Contact-pair neighborhood theorem for submanifolds in symplectic pairs

Abstract

Let M be a 2n-dimensional smooth manifold associated with the structure of symplectic pair which is a pair of closed 2-forms of constant ranks with complementary kernel foliations. Let Q⊂Mbe a codimension 2 compact submanifold. We show some sufficient and necessary conditions on the existence of the structure of contact pair (α,β) on Q,which is a pair of 1-forms of constant classes whose characteristic foliations are transverse and complementary such that α and β restrict to contact forms on the leaves of the characteristic foliations of βand α,respectively. This is a generalization of the neighborhood theorem for contact-type hypersurfaces in symplectic topology.

Keywords

Symplectic pair / neighborhood theorem / contact pair / Liouville vector field

References

[1] Arnold V I. Mathematical Methods of Classical Mechanics. Grad Texts in Math, Vol 60. Berlin: Springer, 1978

[2] Bande G, Ghiggini P, Kotschick D. Stability theorems for symplectic and contact pairs. Int Math Res Not IMRN, 2004, 68: 3673–3688

[3] Bande G, Hadjar A. Contact pairs. Tohoku Math J, 2003, 57(2): 247–260

[4] Bande G, Kotschick D. The geometry of symplectic pairs. Trans Amer Math Soc, 2006, 358: 1643–1655

[5] Bande G, Kotschick D. The geometry of recursion operators. Comm Math Phys, 2008, 280: 737–749

[6] Cartan E. Leçons sur les invariants intégraux. Paris: Hermann Press, 1922

[7] Donaldson S K. Two-forms on four-manifolds and elliptic equations. In: Griffiths P A, ed. Inspired By S. S. Chern: A Memorial Volume in Honor of A Great Mathematician. Nankai Tracts Math, Vol 11. Singapore: World Scientific, 2006, 153–172

[8] Geiges H. An Introduction to Contact Topology. Cambridge: Cambridge Univ Press, 2008

[9] Her H-L. On neighborhood theorems for symplectic pairs. J Geom, 2015, 106: 163–174

[10] Her H-L. Sum of recursion operators. Taiwanese J Math, 2017, 21: 753–766

[11] Her H-L. Almost complex structures for symplectic pairs. Topology Appl, 2018, 235: 35–42

[12] Kotschick D. On products of harmonic forms. Duke Math J, 2001, 107: 521–531

[13] Kotschick D, Morita S. Signatures of foliated surface bundles and the symplectomorphism groups of surfaces. Topology, 2005, 44: 131–149

[14] McDuff D, Salamon D. Introduction to Symplectic Topology. 2nd ed. Oxford: Clarendon Press, 1998

[15] Molino P. Riemannian Foliations. Basel: Birkhäuser Verlag, 1998

RIGHTS & PERMISSIONS

Higher Education Press and Springer-Verlag GmbH Germany, part of Springer Nature