Homotheties and topology of tangent sphere bundles

We prove a Theorem on homotheties between two given tangent sphere bundles $S_rM$ of a Riemannian manifold $M,g$ of $\dim\geq 3$, assuming different variable radius functions $r$ and weighted Sasaki metrics induced by the conformal class of $g$. New examples are shown of manifolds with constant positive or with constant negative scalar curvature, which are not Einstein. Recalling results on the associated almost complex structure $I^G$ and symplectic structure ${\omega}^G$ on the manifold $TM$, generalizing the well-known structure of Sasaki by admitting weights and connections with torsion, we compute the Chern and the Stiefel-Whitney characteristic classes of the manifolds $TM$ and $S_rM$.Comment: 15 pages, to appear in Journal of Geometr

On the characteristic connection of gwistor space

We give a brief presentation of gwistor space, which is a new concept from G_2 geometry. Then we compute the characteristic torsion T^c of the gwistor space of an oriented Riemannian 4-manifold with constant sectional curvature k and deduce the condition under which T^c is \nabla^c-parallel; this allows for the classification of the G_2 structure with torsion and the characteristic holonomy according to known references. The case with the Einstein base manifold is envisaged.Comment: Many changes since first version, including title; Central European Journal of Mathematics, 201

M.T.K. Abbassi and G. Calvaruso ∗ HARMONIC MAPS HAVING TANGENT BUNDLES WITH g-NATURAL METRICS AS SOURCE OR TARGET

Abstract. We produce new examples of harmonic maps, having as either source or target manifold the tangent bundle TM on a Riemannian manifold(M,g), equipped with a Riemannian g-natural metric G. In particular, we study the harmonicity of the canonical projection π:(TM,G)→(M,g), and of the identity map(TM,G)→(TM,g S) and conversely, g S being the Sasaki metric on TM. A corresponding study is made for the unit tangent sphere bundle T 1 M, equipped with a Riemannian g-natural metric ˜G. 1

Some examples of harmonic maps for g-natural metrics

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

Harmonic maps defined by the geodesic flow

Let (M, g) be a Riemannian manifold. We equip the unit tangent sphere bundle T1M of (M, g) and its unit tangent sphere bundle TρT1M of radius ρ> 0 with arbitrary Riemannian g-natural metrics. When (M, g) is two-point homogeneous and both T1M and T1T1M are equipped with the Sasaki metrics, the geodesic flow vector field is harmonic and determines a harmonic map T1M → T1T1M [8]. In this paper we prove that if arbitrary Riemannian g-natural metrics are considered, then the geodesic flow is still a harmonic vector field, and it also defines a harmonic map under some conditions on the g-natural metrics. This permits to exhibit large families of harmonic maps defined in a compact Riemannian manifold and having a target space with a highly nontrivial geometry. In particular, explicit exam-ples are provided on the unit tangent sphere bundle of the sphere Sn and the flat torus Tn. Moreover, the geodesic flow being a Killing vector field is characterized in terms of harmonicity of the corresponding map and of properties of the base manifold

Harmonicity of unit vector fields with respect to Riemannian g-natural metrics

Let $(M, g)$ be a compact Riemannian manifold and $T_1M$ its unit tangent sphere bundle. We equip $T_1M$ with an arbitrary Riemannian g-natural metric $˜G$ , and investigate the harmonicity of a unit vector field V of $M$, thought as a map from $(M, g)$ to $(T_1M, ˜G )$. We then apply this study to characterize unit Killing vector fields and to investigate harmonicity properties of the Reeb vector field of a contact metric manifold

Harmonic sections of tangent bundles equipped with Riemannian gg-natural metrics

Let $(M,g)$ be a Riemannian manifold. When $M$ is compact and the tangent bundle $TM$ is equipped with the Sasaki metric $g^s$, parallel vector fields are the only harmonic maps from $(M,g)$ to $(TM,g^s)$. The Sasaki metric, and other well-known Riemannian metrics on $TM$, are particular examples of $g$-natural metrics. We equip $TM$ with an arbitrary $g$-natural Riemannian metric $G$, and investigate the harmonicity properties of a vector field $V$ of $M$, thought as a map from $(M,g)$ to $(TM,G)$. We then apply this study to the Reeb vector field and, in particular, to Hopf vector fields on odd-dimensional spheres