On characteristic foliations of metric contact-symplectic structures

Amine Hadjar

doi:10.56754/0719-0646.2803.437

Authors

Amine Hadjar Département de Mathématiques, IRIMAS, Université de Haute Alsace, 18 Rue des Frères Lumière, 68093 Mulhouse Cedex, France https://orcid.org/0009-0007-4442-7515

MSC:

53C25, 53B20, 53D10, 53B35, 53C12

DOI:

https://doi.org/10.56754/0719-0646.2803.437

Keywords:

Metric contact-symplectic structure, minimal submanifold, almost contact metric manifold, foliation

Abstract

We study compatible and associated metrics for a contact-symplectic pair \((\eta, \omega)\) on a manifold. We show that the integral curves of the Reeb vector field are geodesics for any compatible metric. We prove that all associated metrics share a common volume element, which we give explicitly. When the characteristic foliations of \(\eta\) and \(\omega\) are orthogonal with respect to an associated metric, their leaves, as well as those of the characteristic foliation of \(d\eta\), are minimal. We construct explicit examples on nilpotent Lie groups and nilmanifolds where the characteristic foliations are not both totally geodesic.

References

[1] G. Bande, P. Ghiggini, and D. Kotschick, "Stability theorems for symplectic and contact pairs," Int. Math. Res. Not., vol. 2004, no. 68, pp. 3673–3688, 2004, doi: 10.1155/S1073792804141974.

[2] G. Bande and D. Kotschick, "The geometry of symplectic pairs," Trans. Am. Math. Soc., vol. 358, no. 4, pp. 1643–1655, 2006, doi: 10.1090/S0002-9947-05-03808-0.

[3] G. Bande, "Contact-symplectic pairs," Trans. Am. Math. Soc., vol. 355, no. 4, pp. 1699–1711, 2003, doi: 10.1090/S0002-9947-02-03209-9.

[4] G. Bande and A. Hadjar, "On normal contact pairs," Int. J. Math., vol. 21, no. 6, pp. 737–754, 2010, doi: 10.1142/S0129167X10006197.

[5] D. E. Blair, Riemannian geometry of contact and symplectic manifolds, 2nd ed., ser. Prog. Math. Boston, MA: Birkhäuser, 2010, vol. 203, doi: 10.1007/978-0-8176-4959-3.

[6] S. Kobayashi and K. Nomizu, Foundations of differential geometry. I, ser. Intersci. Tracts Pure Appl. Math. Interscience Publishers, New York, NY, 1963, vol. 15.

[7] H. Rummler, "Quelques notions simples en géométrie riemannienne et leurs applications aux feuilletages compacts," Comment. Math. Helv., vol. 54, pp. 224–239, 1979, doi: 10.1007/BF02566270.