11 research outputs found
Homotheties and topology of tangent sphere bundles
We prove a Theorem on homotheties between two given tangent sphere bundles
of a Riemannian manifold of , assuming different
variable radius functions and weighted Sasaki metrics induced by the
conformal class of . 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 and
symplectic structure on the manifold , 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 and .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 be a compact Riemannian manifold and its unit tangent sphere bundle. We equip with an arbitrary Riemannian g-natural metric , and investigate the harmonicity of a unit vector field V of , thought as a map from to . 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 -natural metrics
Let be a Riemannian manifold. When is compact and the tangent bundle is equipped with the Sasaki metric , parallel vector fields are the only harmonic maps from to . The Sasaki metric, and other well-known Riemannian metrics on , are particular examples of -natural metrics. We equip with an arbitrary -natural Riemannian metric , and investigate the harmonicity properties of a vector field of , thought as a map from to . We then apply this study to the Reeb vector field and, in particular, to Hopf vector fields on odd-dimensional spheres