On cosymplectic dynamics (original) (raw)
2019, Cornell University - arXiv
Cosymplectic geometry can be viewed as an odd dimensional counterpart of symplectic geometry. Likely in the symplectic case, a related property which is preservation of closed forms ω and η, refers to the theoretical possibility of further understanding a cosymplectic manifold (M, ω, η) from its group of diffeomorphisms. In this paper we study the structures of the group of cosymplectic diffeomorphisms and the group of almost cosymplectic diffeomorphisms of a cosymplectic manifold (M, ω, η) in threefold: first of all, we study cosymplectics counterpart of the Moser isotopy method, a proof of a cosymplectic version of Darboux theorem follows, and we define and present the features of the space of almost cosymplectic vector fields (resp. cosymplectic vector fields), this set forms a Lie group whose Lie algebra is the group of all almost cosymplectic diffeomorphisms (resp. all cosymplectic diffeomorphisms); (II) we prove by a direct method that the identity component in the group of all cosymplectic diffeomorphisms is C 0 −closed in the group Dif f ∞ (M) (a rigidity result: without appealing to similar result from symplectic geometry), while in the almost cosymplectic case, we prove that the Reeb vector field determines the almost cosymplectic nature of the C 0 −limit φ of a sequence of almost cosymplectic diffeomorphisms (a rigidity result). A sufficient condition (based on Reeb's vector field) which guarantees that φ is a cosymplectic diffeomorphism is given (a flexibility condition), and also an attempt to the study almost cosymplectic (resp. cosymplectic) counterpart of flux geometry follows: this gives rise to a group homomorphism whose kernel is path connected; and (III) we study the almost cosymplectic (resp. cosymplectic) analogues of Hofer geometry and Hofer-like geometry: the group of almost co-Hamiltonian diffeomorphisms (resp. co-Hamiltonian diffeomorphisms) carries two bi-invariant norms, the cosymplectic analogues of the usual symplectic capacity-inequality theorem are derived and the cosymplectic analogues of a result that was proved by Hofer-Zehnder follow. This includes further interesting results that will be useful for a subsequent development of the theory.
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