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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 $(M.g)$ be a Riemannian manifold and $D$ is a distribution on $M$. Assume that $D$ is invariant under the action of the isometry group of $M$. Under which conditions such $D$ …

As a related question: Is there a classification of all $1$-dimensional foliations of space tangent to the unit speed vector field $t$ for which the distributions $\{t,n\}$ and $\{t,b\}$ are integrable …

Assume that we have a $1$ dimensional foliation of $\mathbb{R}^2$. Is there a global diffeomorphism of the plane which maps all leaves of the foliation to curves with non zero curvature? One can consi …

Is there an example of a $n$ dimensional manifold $M$ and a natural number $k<n$ with a Lie subalgebra $L$ of $\chi^{\infty}(M)$ with the following property: For every $x\in M$ the space $\{V_x \ …

Is there a Riemannian metric on $M$ such that the leaves of the above foliations are totally geodesic immersed submanifolds? …

What is an example of a Lie group $G$ with two codimension one foliations $F_1 $ and $F_2$ such that they generate two different Godbilon–Vey invariants in $H^3(G)$? …

Let $M$ be a smooth n dimensional manifold. The foliation values of $M$, denoted by $F(M)$, is defined as \begin{equation} F(M)=\{ 1\leq k\leq n\mid \text{there exist an smooth $k$ dimensional fo …

In the following question, we defined the foliation values of an smooth manifold; Foliation values of a manifold Let $S_{i}$'s, $i\in \{0,1,\ldots,27\}$, be the smooth structures of topological $S^{ …

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 …

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 …

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 …

Is there a symplectic manifold $(M,\omega)$ equipped with a Riemannian metric such that $d^* \omega \wedge dd^*\omega=0$ and there is a compact leaf for the foliation of $M\setminus S$ defined by $d^ …