Searcharxiv⌕ Search

arXiv subjects

M. Benyounes

Publications and source records attributed to M. Benyounes.

9 recordsLinked to original sources
arXiv2026

Harmonicity of normal almost contact metric structures

We consider normal almost contact structures on a Riemannian manifold and, through their associated sections of an ad-hoc twistor bundle, study their harmonicity, as sections or as maps. We rewrite these harmonicity equations in terms of the curvature tensor and find conditions relating the harmonicity of the almost contact metric and almost complex structures of the total and base spaces of the Morimoto fibration. We apply these results to homogeneous principal circle bundles over generalised flag manifolds, in particular Aloff-Wallach spaces, to mass-produce harmonic almost contact metric structures.

math.DG↗
arXiv2013

Harmonic morphisms and moment maps on hyper-Kähler manifolds

We characterise the actions, by holomorphic isometries on a Kähler manifold with zero first Betti number, of an abelian Lie group of dim\geq 2, for which the moment map is horizontally weakly conformal (with respect to some Euclidean structure on the Lie algebra of the group). Furthermore, we study the hyper-Kähler moment map $ϕ$ induced by an abelian Lie group T acting by triholomorphic isometries on a hyper-Kähler manifold M, with zero first Betti number, thus obtaining the following: If dim T=1 then $ϕ$ is a harmonic morphism. Moreover, we illustrate this on the tangent bundle of the complex projective space equipped with the Calabi hyper-Kähler structure, and we obtain an explicit global formula for the map. If dim T\geq 2 and either $ϕ$ has critical points, or M is nonflat and dim M=4 dim T then $ϕ$ cannot be horizontally weakly conformal.

math.DG↗
arXiv2013

Harmonic Vector Fields on Space Forms

A vector field s on a Riemannian manifold M is said to be harmonic if there exists a member of a 2-parameter family of generalised Cheeger-Gromoll metrics on TM with respect to which s is a harmonic section. If M is a simply-connected non-flat space form other than the 2-sphere, examples are obtained of conformal vector fields that are harmonic. In particular, the harmonic Killing fields and conformal gradient fields are classified, a loop of non-congruent harmonic conformal fields on the hyperbolic plane constructed, and the 2-dimensional classification achieved for conformal fields. A classification is then given of all harmonic quadratic gradient fields on spheres.

math.DG↗
arXiv2012

Biharmonic holomorphic maps and conformally Kahler geometry

We give conditions on the Lee vector field of an almost Hermitian manifold such that any holomorphic map from this manifold into a (1,2)-symplectic manifold must satisfy the fourth-order condition of being biharmonic, hence generalizing the Lichnerowicz theorem on harmonic maps. These third-order non-linear conditions are shown to greatly simplify on l.c.K. manifolds and construction methods and examples are given in all dimensions.

math.DG↗
arXiv2010

Generalized Cheeger-Gromoll Metrics and the Hopf map

We show, using two different approaches, that there exists a family of Riemannian metrics on the tangent bundle of a two-sphere, which induces metrics of constant curvature on its unit tangent bundle. In other words, given such a metric on the tangent bundle of a two-sphere, the Hopf map is identified with a Riemannian submersion from the universal covering space of the unit tangent bundle onto the two-sphere. A hyperbolic counterpart dealing with the tangent bundle of a hyperbolic plane is also presented.

math.DG↗
arXiv2009

The geometry of generalised Cheeger-Gromoll metrics

We study the geometry of the tangent bundle equipped with a two-parameter family of Riemannian metrics. After deriving the expression of the Levi-Civita connection, we compute the Riemann curvature tensor and the sectional, Ricci and scalar curvatures. Specializing to the case of space forms, we characterise the metrics giving positive sectional curvature and show that one can always find parameters ensuring positive scalar curvature on the tangent space. Under some curvature conditions, this extends to general base manifolds.

math.DG↗
arXiv2008

Harmonic maps and Kaluza-Klein metrics on spheres

This article studies the harmonicity of vector fields on Riemannian manifolds, viewed as maps into the tangent bundle equipped with a family of Riemannian metrics. Geometric and topological rigidity conditions are obtained, especially for surfaces and vector fields of constant norm, and existence is proved on two-tori. Classifications are given for conformal, quadratic and Killing vector fields on spheres. Finally, the class of metric considered on the tangent bundle is enlarged, permitting new vector fields to become harmonic.

math.DG↗
arXiv2007

Harmonic maps and sections on spheres

The absence of interesting harmonic sections for the Sasaki and Cheeger-Gromoll metrics has led to the consideration of alternatives, for example in the form of a two-parameter family of natural metrics shown to relax existence conditions for harmonicity. This article investigates harmonic Killing vector fields, proves their non-existence on S^2, obtains rigidity results for harmonic gradient vector fields on the two-sphere, classifies spherical quadratic gradient fields in all dimensions and determines the tension field, concluding with the discovery of a family of metrics making Hopf vector fields harmonic maps on S^{2n+1}.

math.DG↗