In [7], it is proved that all -natural metrics on tangent bundles of -dimensional Riemannian manifolds depend on arbitrary smooth functions on positive real numbers, whose number depends on 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...
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...
There is a class of metrics on the tangent bundle of a Riemannian manifold (oriented , or non-oriented, respectively), which are ’naturally constructed’ from the base metric [Kow-Sek1]. We call them “-natural metrics" on . 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 from some quadratic forms on to find metrics (not necessary Riemannian)...
We study conformal and Killing vector fields on the unit tangent bundle, over a Riemannian manifold, equipped with an arbitrary pseudo-Riemannian -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.
We produce new examples of harmonic maps, having as source manifold a space of constant curvature and as target manifold its tangent bundle , equipped with a suitable Riemannian -natural metric. In particular, we determine a family of Riemannian -natural metrics on , with respect to which all conformal gradient vector fields define harmonic maps from into .
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