Stability of Tangential Locally Conformal Symplectic Forms

Ida, Cristian

Acta Universitatis Palackianae Olomucensis. Facultas Rerum Naturalium. Mathematica, Tome 53 (2014) no. 1, pp. 81-89

Abstract (VO)

In this paper we firstly define a tangential Lichnerowicz cohomology on foliated manifolds. Next, we define tangential locally conformal symplectic forms on a foliated manifold and we formulate and prove some results concerning their stability.

Détail

Classification : 53C12, 53D99, 57R17, 58A12
Keywords: foliated manifold; tangential Lichnerowicz cohomology; tangential locally conformal symplectic structure; stability

Bibliographie

[1] Banyaga, A.: Some properties of locally conformal symplectic structures. Comment. Math. Helv. 77, 2 (2002), 383–398. | DOI | MR | Zbl

[2] Banyaga, A.: Examples of non $d_{\omega }$-exact locally conformal symplectic forms. Journal of Geometry 87, 1-2 (2007), 1–13. | DOI | MR | Zbl

[3] Bande, G., Kotschick, D.: Moser stability for locally conformally symplectic structures. Proc. Amer. Math. Soc. 137 (2009), 2419–2424. | DOI | MR | Zbl

[4] Bott, R., Tu, L. W.: Differential Forms in Algebraic Topology. Graduate Text in Math. 82 Springer-Verlag, Berlin, 1982. | MR | Zbl

[5] Datta, M., Islam, Md R.: Smooth maps of a foliated manifold in a symplectic manifold. Proc. Indian Acad. Sci. (Math. Sci.) 119, 3 (2009), 333–343. | DOI | MR | Zbl

[6] Giachetta, G., Mangiarotti, L., Sardanashvily, G.: Geometric and Algebraic Toplogical Methods in Quantum Mechanics. World Scientific, Singapore, 2005. | MR

[7] Guedira, F., Lichnerowicz, A.: Geometrie des algebres de Lie locales de Kirillov. J. Math. Pures et Appl. 63 (1984), 407–484. | MR | Zbl

[8] Haller, S., Rybicki, T.: On the Group of Diffeomorphisms Preserving a Locally Conformal Symplectic Structure. Annals of Global Analysis and Geometry 17, 5 (1999), 475–502. | DOI | MR | Zbl

[9] Hector, G., Macias, E., Saralegi, M.: Lemme de Moser feuilleté et classification des variétés de Poisson réguliéres. Publ. Mat. 33 (1989), 423–430. | DOI | MR | Zbl

[10] El Kacimi, A. A.: Sur la cohomologie feuilletée. Compositio Mathematica 49, 2 (1983), 195–215. | MR | Zbl

[11] Lee, H. C.: A kind of even-dimensional differential geometry and its application to exterior calculus. Amer. J. Math. 65 (1943), 433–438. | DOI | MR | Zbl

[12] de León, M., López, B., Marrero, J. C., Padrón, E.: On the computation of the Lichnerowicz–Jacobi cohomology. J. Geom. Phys. 44 (2003), 507–522. | DOI | MR | Zbl

[13] Lichnerowicz, A.: Les variétés de Poisson et leurs algébres de Lie associées. J. Differential Geom. 12, 2 (1977), 253–300. | MR | Zbl

[14] Molino, P.: Riemannian foliations. Progress in Mathematics 73 Birkhäuser, Boston, 1988. | MR | Zbl

[15] Moore, C. C., Schochet, C.: Global Analysis on Foliated Spaces. MSRI Publications 9 Springer-Verlag, New York, 1988. | MR | Zbl

[16] Moser, J.: On the volume elements on a manifold. Trans. Amer. Math. Soc. 120 (1965), 286–294. | DOI | MR | Zbl

[17] Mümken, B.: On Tangential Cohomology of Riemannian Foliations. American J. Math. 128, 6 (2006), 1391–1408. | DOI | MR | Zbl

[18] Novikov, S. P.: The Hamiltonian formalism and a multivalued analogue of Morse theory. Uspekhi Mat. Nauk 37 (1982), 3-49. (1982), 3–49 (in Russian). | MR

[19] Ornea, L., Verbitsky, M.: Morse–Novikov cohomology of locally conformally Kähler manifolds. J. of Geometry and Physics 59 (2009), 295–305. | DOI | MR | Zbl

[20] Rybicki, T.: On foliated, Poisson and Hamiltonian diffeomorphisms. Diff. Geom. Appl. 15 (2001), 33–46. | DOI | MR | Zbl

[21] Tondeur, Ph.: Foliations on Riemannian Manifolds. Universitext, Springer-Verlag, 1988. | MR | Zbl

[22] Vaisman, I.: Variétés riemanniene feuilletées. Czechoslovak Math. J. 21 (1971), 46–75.

[23] Vaisman, I.: Remarkable operators and commutation formulas on locally conformal Kähler manifolds. Compositio Math. 40, 3 (1980), 287–299. | MR | Zbl

[24] Vaisman, I.: Locally conformal symplectic manifolds. Internat. J. Math. and Math. Sci. 8, 3 (1985), 521–536. | MR | Zbl