The sectional curvature of the Sasaki metric of T1Mn

We study the tangent bundle of vectors of fixed length on a Riemannian manifold. We give sufficient conditions for the sectional curvature of the Sasaki metric on the tangent bundle of vectors of fixed length to be nonnegative.Translated from Ukrainskií Geometricheskií Sbornik, No. 30, 1987, pp. 10–17.     

Homotheties and topology of tangent sphere bundles

We prove a Theorem on homotheties between two given tangent sphere bundles S _r M of a Riemannian manifold M, g of \({{\rm dim}\geq3}\) , 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 _r M.     

H-contact unit tangent sphere bundles of Riemannian manifolds

A contact metric manifold is said to be H-contact, if its characteristic vector field is harmonic. We prove that the unit tangent bundle of a Riemannian manifold M equipped with the standard contact metric structure is H-contact if and only if M is 2-stein.     

Geodesic Ricci solitons on unit tangent sphere bundles

In the paper, (Abbassi and Kowalski, Ann Glob Anal Geom, 38: 11–20, 2010) the authors study Einstein Riemannian $g$ natural metrics on unit tangent sphere bundles. In this study, we equip the unit tangent sphere bundle $T_1 M$ of a Riemannian manifold $(M,g)$ with an arbitrary Riemannian $g$ natural metric $\tilde{G}$ and we show that if the geodesic flow $\tilde{\xi }$ is the potential vector field of a Ricci soliton $(\tilde{G},\tilde{\xi },\lambda )$ on $T_1M,$ then $(T_1M,\tilde{G})$ is Einstein. Moreover, we show that the Reeb vector field of a contact metric manifold is an infinitesimal harmonic transformation if and only if it is Killing. Thus, we consider a natural contact metric structure $(\tilde{G}, \tilde{\eta }, \tilde{\varphi }, \tilde{\xi })$ over $T_1 M$ and we show that the geodesic flow $\tilde{\xi }$ is an infinitesimal harmonic transformation if and only if the structure $(\tilde{G}, \tilde{\eta }, \tilde{\varphi },\tilde{\xi })$ is Sasaki $\eta $ -Einstein. Consequently, we get that $(\tilde{G},\tilde{\xi }, \lambda )$ is a Ricci soliton if and only if the structure $(\tilde{G}, \tilde{\eta }, \tilde{\varphi }, \tilde{\xi })$ is Sasaki-Einstein with $\lambda = 2(n-1) >0.$ This last result gives new examples of Sasaki–Einstein structures.     

When are the tangent sphere bundles of a Riemannian manifold reducible?

We determine all Riemannian manifolds for which the tangent sphere bundles, equipped with the Sasaki metric, are local or global Riemannian product manifolds.

     

Notes on the Sasaki metric

We survey on the geometry of the tangent bundle of a Riemannian manifold, endowed with the classical metric established by S. Sasaki 60 years ago. Following the results of Sasaki, we try to write and deduce them by different means. Questions of vector fields, mainly those arising from the base, are related as invariants of the classical metric, contact and Hermitian structures. Attention is given to the natural notion of extension or complete lift of a vector field, from the base to the tangent manifold. Few results are original, but finally new equations of the mirror map are considered.     

On the classification of contact metric ‐spaces via tangent hyperquadric bundles

We classify locally the contact metric ‐spaces whose Boeckx invariant is as tangent hyperquadric bundles of Lorentzian space forms.     

On infinitesimal conformal transformations of the tangent bundles with the synectic lift of a Riemannian metric

The purpose of the present article is to investigate some relations between the Lie algebra of the infinitesimal fibre-preserving conformal transformations of the tangent bundle of a Riemannian manifold with respect to the synectic lift of the metric tensor and the Lie algebra of infinitesimal projective transformations of the Riemannian manifold itself.     

On the geometry of tangent bundles

The main aim of this survey paper is to write a detailed and unified presentation of some of the best known results on the geometry of tangent bundles of Riemannian manifolds.     

Curvatures of tangent hyperquadric bundles

We study geometry of tangent hyperquadric bundles over pseudo-Riemannian manifolds, which are equipped, as submanifolds of the tangent bundles, with the induced Sasaki metric. All kinds of curvatures are calculated, and geometric results concerning the Ricci curvature and the scalar curvature are proved. There exists a hyperquadric bundle whose scalar curvature is a preassigned constant.     

g-natural symmetries on tangent bundles

The study of symmetries is a well known research topic in differential geometry with relevant physical interpretations. Given a Riemannian manifold $(M,g)$, we consider pseudo-Riemannian g-natural metrics G on its tangent bundle $TM$ and characterize conformal, homothetic and Killing vector fields of $(TM,G)$ obtained from natural lifts of vector fields of M.     

On solutions of IHPT equations on tangent bundles with the metric II + III

The main purpose of the present paper is to determine the most general IHPT (Infinitesimal Holomorphically Projective Transformation) on T(M_n) with respect to a Levi-Civita connection of the metric II+III and adapted almost complex structure. Moreover, if T(M_n) admits a non-affine infinitesimal holomorphically projective transformation, then M_n and T(M_n) are locally flat.     

Conformal vector fields with respect to the Sasaki metric tensor

The purpose of this article is to characterize conformal vector fields with respect to the Sasaki metric tensor field on the tangent bundle of a Riemannian manifold of dimension at least three. In particular, if the manifold in question is compact, it is found that the only conformal vector fields are Killing vector fields.