arXiv subjects

S. Tchuiaga

Publications and source records attributed to S. Tchuiaga.

7 recordsLinked to original sources

arXiv2026

Geometric properties and flux of locally conformally symplectic diffeomorphisms

We investigate the geometric and topological properties of the group of locally conformally symplectic (LCS) diffeomorphisms, utilizing the LCS flux homomorphism defined by S. Haller. By analyzing the flux map from the universal cover of the identity component $(\ker \Phi)_0$ to the first Lichnerowicz cohomology group $H_\omega^1(M)$, we establish a short exact sequence characterizing the Hamiltonian subgroup $\Ham_\Omega(M)$ and provide conditions for its topological splitting as a semidirect product. We develop LCS analogues of fundamental symplectic results, including a Weinstein neighborhood theorem, a flux rigidity theorem for homotopies, and a characterization of LCS structures on mapping tori. A central theme of this work is the influence of the Hodge decomposition of the Lee form $\omega = dh + l$. In the exact case ($l=0$), we utilize the global conformal equivalence to symplectic structures to establish energy-capacity inequalities, an LCS Hofer metric, and non-displaceability results. We explicitly analyze the relationship between the LCS Calabi invariant and its symplectic counterpart, showing they are controlled by a multiplicative factor depending on the conformal weight. For the general non-exact case ($l \neq 0$), we introduce a Twisted Calabi invariant that captures the interaction between Hamiltonian dynamics and the harmonic component of the Lee form.

math.SG↗

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, $\Ham_{\eta,\omega}(M)$, 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 $\Ham_{\eta,\omega}(M)$.

math.SG↗

arXiv2025

Arithmetic Aspects of Weil Bundles over $p$-Adic Manifolds

We introduce a systematic theory of Weil bundles over $ p $-adic analytic manifolds, forging new connections between differential calculus over non-archimedean fields and arithmetic geometry. By developing a framework for infinitesimal structures in the $ p $-adic setting, we establish that Weil bundles $ M^A $ associated with a $ p $-adic manifold $ M $ and a Weil algebra $ A $ inherit a canonical analytic structure. Key results include: \text{Lifting theorems :} for analytic functions, vector fields, and connections, enabling the transfer of geometric data from $ M $ to $ M^A $. A \text{Galois-equivariant structure :} on Weil bundles defined over number fields, linking their geometry to arithmetic symmetries. A \text{cohomological comparison isomorphism:} between the Weil bundle $ M^A $ and the crystalline cohomology of $ M $, unifying infinitesimal and crystalline perspectives. Applications to Diophantine geometry and $ p $-adic Hodge theory are central to this work. We show that spaces of sections of Hodge bundles on $ M^A $ parametrize $ p $-adic modular forms, offering a geometric interpretation of deformation-theoretic objects. Furthermore, Weil bundles are used to study infinitesimal solutions of equations on elliptic curves, revealing new structural insights into $ p $-adic deformations.

math.NT↗

arXiv2025

Cosymplectic Lagrangian-like submanifolds

This paper highlights the similarities between even-dimensional geometry (symplectic) and odd-dimensional geometry (cosymplectic). We study the Lagrangian Grassmannian in the cosymplectic setting. The space of compatible co-complex structures is introduced and analyzed. A study of Moser's trick and Lagrangian neighborhood theorems in the cosymplectic context follows. The corresponding Weinstein $1-$form is derived, and its de Rham class is a co-flux.

math.DG↗

arXiv2024

Geometric structures on Weil bundles: Canonical differential-geometric constructions

This paper investigates the transfer of classical geometric structures from a smooth manifold $M$ to its Weil bundle $(M^\mathbf A, \tilde\pi_M, M)$ associated with a Weil algebra $\mathbf A$. We show that various structures including locally conformal symplectic (lcs), locally conformal cosymplectic (lcc), contact, Jacobi, Sasakian, Walker, sub Riemannian, orientation, Riemannian, and K\"ahlerian structures admit canonical lifts to $M^\mathbf A$. Our approach emphasizes the differential geometric properties of these canonical constructions, utilizing the Weil projection $\tilde{\pi}_M$ and related functorial tools. This provides a unified perspective on endowing Weil bundles with rich geometric structure inherited from the base manifold. Furthermore, we highlight a specific construction yielding a cosymplectic manifold on $M^\mathbf{A}$ (for suitable $M$ and $\mathbf{A}$) that is demonstrably not a trivial suspension of a symplectic manifold. We also explicitly show how integrability of almost complex structures is preserved and clarify the nature of lifted characteristic vector fields.

math.DG↗

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↗