×

zbMATH — the first resource for mathematics

Applications of convex integration to symplectic and contact geometry. (English) Zbl 0572.58010
Gromov’s method of convex integration is a powerful tool which permits the construction of closed differential forms on closed manifolds which satisfy appropriately formulated conditions. For example, one can use it to construct closed 2-forms \(\omega\) on \((2m+1)\)-dimensional manifolds \(M^{2m+1}\) such that \(\omega^ m\neq 0\), but not to construct symplectic forms on \(M^{2m}\). In this paper, this method is described, and then is applied to the study of certain problems related to the existence and uniqueness of symplectic and contact structures.
In particular we show that if two symplectic forms on \(M^{2m}\) have the same homotopy theoretic invariants (i.e. they are cohomologous and give rise to homotopic reductions of the structural group of TM to U(m)) then they are concordant, that is there is a 1-dimensional foliation on \(M\times I\) with a transverse symplectic structure which restricts on the two ends to the two given forms. This result enables one to calculate \(\pi_{2m}(B\Gamma^{2m}_{sp})\), where \(\Gamma^{2m}_{sp}\) is Haeflinger’s classifying space for transversally symplectic foliations. Analogous results are proved in the contact case.

MSC:
37J99 Dynamical aspects of finite-dimensional Hamiltonian and Lagrangian systems
58C35 Integration on manifolds; measures on manifolds
PDF BibTeX XML Cite
Full Text: DOI Numdam EuDML
References:
[1] D. BENNEQUIN, Entrelacements et équations de Pfaff, Astérisque, 107-108 (1983), 87-161. · Zbl 0573.58022
[2] E. CARTAN, LES systèmes différentiels extérieurs et leurs applications géométriques, Herman, Paris (1945). · Zbl 0063.00734
[3] J. ELIASHBERG, Rigidity of symplectic and contact structures, Preprint, 1981.
[4] R. GREENE and K. SHIOHAMA, Diffeomorphisms and volume-preserving embeddings of non-compact manifolds, TAMS, 225 (1979), 403-414. · Zbl 0418.58002
[5] M. GROMOV, Stable mappings of foliations into manifolds, Izv. Akad. Nauk SSR, Mat, 33 (1969), 707-734. · Zbl 0197.20404
[6] M. GROMOV, Convex integration of differential relations I, Math USSR Izv., 7 (1973), 329-343. · Zbl 0281.58004
[7] M. GROMOV, Partial differential relations, Springer Verlag, to appear. · Zbl 0651.53001
[8] M. GROMOV, Pseudo-holomorphic curves in symplectic manifolds, Invent. Math., 82 (1985), 307-347. · Zbl 0592.53025
[9] A. HAEFLIGER, Feuilletages sur LES variétés ouvertes, Topology, 9 (1970), 183-194. · Zbl 0196.26901
[10] A. HAEFLIGER, Homotopy and integrability, Manifolds-Amsterdam, Springer Lecture Notes # 197 (1971) 133-163. · Zbl 0215.52403
[11] A. HAEFLIGER, Lectures on the theorem of Gromov, Springer Lecture Notes # 209, (1971), 128-141. · Zbl 0222.57020
[12] J. MARTINET, Formes de contact sur LES variétés de dimension 3, Springer Lecture Notes # 209, (1971), 128-163. · Zbl 0215.23003
[13] D. MCDUFF, On groups of volume-preserving diffeomorphisms and foliations with transverse volume form, Proc. London Math. Soc., 43 (1981), 295-320. · Zbl 0411.57028
[14] D. MCDUFF, Local homology of groups of volume-preserving diffeomorphisms, I, Ann. Sci. Ec. Norm. Sup., 15 (1982), 609-648. · Zbl 0577.58005
[15] D. MCDUFF, Some canonical cohomology classes on groups of volume-preserving diffeomorphisms, TAMS, 275 (1983), 345-356. · Zbl 0522.57029
[16] D. MCDUFF, Examples of simply connected symplectic non-Kählerian manifolds, J. Diff. Geom., 20 (1984), 27. · Zbl 0567.53031
[17] D. MCDUFF, Examples of symplectic structures, to appear in Invent. Math. · Zbl 0625.53040
[18] C. MECKERT, Formes de contact sur la somme connexe de deux variétés de contact, Ann. Inst. Fourier, Grenoble, 32-3 (1982), 251-260. · Zbl 0471.58001
[19] J. MOSER, On the volume elements on a manifolds, TAMS, 120 (1965), 286-294. · Zbl 0141.19407
[20] D. SPRING, Convex integration of non-linear system of partial differential equations, Ann. Inst. Fourier, Grenoble, 33-3 (1983) 121-177. · Zbl 0507.35019
[21] D. SULLIVAN, Cycles for the dynamical study of foliated manifolds and complex manifolds, Invent. Math., 36 (1976), 225-255. · Zbl 0335.57015
[22] C. THOMAS, A classifying space for the contact pseudogroup, Mathematika, 25 (1978), 191-201. · Zbl 0404.53031
[23] W. THURSTON, Some simple examples of symplectic manifolds, Proc. AMS, 55 (1976), 467-468. · Zbl 0324.53031
[24] W. THURSTON and H. WINKELNKEMPER, On the existence of contact forms, Proc. AMS, 52 (1975), 345-347. · Zbl 0312.53028
[25] C. VITERBO, A proof of Weinstein’s conjecture in R2n, preprint, 1986. · Zbl 0631.58013
[26] A. WEINSTEIN, On the hypotheses of Rabinowitz’s periodic orbit theorems, J. Diff. Eq., 33 (1979), 353-358. · Zbl 0388.58020
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.