arXiv subjects
Publications and source records attributed to Emmanuel Gnandi.
In this paper, we study pseudo-Riemannian Hom-Lie algebras. We establish Hom-versions of the classical Bochner theorem in both the Riemannian and Lorentzian settings. Finally, we prove that the space of Killing vectors on a pseudo-Riemannian Hom-Lie algebra forms a totally geodesic Hom-Lie subalgebra.
We study Ricci curvature properties of Hessian metrics on the leaves of the codimension-one foliation $\mathcal{F}_\omega = \ker\,\omega$ generated by the first Koszul form $\omega$ of a closed oriented Hessian manifold. Our main result reveals a striking rigidity phenomenon: non-negative Ricci curvature on a single leaf of $\mathcal{F}_\omega$ compels the Hessian metric to be flat, yields sharp bounds on the first Betti number in terms of the dimension of the Hessian manifold and the topology of the leaves. This rigidity also shows that Koszul-type and radiant affine manifolds admit no leaf carrying non-negative Ricci curvature, reflecting a fundamental incompatibility between affine hyperbolicity and leafwise curvature positivity. In dimension three, we obtain a complete classification of the underlying manifold, extended to the non-orientable setting via the orientation double cover.
This paper investigates timelike conformal vector fields on closed Lorentzian $3$-manifolds and shows that, although these fields form a broader class than Killing fields, their behavior in dimension three is nonetheless remarkably rigid. After performing a conformal change of the metric so that the vector field becomes unit and Killing, we analyze the geometry of the flow it generates through the framework of stable Hamiltonian structures and basic cohomology. Our main result proves that any nowhere-vanishing timelike conformal vector field necessarily arises as the Reeb vector field of either a Sasakian structure or a co-K\"ahler structure. In other words, every such Lorentzian conformal flow is intrinsically "Reeb-like", which forces the underlying geometry to be either contact or cosymplectic. This establishes a striking connection between Lorentzian geometry, Sasakian and co-K\"ahler structures, and the topology of flows in dimension~$3$.
In this work, we revisit quasi-Sasakian geometry in dimension three and examine how these structures interact with the foliation generated by the Reeb vector field and its basic cohomology. Through a deformation-based approach, we show that a closed, orientable $3$-manifold admits a quasi-Sasakian structure precisely when it is either Sasakian or arises as a K\"ahler mapping torus. In particular, every quasi-Sasakian structure in this setting can be deformed into a Sasakian or a co-K\"ahler one. This result leads to a complete classification of quasi-Sasakian manifolds in dimension three and highlights the geometric and topological features that distinguish the two cases.
We investigate the construction of exponential families from statistical manifolds, a central problem in information geometry. We prove that every compact statistical manifold admits a singular foliation whose leaves are Hessian manifolds. In particular, any non-flat, compact, orientable 3-dimensional leaf arises as a quotient of an exponential family and has only odd Betti numbers. Our approach is constructive: we explicitly describe the foliation and analyze the geometric and topological properties of its leaves. We show that compact orientable leaves are either finite quotients of flat torus or mapping torus with periodic monodromy. In three dimensions, non-flat leaves admit a co-K\"ahler structure, which allows us to realize them as explicit exponential families parametrized by a Lorentz cone. These results establish a concrete bridge between abstract statistical manifolds and exponential families, highlighting deep connections between information geometry, differential geometry, and the topology of 3-manifolds.
The question of whether a closed, orientable manifold can admit a nontrivial vector field that is parallel with respect to some Riemannian metric is a classical problem in Differential Geometry, first posed by S. S. Chern [11]. In this work, we provide a complete answer to Chern's question in dimension three. Specifically, we show that a closed, orientable 3-manifold admits a nontrivial parallel vector field with respect to some Riemannian metric if and only if it is a K\"ahler mapping torus. Furthermore, we prove that the Betti numbers of any such 3-manifold are necessarily odd. A full classification of these manifolds is also obtained. Similar results are established for compact orientable Lorentzian 3-manifolds admitting either a parallel timelike vector field or a parallel lightlike vector field.
We investigate the global topology of 3-dimensional Hessian manifolds. We prove that any compact, orientable 3-dimensional Hessian manifold is either a Hantzsche-Wendt manifold or admits the structure of a K\"ahler mapping torus. We analyze the parity of Betti numbers for compact, orientable 3-dimensional Hessian manifolds, with special focus on those of Koszul type (hyperbolic manifolds). Moreover, we show that the product of two compact, orientable, 3-dimensional Hessian manifolds of Koszul type naturally carries a K\"ahler structure. Finally, we establish that every compact, orientable, 3-dimensional Hessian manifold is a Seifert manifold with trivial Euler number, whose underlying orbifold has either vanishing or negative Euler characteristic, thus providing a complete topological classification.
The Fisher information metric or the Fisher-Rao metric corresponds to a natural Riemannian metric defined on a parameterized family of probability density functions. As in the case of Riemannian geometry, we can define a distance in terms of the Fisher information metric, called the Fisher-Rao distance. The Fisher information metric has a wide range of applications in estimation and information theories. Indeed, it provides the most informative Cramer-Rao bound for an unbiased estimator. The Goldberg conjecture is a well-known unsolved problem which states that any compact Einstein almost K\"ahler manifold is necessarily a K\"ahler-Einstein. Note that, there is also a known odd-dimensional analog of the Goldberg conjecture in the literature. The main objective of this paper is to establish a new characterization of coK\"ahler manifolds and K\"ahler manifolds; our characterization is statistical in nature. Finally, we corroborate that every, K\"ahler and co-K\"ahler manifolds, can be viewed as being a parametric family of probability density functions, whereas K\"ahler and coK\"ahler metrics can be regarded as Fisher information metrics. In particular, we prove that, when the K\"ahler metric is real analytic, it is always locally the Fisher information of an exponential family. We also tackle the link between K\"ahler potential and Kullback-Leibler divergence.