## 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)].

### MSC:

 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)

### Citations:

Zbl 0141.19407; Zbl 1020.53050
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### References:

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