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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.

37J99 Dynamical aspects of finite-dimensional Hamiltonian and Lagrangian systems
58C35 Integration on manifolds; measures on manifolds
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