Toric actions in cosymplectic geometry
-
Giovanni Bazzoni
und
Oliver Goertsches
Abstract
We show that compact toric cosymplectic manifolds are mapping tori of equivariant symplectomorphisms of toric symplectic manifolds.
Funding statement: The first author is supported by a Juan de la Cierva – Incorporación Grant of Ministerio de Ciencia, Innovación y Universidades (Spain).
Acknowledgements
We would like to thank Eva Miranda and Álvaro Pelayo for useful discussions. The authors are also indebted to the anonymous referee for her/his help in improving the exposition of the paper and for pointing out to us relevant literature.
References
[1] C. Albert, Le théorème de réduction de Marsden–Weinstein en géométrie cosymplectique et de contact, J. Geom. Phys. 6 (1989), no. 4, 627–649. 10.1016/0393-0440(89)90029-6Suche in Google Scholar
[2] M. F. Atiyah and R. Bott, The Yang–Mills equations over Riemann surfaces, Philos. Trans. A 308 (1983), no. 1505, 523–615. 10.1098/rsta.1983.0017Suche in Google Scholar
[3] G. Bazzoni, M. Fernández and V. Muñoz, Non-formal co-symplectic manifolds, Trans. Amer. Math. Soc. 367 (2015), no. 6, 4459–4481. 10.1090/S0002-9947-2014-06361-7Suche in Google Scholar
[4] G. Bazzoni and O. Goertsches, K-cosymplectic manifolds, Ann. Global Anal. Geom. 47 (2015), no. 3, 239–270. 10.1007/s10455-014-9444-ySuche in Google Scholar
[5] G. Bazzoni and J. C. Marrero, On locally conformal symplectic manifolds of the first kind, Bull. Sci. Math. 143 (2018), 1–57. 10.1016/j.bulsci.2017.10.001Suche in Google Scholar
[6] G. Bazzoni and J. Oprea, On the structure of co-Kähler manifolds, Geom. Dedicata 170 (2014), 71–85. 10.1007/s10711-013-9869-7Suche in Google Scholar
[7] B. Cappelletti-Montano, A. De Nicola and I. Yudin, A survey on cosymplectic geometry, Rev. Math. Phys. 25 (2013), no. 10, Article ID 1343002. 10.1142/S0129055X13430022Suche in Google Scholar
[8] D. Chinea, M. de León and J. C. Marrero, Topology of cosymplectic manifolds, J. Math. Pures Appl. (9) 72 (1993), no. 6, 567–591. Suche in Google Scholar
[9] D. Conti and M. Fernández, Einstein almost cokähler manifolds, Math. Nachr. 289 (2016), no. 11–12, 1396–1407. 10.1002/mana.201400412Suche in Google Scholar
[10] T. Delzant, Hamiltoniens périodiques et images convexes de l’application moment, Bull. Soc. Math. France 116 (1988), no. 3, 315–339. 10.24033/bsmf.2100Suche in Google Scholar
[11] J. Duflot, Smooth toral actions, Topology 22 (1983), no. 3, 253–265. 10.1016/0040-9383(83)90012-5Suche in Google Scholar
[12] P. Frejlich, D. M. Torres and E. Miranda, A note on the symplectic topology of b-manifolds, J. Symplectic Geom. 15 (2017), no. 3, 719–739. 10.4310/JSG.2017.v15.n3.a4Suche in Google Scholar
[13] M. Gualtieri, S. Li, A. Pelayo and T. S. Ratiu, The tropical momentum map: A classification of toric log symplectic manifolds, Math. Ann. 367 (2017), no. 3–4, 1217–1258. 10.1007/s00208-016-1427-9Suche in Google Scholar
[14] V. Guillemin, E. Miranda and A. R. Pires, Codimension one symplectic foliations and regular Poisson structures, Bull. Braz. Math. Soc. (N. S.) 42 (2011), no. 4, 607–623. 10.1007/s00574-011-0031-6Suche in Google Scholar
[15] V. Guillemin, E. Miranda and A. R. Pires, Symplectic and Poisson geometry on b-manifolds, Adv. Math. 264 (2014), 864–896. 10.1016/j.aim.2014.07.032Suche in Google Scholar
[16] V. Guillemin, E. Miranda, A. R. Pires and G. Scott, Toric actions on b-symplectic manifolds, Int. Math. Res. Not. IMRN 2015 (2015), no. 14, 5818–5848. 10.1093/imrn/rnu108Suche in Google Scholar
[17] Z. He, Odd dimensional symplectic manifolds, PhD thesis, Massachusetts Institute of Technology, 2010, https://core.ac.uk/download/pdf/4424560.pdf. Suche in Google Scholar
[18] E. Lerman, Contact toric manifolds, J. Symplectic Geom. 1 (2003), no. 4, 785–828. 10.4310/JSG.2001.v1.n4.a6Suche in Google Scholar
[19] E. Lerman and S. Tolman, Hamiltonian torus actions on symplectic orbifolds and toric varieties, Trans. Amer. Math. Soc. 349 (1997), no. 10, 4201–4230. 10.1090/S0002-9947-97-01821-7Suche in Google Scholar
[20] H. Li, Topology of co-symplectic/co-Kähler manifolds, Asian J. Math. 12 (2008), no. 4, 527–543. 10.4310/AJM.2008.v12.n4.a7Suche in Google Scholar
[21] P. Libermann, Sur les automorphismes infinitésimaux des structures symplectiques et des structures de contact, Colloque Géométrie Différentielle Globale (Bruxelles 1958), Centre Belge Rech. Math., Louvain (1959), 37–59. Suche in Google Scholar
[22] Y. Lin and R. Sjamaar, Convexity properties of presymplectic moment maps, preprint (2017), https://arxiv.org/abs/1706.00520. 10.4310/JSG.2019.v17.n4.a6Suche in Google Scholar
[23] M. Masuda, Symmetry of a symplectic toric manifold, J. Symplectic Geom. 8 (2010), no. 4, 359–380. 10.4310/JSG.2010.v8.n4.a1Suche in Google Scholar
[24] A. L. Onishchik, Topology of Transitive Transformation Groups, Johann Ambrosius Barth, Leipzig, 1994. Suche in Google Scholar
[25] M. Pinsonnault, Maximal compact tori in the Hamiltonian group of 4-dimensional symplectic manifolds, J. Mod. Dyn. 2 (2008), no. 3, 431–455. 10.3934/jmd.2008.2.431Suche in Google Scholar
[26]
D. Tischler,
On fibering certain foliated manifolds over
© 2019 Walter de Gruyter GmbH, Berlin/Boston
Artikel in diesem Heft
- Frontmatter
- Characteristic functions of semigroups in semi-simple Lie groups
- Damping estimates for oscillatory integral operators with real-analytic phases and its applications
- Formality properties of finitely generated groups and Lie algebras
- Toric actions in cosymplectic geometry
- On classification of typical representations for GL3(F)
- Integrable generators of Lie algebras of vector fields on ℂn
- Expanding phenomena over matrix rings
- On sup-norm bounds part II: GL(2) Eisenstein series
- Sublinear elliptic problems under radiality. Harmonic NA groups and Euclidean spaces
- Partial and full boundary regularity for non-autonomous functionals with Φ-growth conditions
- Radial averaging operator acting on Bergman and Lebesgue spaces
- The 𝑝-adic variation of the Gross–Kohnen–Zagier theorem
Artikel in diesem Heft
- Frontmatter
- Characteristic functions of semigroups in semi-simple Lie groups
- Damping estimates for oscillatory integral operators with real-analytic phases and its applications
- Formality properties of finitely generated groups and Lie algebras
- Toric actions in cosymplectic geometry
- On classification of typical representations for GL3(F)
- Integrable generators of Lie algebras of vector fields on ℂn
- Expanding phenomena over matrix rings
- On sup-norm bounds part II: GL(2) Eisenstein series
- Sublinear elliptic problems under radiality. Harmonic NA groups and Euclidean spaces
- Partial and full boundary regularity for non-autonomous functionals with Φ-growth conditions
- Radial averaging operator acting on Bergman and Lebesgue spaces
- The 𝑝-adic variation of the Gross–Kohnen–Zagier theorem
