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Bertuel Tangue Ndawa

Publications and source records attributed to Bertuel Tangue Ndawa.

12 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

Dynamics on Bi-Lagrangian Structures and Cherry maps

We consider a bi-Lagrangian structure $(ω,\mathcal{F}_1,\mathcal{F}_2)$ on a manifold $M$; that is, $(M,ω,\mathcal{F}_1,\mathcal{F}_2)$ is a bi-Lagrangian manifold. We prolong bi-Lagrangian structures on $M$ and lift a given dynamics to its tangent and cotangent bundles in several ways. In some cases, we show that the lifted structures are affine. In the case of the two-dimensional torus $\mathbb{T}^2$, we prove that an extension of the same dynamics to pairs of so-called Cherry vector fields induces a conjugation action on a subset of Cherry maps, namely circle maps with a flat interval. Furthermore, we define linear connections for certain Cherry maps using bi-Lagrangian data. Finally, we show that maps generated by a bi-Lagrangian structure belong a priori to different Hölder homeomorphism classes.

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

Infinite Lifting of an Action of Symplectomorphism Group on the set of Bi-Lagrangian Structures

$(ω,\mathcal{F}_{1},\mathcal{F}_{2})$, where $ω$ is symplectic and $(\mathcal{F}_{1},\mathcal{F}_{2})$ is a pair of transversal Lagrangian foliations. Such structures admit a natural geometric object, the Hess connection, which is central to the classification of affine bi-Lagrangian structures. We show that any bi-Lagrangian structure on $M$ lifts to a bi-Lagrangian structure on the trivial bundle $M\times\mathbb{R}^{2n}$. Moreover, the lift of an affine bi-Lagrangian structure is again affine. We then define a dynamical system in terms of an action of the symplectomorphism group on the space of bi-Lagrangian structures. This dynamics is compatible with Hess connections, preserves affine bi-Lagrangian structures, and can be lifted to $M\times\mathbb{R}^{n}$, and iterated to $\left(M\times\mathbb{R}^{2n}\right)\times\mathbb{R}^{4n}$. The resulting lifted dynamics agrees with the initial one on $M\times\mathbb{R}^{n}$ for suitable structures. Finally, the same results hold when $M\times\mathbb{R}^{2n}$ is replaced by $TM$ or $T^{*}M$, provided that $M$ is parallelizable.

math.DS↗
arXiv2026

Factorization of Isomorphisms of $(H,θ)$-twisted Lie algebroids

We study the isomorphism groupoid $\mathcal{T}(M)$ of $θ$-almost twisted Poisson ($θ$-atP) structures on a smooth manifold $M$, focusing on the internal structure of its morphisms. A morphism in $\mathcal{T}(M)$ is a $C^\infty(M)$-linear isomorphism $Φ:\gO^1(M)\to\gO^1(M)$ that simultaneously intertwines the anchor maps and the $(H,θ)$-twisted Koszul brackets associated with two $θ$-atP structures. Every such morphism induces a canonical isomorphism in $θ$-atP cohomology. We define a classifying functor $$ Δ: \mathrm{Mor}(\mathcal{T}(M)) \longrightarrow (Z^1_{\mathrm{dR}}(M) ,+), \qquad Δ(Φ)=θ' - θ, $$ which is additive under composition and partitions the morphisms into two complementary families: the sub-groupoid $\mathcal{T}_{\mathrm{fix}}=\kerΔ$ of isomorphisms preserving $θ$, and the family $\mathcal{T}_{\mathrm{mod}}$ of isomorphisms shifting $θ$. We describe each element in this partition.

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↗
arXiv2024

$C^1$-Diffeomorphism Class of some Circle Maps with a Flat Interval

We study a certain class circle maps which are constant on one interval (called flat piece), and such that the degrees of the singularities at the boundary of the flat piece are different. In this paper, we show that if the topological conjugacy between two maps of my class is a bi-Lipschitz homeomorphism, then it is a $C^1$ diffeomorphism; that is, the bi-Lipschitz homeomorphism class and $C^1$ diffeomorphism class of a map in our class are equivalent.

math.DS↗
arXiv2023

Dynamic of Pair of some Distributions: Bi-lagrangian structure and its prolongations on the (co)tangent bundles, and Cherry flow

We consider a bi-Lagrangian manifold $(M,ω,\mathcal{F}_{1},\mathcal{F}_{2})$. That is, $ω$ is a 2-form, closed and non-degenerate (called symplectic form) on $M$, and $(\mathcal{F}_{1},\mathcal{F}_{2})$ is a pair of transversal Lagrangian foliations on the symplectic manifold $(M,ω)$. In this case, $(ω, \mathcal{F}_{1},\mathcal{F}_{2})$ is a bi-Lagrangian structure on $M$. In this paper, we prolong a bi-Lagrangian structure on $M$ on its tangent bundle $TM$ and its cotangent bundle $T^{*}M$ in different ways. As a consequence some dynamics on the bi-Lagrangian structure of $M$ can be prolonged as dynamics on the bi-Lagrangian structure of $TM$ and $T^{*}M$. Observe that a pair of transversal vector fields without singularity on the 2-torus $\mathbb{T}^2=\mathbb{S}^1\times\mathbb{S}^1$ endowed with a symplectic form defines a bi-Lagrangian structure on $\mathbb{T}^2$. This sparked our curiosity. By studying the dynamic of pairs of vector fields on $\mathbb{T}^2$, we found that some circle maps with a flat piece (called Cherry maps) can be generated by a pair of vector fields. Moreover, the push forward action of the set of diffeomorphisms $\mathbb{T}^2$ on the set of its vector fields induces a conjugation action on the set of generated Cherry maps.

math.DS↗
arXiv2022

Rigidity of Fibonacci Circle Maps with a Flat Piece and Different Critical Exponents

We consider order preserving $C^3$ circle maps with a flat piece, Fibonacci rotation number, critical exponents $(\ell_1, \ell_2)$ and negative shwarzian derivative. This paper treat the geometry characteristic of the non-wondering (cantor (fractal)) set from a map of our class. We prove that, for $(\ell_1, \ell_2)$ in $(1,2)^2$, the geometry of system is degenerate (double exponentially fast). As consequences, the renormalization diverges and the geometric (rigidity) class depends on the three couples $(c_u(f), c'_u(f) )$, $( c_+(f), c'_+(f))$ and $(c_s(f), c'_s(f) )$.\vspace{0.5cm}

math.DS↗
arXiv2021

Cherry Maps with Different Critical Exponents: Bifurcation of Geometry

We consider order preserving $C^3$ circle maps with a flat piece, irrational rotation number and critical exponents $(\ell_1, \ell_2)$. We detect a change in the geometry of the system. For $(\ell_1, \ell_2) \in [1,2]^2$ the geometry is degenerate and it becomes bounded for $(\ell_1, \ell_2) \in [2,\infty)^2 \setminus \{(2,2)\}$. When the rotation number is of the form $[abab\cdots]$; for some $a,b\in\mathbb{N}^*$, the geometry is bounded for $(\ell_1, \ell_2)$ belonging above a curve defined on $]1, +\infty [^2$. As a consequence we estimate the Hausdorff dimension of the non-wandering set $K_f= \mathcal{S}^1 \setminus \bigcup_{i=0}^\infty f^{-i}(U)$. Precisely, the Hausdorff dimension of this set is equal to zero when the geometry is degenerate and it is strictly positive when the geometry is bounded.

math.DS↗