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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 …

Is there a Riemannian metric on $\mathbb{R}^{2}=\mathbb{R} \times \mathbb{R}$ which is not conformally equivalent to a product metric? More generally, assume that $M$ and $N$ are two manifolds. Wha …

If a vector field $V$ has a semi stable limit cycle then there is no any volum form $\Omega$ such that $Div_{\Omega} X=C$, a constant. Explanation: A semi stable limit cycle is an isolated perio …

To have a wide familly of counter example lets consider the Yamabe problem which says every compact manifold admite a metric of constant scalar curvature. So every manifold which does not …

Edit: We thank Vladimir Matveev for his comment on this post which leeds us to revise the question as follows: Assume that $M_{g}$ is a compact Riemann surface with constant negative cuvature (That …

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 …

What is an example of a three-dimensional smooth distribution $D$ of $\mathbb{R}^4$ with this property: Not only $D$ is not integrable but also there is no a two-dimensional foli …

Is there a $2$- dimensional foliation of $\mathbb{R}^4\setminus \{0\}$ whose tangent space is contained in $\ker \alpha$ where $\alpha$ is the following non integrable $1$-form? $$\alpha=(x^2+y^2)dx+ …

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 tang …

Let $S\subset \mathbb{R}^3$ be a smooth surface with the Gauss normal map $N:S\to S^2$. Then for every $x\in S$, the differential $(dN)_x:T_xS\to T_{N(x)}S^2$ can be considered as an endom …

What is an example of an orientable compact $2n$ dimensional manifold $M$ whose all even dimensional De Rham cohomology groups $H_{\mathrm{DeR}}^{2i}(M)$ are nonzero, but $M$ does not admit any symple …

Let $(M,g)$ be a Riemannian manifold. We equip the tangent bundle $TM$ with the Sasaki metric $g_s$. Assume that $X: M \to TM$ is a vector field on $M$. We say that $X$ is an orthonormal vector field …

Edit: According to the comment of Ycor we remove the phrase "Naturally arises from a left invariant metric' Let $G$ be a Lie group with Lie algebra $\mathfrak{g}$. We fix an orientation for $G$. We d …

Let $S$ be a surface in $\mathbb{R}^{3}$ such that every regular curve $\gamma\subset S$ has nowhere vanishing curvature, that is $\kappa(z)\neq 0$ for all $z\in \gamma$. Does this imply that …

Let $M$ be a Riemannian manifold and $\gamma$ be a non closed geodesic. Is there an isometric embedding of $M$ into some $\mathbb{R}^{n}$ which send $\gamma$ into an straight line? The second questio …