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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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