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BY 4.0 license Open Access Published by De Gruyter Open Access May 2, 2022

On Cosymplectic Dynamics I

Stephane Tchuiaga, Franck Houenou and Pierre Bikorimana
From the journal Complex Manifolds

Abstract

This paper is an introduction to cosymplectic topology. Through it, we study the structures of the group of cosymplectic diffeomorphisms and the group of almost cosymplectic diffeomorphisms of a cosymplectic manifold (M, ω, η) : (i)− we define and present the features of the space of almost cosymplectic vector fields (resp. cosymplectic vector fields); (ii)− we prove by a direct method that the identity component in the group of all cosymplectic diffeomorphisms is C0−closed in the group Diff (M) (a rigidity result), while in the almost cosymplectic case, we prove that the Reeb vector field determines the almost cosymplectic nature of the C0−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 ˛exibility condition), the cosymplectic analogues of the usual symplectic capacity-inequality theorem are derived and the cosymplectic analogue of a result that was proved by Hofer-Zehnder follows.

MSC 2010: 53C24; 54A20; 37C05; 37B02

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Received: 2022-01-28
Accepted: 2022-03-24
Published Online: 2022-05-02

© 2022 Stephane Tchuiaga et al., published by De Gruyter

This work is licensed under the Creative Commons Attribution 4.0 International License.

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