Abstract
In this paper we study K-cosymplectic manifolds, i.e., smooth cosymplectic manifolds for which the Reeb field is Killing with respect to some Riemannian metric. These structures generalize coKähler structures, in the same way as K-contact structures generalize Sasakian structures. In analogy to the contact case, we distinguish between (quasi-)regular and irregular structures; in the regular case, the K-cosymplectic manifold turns out to be a flat circle bundle over an almost Kähler manifold. We investigate de Rham and basic cohomology of K-cosymplectic manifolds, as well as cosymplectic and Hamiltonian vector fields and group actions on such manifolds. The deformations of type I and II in the contact setting have natural analogues for cosymplectic manifolds; those of type I can be used to show that compact K-cosymplectic manifolds always carry quasi-regular structures. We consider Hamiltonian group actions and use the momentum map to study the equivariant cohomology of the canonical torus action on a compact K-cosymplectic manifold, resulting in relations between the basic cohomology of the characteristic foliation and the number of closed Reeb orbits on an irregular K-cosymplectic manifold.
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4096, to be precise.
References
Albert, C.: Le théorème de réduction de Marsden-Weinstein en géométrie cosymplectique et de contact. J. Geom. Phys. 6(4), 627–649 (1989)
Audin, M.: Torus Actions on Symplectic Manifolds, volume 93 of Progress in Mathematics. Birkhäuser Verlag, Basel (2004). (Revised edition)
Bazzoni, G., Fernández, M., Muñoz, V.: Non-formal co-symplectic manifolds. Trans. Am. Math. Soc. (2013). doi:10.1090/S0002-9947-2014-06361-7
Bazzoni, G., Lupton, G., Oprea, J.: Hereditary properties of co-Kähler manifolds (2013). arXiv:1311.5675
Bazzoni, G., Oprea, J.: On the structure of co-Kähler manifolds. Geom. Dedicata 170, 71–85 (2014)
Benson, C., Gordon, C.S.: Kähler and symplectic structures on nilmanifolds. Topology 27(4), 513–518 (1988)
Blair, D.E.: The theory of quasi-Sasakian structures. J. Differ. Geom. 1, 331–345 (1967)
Blair, D.E.: Riemannian Geometry of Contact and Symplectic Manifolds, volume 203 of Progress in Mathematics. Birkhäuser Boston Inc., Boston (2002)
Boyer, C.P., Galicki, K.: Sasakian Geometry. Oxford Mathematical Monographs. Oxford University Press, Oxford (2008)
Cappelletti-Montano, B., de Nicola, A., Yudin, I.: A survey on cosymplectic geometry. Rev. Math. Phys. 25(10), 1343002 (2013)
Cavalcanti, G.: Examples and counter-examples of log-symplectic manifolds (2013). arXiv:1303.6420
Chinea, D., de León, M., Marrero, J.C.: Topology of cosymplectic manifolds. J. Math. Pures Appl. (9) 72(6), 567–591 (1993)
Chinea, D., González, C.: A classification of almost contact metric manifolds. Ann. Mat. Pura Appl. 4(156), 15–36 (1990)
Chinea, D., Marrero, J.C.: Classification of almost contact metric structures. Rev. Roumaine Math. Pures Appl. 37(3), 199–211 (1992)
El Kacimi-Alaoui, A., Sergiescu, V., Hector, G.: La cohomologie basique d’un feuilletage riemannien est de dimension finie. Math. Z. 188(4), 593–599 (1985)
Fernández, M., Muñoz, V.: An 8-dimensional nonformal, simply connected, symplectic manifold. Ann. Math. (2) 167(3), 1045–1054 (2008)
Fino, A., Vezzoni, L.: Some results on cosymplectic manifolds. Geom. Dedicata 151, 41–58 (2011)
Frejlich, P., Martínez Torres, D., Miranda, E.: Symplectic topology of \(b\)-symplectic manifolds (2013). arXiv:1312.7329
Goertsches, O., Nozawa, H., Töben, D.: Equivariant cohomology of \(K\)-contact manifolds. Math. Ann. 354(4), 1555–1582 (2012)
Goldberg, S.I., Yano, K.: Integrability of almost cosymplectic structures. Pacif. J. Math. 31, 373–382 (1969)
Gompf, R.E.: A new construction of symplectic manifolds. Ann. Math. (2) 142(3), 527–595 (1995)
Gray, A., Hervella, L.M.: The sixteen classes of almost Hermitian manifolds and their linear invariants. Ann. Mat. Pura Appl. 4(123), 35–58 (1980)
Greub, W., Halperin, S., Vanstone, R.: Connections, Curvature, and Cohomology. Vol. II: Lie Groups, Principal Bundles, and Characteristic Classes. Pure and Applied Mathematics, vol. 47-II. Academic Press [A subsidiary of Harcourt Brace Jovanovich, Publishers], New York, London (1973)
Greub, W., Halperin, S., Vanstone, R.: Connections, Curvature, and Cohomology. Volume III: Cohomology of Principal Bundles and Homogeneous Spaces, Pure and Applied Mathematics, vol. 47-III. Academic Press [Harcourt Brace Jovanovich, Publishers], New York, London (1976)
Guillemin, V., Ginzburg, V., Karshon, Y.: Moment Maps, Cobordisms, and Hamiltonian Group Actions, Volume 98 of Mathematical Surveys and Monographs. American Mathematical Society, Providence (2002) (Appendix J by Maxim Braverman)
Guillemin, V., Miranda, E., Pires, A.R.: Codimension one symplectic foliations and regular Poisson structures. Bull. Braz. Math. Soc. (N.S.) 42(4), 607–623 (2011)
Guillemin, V., Miranda, E., Pires, A.R.: Symplectic and Poisson geometry on \(b\)-manifolds (2012). arXiv:1206.2020
Guillemin, V., Miranda, E., Pires, A.R., Scott, G.: Toric actions on \(b\)-symplectic manifolds (2013). arXiv:1309.1897v3
Guillemin, V., Sternberg, S.: Symplectic Techniques in Physics, 2nd edn. Cambridge University Press, Cambridge (1990)
Husemoller, D.: Fibre Bundles. McGraw-Hill Book Co., New York, London, Sydney (1966)
Kirwan, F.C.: Cohomology of Quotients in Symplectic and Algebraic Geometry, Volume 31 of Mathematical Notes. Princeton University Press, Princeton (1984)
Kobayashi, S., Nomizu, K.: Foundations of Differential Geometry, vol. II. Wiley Classics Library. Wiley, New York (1996) (reprint of the 1969 original, A Wiley-Interscience Publication)
Li, H.: Topology of co-symplectic/co-Kähler manifolds. Asian J. Math. 12(4), 527–543 (2008)
Libermann, P.: Sur les automorphismes infinitésimaux des structures symplectiques et des structures de contact. In: Colloque Géom. Diff. Globale (Bruxelles, 1958), pp. 37–59. Centre Belge Rech. Math., Louvain (1959)
Martínez Torres, D.: Codimension-one foliations calibrated by nondegenerate closed 2-forms. Pacif. J. Math. 261(1), 165–217 (2013)
Myers, S.B., Steenrod, N.E.: The group of isometries of a Riemannian manifold. Ann. Math.(2) 40(2), 400–416 (1939)
Onishchik, A.L.: Topology of Transitive Transformation Groups. Johann Ambrosius Barth Verlag GmbH, Leipzig (1994)
Oprea, J., Tralle, A.: Symplectic Manifolds with No Kähler Structure. Lecture Notes in Mathematics, vol. 1661. Springer, Berlin (1997)
Rukimbira, P.: Vertical sectional curvature and \(K\)-contactness. J. Geom. 53(1–2), 163–166 (1995)
Rukimbira, P.: On \(K\)-contact manifolds with minimal number of closed characteristics. Proc. Am. Math. Soc. 127(11), 3345–3351 (1999)
Steenrod, N.: The Topology of Fibre Bundles, vol. 14. Princeton Mathematical Series, Princeton University Press, Princeton (1951)
Takahashi, T.: Deformations of Sasakian structures and its application to the Brieskorn manifolds. Tôhoku Math. J. (2) 30(1), 37–43 (1978)
Tanno, S.: The topology of contact Riemannian manifolds. Ill. J. Math. 12, 700–717 (1968)
Tischler, D.: On fibering certain foliated manifolds over \(S^{1}\). Topology 9, 153–154 (1970)
Yamazaki, T.: A construction of \(K\)-contact manifolds by a fiber join. Tohoku Math. J. (2) 51(4), 433–446 (1999)
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Bazzoni, G., Goertsches, O. K-Cosymplectic manifolds. Ann Glob Anal Geom 47, 239–270 (2015). https://doi.org/10.1007/s10455-014-9444-y
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DOI: https://doi.org/10.1007/s10455-014-9444-y