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.


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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