Tangent Sphere Bundle Research Articles

A REMARK ON H-CONTACT UNIT TANGENT SPHERE BUNDLES

We shall give some curvature conditions for the unit tangent sphere bundle of an n( 4)-dimensional Riemannian manifold to be H- contact. Furthermore, we provide an example illustrating Main Theorem.

WEIGHTED METRICS ON TANGENT SPHERE BUNDLES

Natural metric structures on the tangent bundle and tangent sphere bundles $S_rM$ of a Riemannian manifold $M$ with radius function $r$ enclose many important unsolved problems. Admitting metric connections on $M$ with torsion, we deduce the equations of induced metric connections on those bundles. Then the equations of reducibility of $TM$ to the almost Hermitian category. Our purpose is the study of the natural contact structure on $S_rM$ and the $G_2$-twistor space of any oriented Riemannian 4-manifold.

The Holomorphic φ-Sectional Curvature of Tangent Sphere Bundles with Sasakian Structures

The Holomorphic φ-Sectional Curvature of Tangent Sphere Bundles with Sasakian Structures

Harmonic morphisms and Riemannian geometry of tangent bundles

Let (TM, G) and \({(T_1 M,\tilde G)}\) respectively denote the tangent bundle and the unit tangent sphere bundle of a Riemannian manifold (M, g), equipped with arbitrary Riemannian g-natural metrics. After studying the geometry of the canonical projections π : (TM, G) → (M, g) and \({\pi_1:(T_1 M,\tilde G) \rightarrow (M,g)}\), we give necessary and sufficient conditions for π and π 1 to be harmonic morphisms. Some relevant classes of Riemannian g-natural metrics will be characterized in terms of harmonicity properties of the canonical projections. Moreover, we study the harmonicity of the canonical projection \({\Phi:(TM-\{0\},G)\to (T_1 M,\tilde G)}\) with respect to Riemannian g-natural metrics \({G,\tilde G}\) of Kaluza–Klein type.

MINIMALITY, HARMONICITY AND CR GEOMETRY FOR REEB VECTOR FIELDS

Let (M, g) be a Riemannian manifold and T1 M its unit tangent sphere bundle. Minimality and harmonicity of unit vector fields have been extensively studied by considering on T1M the Sasaki metric [Formula: see text]. This metric, and other well-known Riemannian metrics on T1 M, are particular examples of Riemannian natural metrics. In this paper we equip T1 M with a Riemannian natural metric [Formula: see text] and in particular with a natural contact metric structure. Then, we study the minimality for Reeb vector fields of contact metric manifolds and of quasi-umbilical hypersurfaces of a Kähler manifold. Several explicit examples are given. In particular, the Reeb vector field ξ of a K-contact manifold is minimal for any [Formula: see text] that belongs to a family depending on two parameters of metrics of the Kaluza–Klein type. Next, we show that the Reeb vector field ξ of a K-contact manifold defines a harmonic map [Formula: see text] for any Riemannian natural metric [Formula: see text]. Besides this, if the Reeb vector ξ of an almost contact metric manifold is a CR map then the induced almost CR structure on M is strictly pseudoconvex and ξ is a pseudo-Hermitian map; if in addition ξ is geodesic then [Formula: see text] is a harmonic map. Moreover, the Reeb vector field ξ of a contact metric manifold is a CR map iff ξ is Killing and [Formula: see text] is a special metric of the Kaluza–Klein type. Finally, in the final section, we obtain that there is a family of strictly pseudoconvex CR structures on T1S2n+1 depending on one parameter, for which a Hopf vector field ξ determines a pseudo-harmonic map (in the sense of Barletta–Dragomir–Urakawa [8]) from S2n+1 to T1S2n+1.

HOMOGENEOUS AND H-CONTACT UNIT TANGENT SPHERE BUNDLES

AbstractWe prove that all g-natural contact metric structures on a two-point homogeneous space are homogeneous contact. The converse is also proved for metrics of Kaluza–Klein type. We also show that if (M,g) is an Einstein manifold and $\tilde G$ is a Riemannian g-natural metric on T1M of Kaluza–Klein type, then $(T_1 M,\tilde \eta ,\tilde G)$ is H-contact if and only if (M,g) is 2-stein, so proving that the main result of Chun et al. [‘H-contact unit tangent sphere bundles of Einstein manifolds’, Q. J. Math., to appear. DOI: 10.1093/qmath/hap025] is invariant under a two-parameter deformation of the standard contact metric structure on T1M. Moreover, we completely characterize Riemannian manifolds admitting two distinct H-contact g-natural contact metric structures, with associated metric of Kaluza–Klein type.

On the [formula omitted] bundle of a Riemannian 4-manifold

We study the natural G 2 structure on the unit tangent sphere bundle S M of any given orientable Riemannian 4-manifold M , as was discovered in Albuquerque and Salavessa (2009,2010) [9,10]. A name is proposed for the space. We work in the context of metric connections, or so-called geometry with torsion, and describe the components of the torsion of the connection which imply certain equations of the G 2 structure. This article is devoted to finding the G 2 torsion tensors which classify our structure according to the theory in Fernandez and Gray (1982) [4].

On Einstein Riemannian g-natural metrics on unit tangent sphere bundles

We prove that there is always a locally homogeneous Einstein g-natural metric on the unit tangent sphere bundle over any Riemannian space of constant positive sectional curvature. Furthermore, using the (1–1) correspondence between all SO(m + 1)-invariant homogeneous metrics on the Stiefel manifold \({V_2 \mathbb{R}^{m+1} = {{SO}}(m+1)/{{SO}}(m-1)}\) and all g-natural metrics on \({T_1 S^m}\) (Abbassi and Kowalski, Diff. Geom. Appl., to appear [7]), we reconstruct, by purely local procedure, the same well-known unique SO(m + 1)-invariant homogeneous Einstein metric on \({V_2 \mathbb{R}^{m+1}, m \neq 3}\), initially constructed by Kobayashi.

NOTES ON TANGENT SPHERE BUNDLES OF CONSTANT RADII

We show that the Riemannian geometry of a tangent sphere bundle of a Riemannian manifold (M, g) of constant radius <TEX>$\gamma$</TEX> reduces essentially to the one of unit tangent sphere bundle of a Riemannian manifold equipped with the respective induced Sasaki metrics. Further, we provide some applications of this theorem on the <TEX>$\eta$</TEX>-Einstein tangent sphere bundles and certain related topics to the tangent sphere bundles.

η-EINSTEIN TANGENT SPHERE BUNDLES OF CONSTANT RADII

We study the geometric properties of the base manifold for the tangent sphere bundle of radius r satisfying the η-Einstein condition with the standard contact metric structure. One of the main theorems is that the tangent sphere bundle of the n(≥3)-dimensional locally symmetric space, equipped with the standard contact metric structure, is an η-Einstein manifold if and only if the base manifold is a space of constant sectional curvature [Formula: see text] or [Formula: see text].

Naturality of homogeneous metrics on Stiefel manifolds [formula omitted

It is well known that the unit tangent sphere bundle T 1 S m of the standard sphere S m can be naturally identified with the Stiefel manifold V 2 R m + 1 = SO ( m + 1 ) / SO ( m − 1 ) . In this paper, we construct the ( 1 – 1 ) correspondence between all SO ( m + 1 ) -invariant homogeneous metrics on V 2 R m + 1 and all so-called g-natural metrics on T 1 S m .

η-PARALLEL CONTACT 3-MANIFOLDS

In this paper, we give a classification of contact 3-manifolds whose Ricci tensors are ·-parallel. It is well-known that for a 3-dimensional Riemannian manifold its curvature tensor R is expressed only in terms of the Ricci tensor S, the metric tensor g and the scalar curvature r. Hence, in studying the 3-dimensional Riemannian geometry, we see at once that the condition of local symmetry (rR = 0) is equivalent to the Ricci-parallel condition (rS = 0). Recently Boeckx and the present author (7) proved that a locally symmetric contact Riemannian manifold is either Sasakian and of constant curvature 1 or locally isometric to the unit tangent sphere bundle (with its standard con- tact metric structure) of a Euclidean space. (In the 3-dimensional case, we may also refer to (5).) This result says that the local symmetry is rather a strong condition in contact Riemannian geometry and hence, it is natural to consider a weaker condition, that is ·-parallel. Let M = (M;·,g,',») be a contact Riemannian manifold. Then the contact form · determines the con- tact distribution D which is given by the kernel of ·. We say that the Ricci tensor S is ·-parallel if S satisfies g((rXS)Y,Z) = 0 for any X,Y,Z 2 D. In this paper, we shall study 3-dimensional contact Riemannian manifolds whose Ricci-tensors are ·-parallel. On the other hand, given a contact manifold M = (M;·) one may raise a following natural question (cf. (16)): Which metric is most proper among Rie- mannian metrics associated with ·? One method of finding the nice Riemann- ian metrics is to study the criticality in the variational sense. In particular, in (2) and (16) the authors showed that M satisfies r»h = 2h' if and only if it has the critical metric of the Dirichlet energy functional E(g) = R M kL»gk 2 dM defined on the set of all Riemannian metrics associated with the given contact form ·, where h = 1 L»' and L» is the Lie derivative with respect to ». (Chern- Hamilton (10) first studied critical metrics of the Dirichlet energy functional in

When are the tangent sphere bundles of a Riemannian manifold η-Einstein?

We study the geometry of a tangent sphere bundle of a Riemannian manifold (M, g). Let M be an n-dimensional Riemannian manifold and TrM be the tangent bundle of M of constant radius r. The main theorem is that TrM equipped with the standard contact metric structure is η-Einstein if and only if M is a space of constant sectional curvature \({\frac{1}{r^2}}\) or \({\frac{n-2}{r^2}}\).

AN INTEGRAL FORMULA AND ITS APPLICATION TO NORMAL EULER NUMBER

We establish a general integral formula over sphere, and then apply it to give a geometrical proof of the celebrated topological theorem of Lashof and Smale, which asserts that the tangential degree of the tangent sphere bundle coincides with the normal Euler number for an immersion Mn → E2n of an oriented closed manifold into Euclidean space of twice dimension.

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

Let ( M , g ) be a compact Riemannian manifold and T 1 M its unit tangent sphere bundle. Unit vector fields defining harmonic maps from ( M , g ) to ( T 1 M , g ˜ s ) , g ˜ s being the Sasaki metric on T 1 M , have been extensively studied. The Sasaki metric, and other well known Riemannian metrics on T 1 M , are particular examples of g-natural metrics. We equip T 1 M 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 1 M , 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.

Some Aspects on the Geometry of the Tangent Bundles and Tangent Sphere Bundles of a Riemannian Manifold

In this paper we study a Riemanian metric on the tangent bundle $T(M)$ of a Riemannian manifold $M$ which generalizes Sasaki metric and Cheeger Gromoll metric and a compatible almost complex structure which together with the metric confers to $T(M)$ a structure of locally conformal almost K\"ahlerian manifold. This is the natural generalization of the well known almost K\"ahlerian structure on $T(M)$. We found conditions under which $T(M)$ is almost K\"ahlerian, locally conformal K\"ahlerian or K\"ahlerian or when $T(M)$ has constant sectional curvature or constant scalar curvature. Then we will restrict to the unit tangent bundle and we find an isometry with the tangent sphere bundle (not necessary unitary) endowed with the restriction of the Sasaki metric from $T(M)$. Moreover, we found that this map preserves also the natural almost contact structures obtained from the almost Hermitian ambient structures on the unit tangent bundle and the tangent sphere bundle, respectively.

Contact metric manifolds with η-parallel torsion tensor

We show that a non-Sasakian contact metric manifold with η-parallel torsion tensor and sectional curvatures of plane sections containing the Reeb vector field different from 1 at some point, is a (k, μ)-contact manifold. In particular for the standard contact metric structure of the tangent sphere bundle the torsion tensor is η-parallel if and only if M is of constant curvature, in which case its associated pseudo-Hermitian structure is CR- integrable. Next we show that if the metric of a non-Sasakian (k, μ)-contact manifold (M, g) is a gradient Ricci soliton, then (M, g) is locally flat in dimension 3, and locally isometric to En+1 × Sn(4) in higher dimensions.

Some results on the geometry of tangent bundle of Finsler manifolds

Abstract. In this paper we study the geometry of tangent bundle of a Finslermanifold endowed with Sasaki metric and obtain some results. We characterize theRiemannian manifold as the Finsler manifold such that the vertical lift of any vectorfield is divergence-free or equivalently, such that the horizontal distribution is minimalin the tangent bundle of the slit tangent bundle, and prove that the almost complexstructure on the slit tangent bundle is integrable if and only if the base manifold haszero flag curvature. In that case, the slit tangent bundle is K¨ahlerian. We also provethat the slit tangent bundle is locally symmetric if and only if the base manifold islocally Euclidean. Our results generalize the corresponding results for the Riemanniansetting in the literature. 1. IntroductionThe geometry of tangent bundle or tangent sphere bundle of a Riemannianmanifold has been well developed. In the general case, however, the geometry oftangent bundle or the indicatrix bundle of a Finsler manifold has not been studiedat the same pace. Several people have made some fundamental contributions tothis subject from various points of view. For instance, Hasegawa, Yamauchi,and Shimada proved that the indicatrix bundle of a Finsler manifold (M,F) withthe induced almost contact metric structure is Sasakian if and only if (M,F) is ofconstant flag curvature 1 [8], while Bejancu and Farran obtained that (M,F)

Eta-Einstein condition on twistor spaces of odd-dimensional Riemannian manifolds

In this note, we find the conditions on an odd-dimensional Riemannian manifolds under which its twistor space is eta-Einstein. This can be used to yield an Einstein metric on the tangent sphere bundle of any 3-dimensional manifold of positive constant curvature.

G-Natural Contact Metrics on Unit Tangent Sphere Bundles

We construct a three-parameter family of contact metric structures on the unit tangent sphere bundle T1M of a Riemannian manifold M and we study some of their special properties related to the Levi-Civita connection. More precisely, we give the necessary and sufficient conditions for a constructed contact metric structure to be K-contact, Sasakian, to satisfy some variational conditions or to define a strongly pseudo-convex CR-structure. The obtained results generalize classical theorems on the standard contact metric structure of T1M.