Moser stability for locally conformally symplectic structures. (English) Zbl 1176.53081

A non-degenerate 2-form \(\omega\) on a manifold \(M\) is said to be locally conformally symplectic (lcs) if there exists an open covering \(\{U_i\}\) of \(M\) and smooth positive function \(f_i\) on each \(U_i\) such that \(f_i\omega|_{U_i}\) is symplectic on \(U_i\). The authors give a necessary and sufficient condition for the existence of an isotopy making a smooth family of lcs structures constant on a closed manifold. This result is the lcs analogue of a theorem of J. Moser [Trans. Am. Math. Soc. 120, 286–294 (1965; Zbl 0141.19407)] and is described in the lcs category by Lichnerowicz cohomology deformed from the de Rham cohomology by the Lee form \(\theta_t\) for the family \(\omega_t\) of lcs form on a closed manifold \(M\) depending smoothly on \(t\in[0, 1]\). As corollaries, the authors reprove some results on lcs forms of A. Banyaga [Comment. Math. Helv. 77, No. 2, 383–398 (2002; Zbl 1020.53050)].


53D35 Global theory of symplectic and contact manifolds
53D99 Symplectic geometry, contact geometry
57R17 Symplectic and contact topology in high or arbitrary dimension
53D05 Symplectic manifolds (general theory)
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