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Note on the classification theorems of g -natural metrics on the tangent bundle of a Riemannian manifold ( M , g )

Mohamed Tahar Kadaoui Abbassi — 2004

Commentationes Mathematicae Universitatis Carolinae

In [7], it is proved that all g -natural metrics on tangent bundles of m -dimensional Riemannian manifolds depend on arbitrary smooth functions on positive real numbers, whose number depends on m and on the assumption that the base manifold is oriented, or non-oriented, respectively. The result was originally stated in [8] for the oriented case, but the smoothness was assumed and not explicitly proved. In this note, we shall prove that, both in the oriented and non-oriented cases, the functions generating...

On g -natural conformal vector fields on unit tangent bundles

Mohamed Tahar Kadaoui Abbassi;  Noura Amri — 2021

Czechoslovak Mathematical Journal

We study conformal and Killing vector fields on the unit tangent bundle, over a Riemannian manifold, equipped with an arbitrary pseudo-Riemannian g -natural metric. We characterize the conformal and Killing conditions for classical lifts of vector fields and we give a full classification of conformal fiber-preserving vector fields on the unit tangent bundle endowed with an arbitrary pseudo-Riemannian Kaluza-Klein type metric.

On the completeness of total spaces of horizontally conformal submersions

Mohamed Tahar Kadaoui Abbassi;  Ibrahim Lakrini — 2021

Communications in Mathematics

In this paper, we address the completeness problem of certain classes of Riemannian metrics on vector bundles. We first establish a general result on the completeness of the total space of a vector bundle when the projection is a horizontally conformal submersion with a bound condition on the dilation function, and in particular when it is a Riemannian submersion. This allows us to give completeness results for spherically symmetric metrics on vector bundle manifolds and eventually for the class...

On natural metrics on tangent bundles of Riemannian manifolds

Mohamed Tahar Kadaoui Abbassi;  Maâti Sarih — 2005

Archivum Mathematicum

There is a class of metrics on the tangent bundle T M of a Riemannian manifold ( M , g ) (oriented , or non-oriented, respectively), which are ’naturally constructed’ from the base metric g [Kow-Sek1]. We call them “ g -natural metrics" on T M . To our knowledge, the geometric properties of these general metrics have not been studied yet. In this paper, generalizing a process of Musso-Tricerri (cf. [Mus-Tri]) of finding Riemannian metrics on T M from some quadratic forms on O M × ℝ m to find metrics (not necessary Riemannian)...

Some examples of harmonic maps for g -natural metrics

Mohamed Tahar Kadaoui Abbassi;  Giovanni Calvaruso;  Domenico Perrone — 2009

Annales mathématiques Blaise Pascal

We produce new examples of harmonic maps, having as source manifold a space ( M , g ) of constant curvature and as target manifold its tangent bundle T M , equipped with a suitable Riemannian g -natural metric. In particular, we determine a family of Riemannian g -natural metrics G on T 𝕊 2 , with respect to which all conformal gradient vector fields define harmonic maps from 𝕊 2 into ( T 𝕊 2 , G ) .

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