arXiv subjects

Nasser Saipele Nansidi

Publications and source records attributed to Nasser Saipele Nansidi.

4 recordsLinked to original sources

arXiv2026

Diffeomorphism Group Actions on Para-Kähler Structures: Lifts and Induced Dynamics on 3-Webs

Let $M$ be a smooth manifold equipped with a para-Kähler structure. We define an action of the diffeomorphism group $\operatorname{Diff}(M)$ on the space of para-Kähler structures and prove that this action preserves the associated affine structures. Although this result is well known, we provide a proof based on this action, showing in particular that an isometry commutes with the Levi--Civita connection. We then construct several natural liftings of this action to the tangent bundle $TM$, the cotangent bundle $T^*M$, and the Whitney sum $E=TM\oplus T^*M$. In certain cases, these lifted actions induce natural dynamical systems on a distinguished class of $3$-webs.

math.DS↗

arXiv2026

Bi-Lagrangian and transverse Dirac structures induced on the Whitney sum

Let $(M,ω,\mathcal{F}_1,\mathcal{F}_2)$ be a bi-Lagrangian manifold. For a distribution $\mathcal{D}\subseteq TM$, we characterize when the subbundle $N^{*t}\mathcal{D}:=\mathcal{D}\oplus N^*\mathcal{D}$ of the Whitney sum $W=TM\oplus T^*M$ is a Dirac structure for both the untwisted and $(H,θ)$-twisted Courant brackets. Consequently, every bi-Lagrangian structure canonically determines a pair of transverse Dirac structures. the associated bi-Lagrangian connection, also called the Hess connection $\nabla$ induces a linear connection on $W$ with respect to which both Dirac subbundles are parallel, thereby establishing a natural link between bi-Lagrangian geometry and Courant Dirac geometry. We also construct induced bi-Lagrangian structures on $TM$, $T^*M$, and $W$, and study the relation between their affineness and that of the original structure. We also lift to these bundles the canonical action of the symplectomorphism group on bi-Lagrangian structures.

math.DS↗

arXiv2026

On the geometric quantization of $θ$-almost twisted Poisson manifold

We introduce and investigate the concept of a $θ$-almost twisted Poisson manifold $(M,Λ,φ,θ)$. This structure consists of a smooth manifold $M$ equipped with a bivector field $Λ$, a 3-form $φ$, and a closed 1-form $θ$, satisfying the following conditions: the exterior derivative $dφ$ of $φ$ equals the wedge product $θ\wedge φ$; the anchor $Λ^\#(θ)$ of $θ$ vanishes identically; and one-half of the Schouten-Nijenhuis bracket $[Λ, Λ]$ equals the anchor $Λ^\#(φ)$ of $φ$. This structure generalizes both Poisson and twisted Poisson manifolds, permitting the 3-form $φ$ to be non-closed in a way controlled by the 1-form $θ$. We construct a Lie-Rinehart algebra on the module of 1-forms $Ω^1(M)$, giving rise to a cochain complex and an associated cohomology theory called $θ$-almost twisted Poisson cohomology. Moreover, we develop the geometric quantization of these manifolds by defining a suitable contravariant derivative, establishing a prequantization condition in terms of the cohomology, and constructing a quantum Hilbert space via polarization. We illustrate our results with several examples, including the computation of the cohomology and quantization on $\mathbb{R}^5$.

math.DS↗

arXiv2025

$θ$-almost twisted Poisson cohomology

We introduce the notion of a $θ$-almost twisted Poisson structure on manifolds, which involves incorporating a closed $1$-form $θ$ into twisted Poisson structures under specific conditions. We provide a characterization of this structure on low-dimensional manifolds and construct the Lie-Rinehart algebra on the module of $1$-forms on manifolds equipped with this structure. This construction leads to a cochain complex and its associated cohomology, which we refer to as $θ$-almost twisted Poisson cohomology. An example illustrating this cohomology is also presented on $\mathbb{R}^5$.

math.DG↗