $P$ and base space $M$ are compact connected manifolds (the fiber $G$ is a compact Lie group) such that $$H^*(P,\ …

Is there a simple smooth closed curve $\gamma$ in $\mathbb{R}^{3}$ such that for all $x,y\in \gamma$ with $x \neq y $, $l_{x}$ and $l_{y}$ are skew lines, where $l_{x} $ and $l_{y}$ are s …

Consider the vector field $$X=(y-10x)\partial_x-x\partial_y$$ it is not a gradient vector field with respect to the standard Riemannian metric of $\mathbb{R}^2$ but it is a gradient …

For a connected $n$-manifold $M$, the Lie algebra of all smooth vector fields is denoted by $\chi^{\infty}(M)$. For a point $p\in M$ we define $L_{p}=\{X\in \chi^{\infty}(M)\mid X(p)=0\}$. Of course …

According to Jairo comment on the first version of this question I revise the question as follows; Let $g$ be a real analytic Riemannan metric on $S^{2}$. Is it true to say that: There are at most a …