arXiv subjects

Ferdinand Ngakeu

Publications and source records attributed to Ferdinand Ngakeu.

4 recordsLinked to original sources

arXiv2026

Diffeomorphism Group Actions on Para-K\"ahler Structures: Lifts and Induced Dynamics on 3-Webs

Let $M$ be a smooth manifold equipped with a para-K\"ahler structure. We define an action of the diffeomorphism group $\operatorname{Diff}(M)$ on the space of para-K\"ahler 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 Lagrangian $3$-webs.

math.DS↗

arXiv2026

Lifting Symplectomorphism Group Actions on Bi-Lagrangian Structures to the Whitney Sum

Let $M$ be a manifold endowed with a bi-Lagrangian structure $(\omega, \mathcal{F}_1, \mathcal{F}_2)$. Thus, $\omega$ is a symplectic form, and $(\mathcal{F}_1, \mathcal{F}_2)$ is a pair of transverse Lagrangian foliations on the symplectic manifold $(M, \omega)$. A bi-Lagrangian structure is said to be \textbf{affine} if the associated linear connection is curvature-free. We prove that, if $M$ is parallelizable, then every bi-Lagrangian structure on $M$ naturally induces two bi-Lagrangian structures on the tangent bundle $TM$ and on the cotangent bundle $T^*M$, and hence on the Whitney sum $W = TM \oplus T^*M$. The first way to lift a bi-Lagrangian structure yields an affine bi-Lagrangian structure. For the second way, we prove that the lifted bi-Lagrangian structure is affine if and only if the initial one is affine. We also show that, if the bi-Lagrangian structures on $M$ can be lifted to $TM$ or $T^*M$, then the action of the symplectomorphism group on the set of bi-Lagrangian structures defined in \cite{TNB} admits natural lifts to $TM$, $T^*M$, and hence to $W = TM \oplus T^*M$.

math.DS↗

arXiv2019

Null hypersurfaces and trapping horizons

The purpose of the present work is to study (marginally) trapped submanifolds lying in a null hypersurface. Let $(M,g,N)\to\Bm(c)$ be a null hypersurface of a space-time with constant sectional curvature $c$, endowed with a Screen Integrable and Conformal rigging $N$. The (Marginally) Trapped Submanifolds we are interested with are particular leaves of the screen distribution according to the sign of their expansions. We prove that if $c$ is non-positive, then $\Bm$ cannot contain a null non-expanding horizon. In the case $c$ is positive, we show that if $\Bm$ satisfies Einstein's equation and dominant energy condition holds, then any null trapping horizon of $\Bm$ is a null non-expanding horizon. More generally we prove that in a spacetime $\Bm(c)$ with constant sectional curvature $c$, cross-sections of a marginally outer trapped tube are Riemann manifold with the same constant sectional curvature $c$.

math.DG↗

arXiv2018

$α$-associated Metric On Rigged Null hypersurfaces

Let $x:M\to\Bm$ be the canonical injection of a Null Hypersurface $(M,g)$ in a semi-Riemannian manifold $(\overline{M},\bar g)$. A rigging for $M$ is a vector field $L$ defined on some open set of $\overline{M}$ containing $M$ such that $L_p\notin T_pM$ for each $p\in M$. Such a vector field induces a null rigging $N$. Let $\bar η$ be the 1-form which is $\bar g$-metrically equivalent to $N$ and $η=x^\star\barη$ its pull back on $M$. We introduce and study for a given non vanishing function $α$ on $M$ the so-called $α$-associated (semi-)Riemannian metric $ g_α=g+αη\otimes η$. For a closed rigging $N$ we give a constructive method to find an $α$-associated metric whose Levi-Civita connection coincides with the connection $\nabla$ induced on $M$ by the Levi-Civita connection $\overline{\nabla}$ of $\overline{M}$ and the null rigging $N$. We relate geometric objects of ${g}_α$ to those of $g$ and $\overline{g}$. As application, we show that given a null Monge hypersurface $M$ in $\R_q^{n+1},$ there always exists a rigging and an $α$-associated metric whose Levi-Civita connection coincides with the induced connection on $M$.

math.DG↗