## Some properties of locally conformal symplectic structures.(English)Zbl 1020.53050

A 2-form $$\Omega$$ on a manifold $$M$$ is called a locally conformal symplectic structure if there exist an open cover $$M=\cup U_i$$ and positive functions $$\lambda_i$$ on $$U_i$$ such that $$\Omega_i=\lambda_i\Omega$$ are symplectic forms on $$U_i$$. Then $$d\Omega=-\omega\wedge \Omega$$ for a certain closed 1-form $$\omega$$ (the Lee form) and $$d_\omega$$-cohomologies may be introduced by using the coboundary operator $$d_\omega \gamma=d\gamma+ \omega\wedge \gamma$$. If $$\chi_\Omega(M)$$ is the Lie algebra of all vector fields $$X$$ on $$M$$ such that $$L_X\Omega=0$$, then $$\Omega$$ is said to be of the first kind if $$\omega(X)\neq 0$$ for appropriate $$X\in \chi_\Omega(M)$$, otherwise, $$\Omega$$ is of the second kind. The author very clearly recalls all such fundamental concepts (and many others). Then the article is devoted to a large spectrum of interesting topics: comparison of $$d_\omega$$-cohomologies and the de Rham cohomologies, a Moser theorem concerning the equivalence and deformation of structures, connections between first kind, second kind, essential and inessential, local and global conformal structures, conformal invariants. The Gelfand-Fuks cohomologies and the Hodge-de Rham theory are employed and several open problems are stated.

### MSC:

 53D05 Symplectic manifolds (general theory) 53D10 Contact manifolds (general theory) 53C15 General geometric structures on manifolds (almost complex, almost product structures, etc.) 53C12 Foliations (differential geometric aspects)
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