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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)
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References:

[1] Augustin Banyaga, Some properties of locally conformal symplectic structures, Comment. Math. Helv. 77 (2002), no. 2, 383 – 398. · Zbl 1020.53050
[2] Augustin Banyaga, Examples of non \?_{\?}-exact locally conformal symplectic forms, J. Geom. 87 (2007), no. 1-2, 1 – 13. · Zbl 1157.53040
[3] Sorin Dragomir and Liviu Ornea, Locally conformal Kähler geometry, Progress in Mathematics, vol. 155, Birkhäuser Boston, Inc., Boston, MA, 1998. · Zbl 0887.53001
[4] Fouzia Guedira and André Lichnerowicz, Géométrie des algèbres de Lie locales de Kirillov, J. Math. Pures Appl. (9) 63 (1984), no. 4, 407 – 484 (French). · Zbl 0562.53029
[5] Stefan Haller and Tomasz Rybicki, On the group of diffeomorphisms preserving a locally conformal symplectic structure, Ann. Global Anal. Geom. 17 (1999), no. 5, 475 – 502. · Zbl 0940.53044
[6] Hwa-Chung Lee, A kind of even-dimensional differential geometry and its application to exterior calculus, Amer. J. Math. 65 (1943), 433 – 438. · Zbl 0060.38302
[7] Jürgen Moser, On the volume elements on a manifold, Trans. Amer. Math. Soc. 120 (1965), 286 – 294. · Zbl 0141.19407
[8] Izu Vaisman, Locally conformal symplectic manifolds, Internat. J. Math. Math. Sci. 8 (1985), no. 3, 521 – 536. · Zbl 0585.53030
[9] Frank W. Warner, Foundations of differentiable manifolds and Lie groups, Scott, Foresman and Co., Glenview, Ill.-London, 1971. · Zbl 0241.58001
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