arXiv subjects

P. Bikorimana

Publications and source records attributed to P. Bikorimana.

3 recordsLinked to original sources

arXiv2025

The Simplicity of the Group of Weakly Hamiltonian Diffeomorphisms on Cosymplectic Manifolds

We establish a cosymplectic counterpart of Banyaga's theorem by proving that the group of weakly Hamiltonian diffeomorphisms, $\Ham_{η,ω}(M)$, is simple on any closed cosymplectic manifold. A key structural result, derived from Lie group theory, provides the foundation for our argument: the Reeb flow on any closed cosymplectic manifold is always periodic. This property, in turn, forces the associated flux group to be discrete. Building on this discrete invariant, we develop the essential fragmentation and transitivity principles needed to prove perfectness and simplicity. Beyond this algebraic framework, we recover Li's result realizing closed cosymplectic manifolds as symplectic mapping tori, and we establish a Liouville-type integrability theorem for Hamiltonian systems invariant under the Reeb flow, producing $(n+1)$-dimensional invariant tori. Finally, we characterize the commutator subgroup of the full cosymplectomorphism group as $\Ham_{η,ω}(M)$.

math.SG↗

arXiv2020

On cosymplectic dynamics

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 present the features of the space of almost cosymplectic vector fields, this set forms a Lie group whose Lie algebra is the group of all almost 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 $Diff^\infty(M)$, 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 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 analogues of Hofer geometry and Hofer-like geometry: the group of almost co-Hamiltonian diffeomorphisms carries two bi-invariant norms, the cosymplectic analogues of the usual symplectic capacity-inequality are derived and the cosymplectic analogues of a result that was proved by Hofer-Zehnder follow.

math.DG↗

arXiv2020

On the geometry of co-Hamiltonian diffeomorphisms

This paper studies the geometry of the group of all co-Hamiltonian diffeomorphisms of a compact cosymplectic manifold $(M, ω, η)$. The fix-point theory for co-Hamiltonian diffeomorphisms is studied, and we use Arnold's conjecture to predict the exact minimum number of fix point that such a diffeomorphism must have (this minimum number is at least $1$). It follows that the generating function of any co-Hamiltonian isotopy is a constant function along it orbits. Therefore, we study the co-Hofer norms for co-Hamiltonian isotopies, and establish several co-Hamiltonian and almost co-Hamiltonian analogues of some approximations lemmas and reparameterizations lemmas found in the theory of Hamiltonian dynamics, we define two $C^0-$co-Hamiltonian topologies, and use these topologies to define the spaces of cohameomorphisms, and almost cohameomorphisms. Finally, we raise several important questions for future studies.

math.DG↗