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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 …

According to the interesting comment of Mohammad F Tehrani, I revise the question as follows: Assume $n>2$. For what type of compact n dimensional manifolds $M$ we can say: For every smooth em …

Let $(M,g)$ be a Riemannian manifold which admit a non vanishing vector field.(That is $\chi(M)=0$ when $M$ is a compact manifold). We pull back The symplectic structure of the cotange …

It is well known that $T\mathbb{S}^{n-1}$ is diffeomorphic to $M= f^{-1}(1)$ where $f:\mathbb{C}^n\rightarrow \mathbb{C}$ is $f(z):=\sum_{i=1}^{n} z_{i}^{2}$. Two questions: 1) Is $M$ a symple …

Let $M$ be a smooth n dimensional manifold. Is there an smooth embedding $f:M \to \mathbb{R}^{2n}$ whose image is a Lagrangian submanifold of $\mathbb{R}^{2n}$?

Assume that $G$ is a Lie group and at the same time it admits a symplectic structure. Does $G$ necessarily admit a symplectic structure such that the right multiplication preserves the symplectic …

What is a smooth two dimensional real vector bundle $E$ over $S^{2}$ such that the total space $E$, as a smooth four manifold, does not admit a symplectic structure? To what extent a …

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 $2$ …

Let $(M,\omega)$ be a symplectic manifold of dimension $2n$ with the volume form $\omega^n.$ In this question we associate a Lie algebra $L(M,\omega)$ to $(M,\omega)$. Then we are interested …

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^ …

Let $(M,g)$ be a Riemannian manifold. The $LC$ connection associated to the metric gives an $n$ dimensional distribution $D$ for $TM$. Let $\omega$ be the symplectic structure of $TM$ which is obtain …

This post is an expanded version of this MSE post. Assume that $(M, \omega)$ is a symplectic manifold which is equiped with a Riemannian metric. Is there a symplectic structure $ …

Assume that $n$ is an even number. What is a natural symplectic structure on $M_{n}(\mathbb{R})$, such that for every $1\leq k \leq n$ the manifold of $k$-rank matrices would be invariant under th …

The curl of a vector field $X=P\partial_x+Q\partial_y+R\partial_z$ is equal to $$ \mathrm{Curl}(X)= (R_y-Q_z)\,\partial_x +(P_z-R_x)\,\partial_y+ (Q_x-P_y)\,\partial_z $$ For the moment we repla …

Let $(M,\omega, g)$ be a Riemannian symplectic manifold. The Poisson bracket of two functions $f,g$ is denoted by $\{f,g\}$. The Hamiltonian vector field and gradient vector field …