Tangent Sphere Bundle Research Articles

Notes on the geometry of cotangent bundle and unit cotangent sphere bundle

Let $(\mathtt{N},\mathfrak{g})$ be a Riemannian manifold, by using the musical isomorphisms ♪ and $\natural$ induced by $\mathfrak{g}$, we built a bridge between the geometry of the tangent bundle $\mathtt{TN}$ (resp. the unit tangent sphere bundle $\mathtt{T}_{1}\mathtt{N}$) equipped with the Sasaki metric $\mathfrak{g}_{S}$ (resp. the induced Sasaki metric $\bar{\mathfrak{g}}_{S}$) and that of the cotangent bundle $\mathtt{T}^{\ast}\mathtt{N}$ (resp. the unit cotangent sphere bundle $\mathtt{T}_{1}^{\ast}\mathtt{N}$) endowed with the Sasaki metric $\mathfrak{g}_{\widetilde{S}}$ (resp. the induced Sasaki metric $\tilde{\mathfrak{g}}_{\widetilde{S}}$). Moreover, we prove that $\mathtt{T}_{1}^{\ast}\mathtt{N}$ carries a contact metric structure and study some of its proprieties.

Minimal Legendrian surfaces in the tangent sphere bundle of 𝕊³

In this paper we study minimal Legendrian surfaces Σ \Sigma immersed in tangent sphere bundle T 1 S 3 T_1{\mathbb {S}}^3 . We classify (1) totally geodesic Legendrian surfaces, (2) closed minimal Legendrian surfaces of genus smaller than or equal to one and complete minimal Legendrian surfaces with non-negative Gauss curvature.

Invariant contact metric structures on tangent sphere bundles of compact symmetric spaces

A new characterization is provided for the class of compact rank-one symmetric spaces. Such spaces are the only symmetric spaces of compact type for which the standard vector field xi ^{S} on their sphere bundles is Killing with respect to some invariant Riemannian metric. The set of all these metrics is determined, as well as the set of all those invariant contact metric structures with characteristic vector field xi ^{S}. Moreover, on tangent sphere bundles of compact symmetric spaces with rank greater than or equal to two, a family of invariant contact metric structures, which contains the standard structure, is obtained.

A Schur type lemma for the mean Berwald curvature in Finsler geometry

In this short paper, we study a symmetric covariant tensor in Finsler geometry, which is called the mean Berwald curvature. We first investigate the geometry of the fibres as the submanifolds of the tangent sphere bundle on a Finsler manifold. Then we prove that if the mean Berwald curvature is isotropic along fibres, then the Berwald scalar curvature is constant along fibres.

Magnetic unit vector fields

We show that a unit vector field on an oriented Riemannian manifold is a critical point of the Landau Hall functional if and only if it is a critical point of the Dirichlet energy functional. Therefore, we provide a characterization for a unit vector field to be a magnetic map into its unit tangent sphere bundle. Then, we classify all magnetic left invariant unit vector fields on $3$-dimensional Lie groups.

Sasakian Structures on Tangent Sphere Bundles of Compact Rank-One Symmetric Spaces

A positive answer is given to the existence of Sasakian structures on the tangent sphere bundle of some Riemannian manifold whose sectional curvature is not constant. Among other results, it is proved that the tangent sphere bundle T_{r}(G/K), for any r> 0, of a compact rank-one symmetric space G/K, not necessarily of constant sectional curvature, admits a unique G-invariant K-contact structure whose characteristic vector field is the standard field of T(G/K). Such a structure is in fact Sasakian and it can be expressed as an induced structure from an almost Hermitian structure on the punctured tangent bundle T(G/K){setminus } {text{ zero } text{ section }}.

On the Classifying of the Tangent Sphere Bundle with Almost Contact B-Metric Structure

On the Classifying of the Tangent Sphere Bundle with Almost Contact B-Metric Structure

On the completeness of total spaces of horizontally conformal submersions

Abstract 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 of Cheeger-Gromoll and generalized Cheeger-Gromoll metrics on vector bundle manifolds. Moreover, we study the completeness of a subclass of g-natural metrics on tangent bundles and we extend the results to the case of unit tangent sphere bundles. Our proofs are mainly based on techniques of metric topology and on the Hopf-Rinow theorem.

Unit tangent sphere bundles of conformally flat manifolds

In this paper, we characterize conformally flat Riemannian manifolds with constant scalar curvature by the standard contact metric structure of their unit tangent sphere bundles.

Magnetic trajectories on tangent sphere bundle with g-natural metrics

We study magnetic trajectories in the unit tangent sphere bundle with pseudo-Riemannian g-natural metrics of a Riemannian manifold. A high interest is dedicated to the case when the base manifold is a space form and when the metric is of Kaluza–Klein type. Slant curves are obtained when a certain conservation law is satisfied. We give a complete classification of slant magnetic curves (respectively, geodesics) on $$T_1M$$ , when M is a two-dimensional Riemannian manifold of constant curvature.

Natural SU(2)-structures on tangent sphere bundles

We define and study natural $\mathrm{SU}(2)$-structures, in the sense of Conti-Salamon, on the total space $\cal S$ of the tangent sphere bundle of any given oriented Riemannian 3-manifold $M$. We recur to a fundamental exterior differential system of Riemannian geometry. Essentially, two types of structures arise: the contact-hypo and the non-contact and, for each, we study the conditions for being hypo, nearly-hypo or double-hypo. We discover new double-hypo structures on $S^3\times S^2$, of which the well-known Sasaki-Einstein are a particular case. Hyperbolic geometry examples also appear. In the search of the associated metrics, we find a theorem, useful for explicitly determining the metric, which applies to all $\mathrm{SU}(2)$-structures in general. Within our application to tangent sphere bundles, we discover a whole new class of metrics specific to 3d-geometry. The evolution equations of Conti-Salamon are considered; leading to a new integrable $\mathrm{SU}(3)$-structure on ${\cal S}\times\mathbb{R}_+$ associated to any flat $M$.

A fundamental differential system of Riemannian geometry

We discover a fundamental exterior differential system of Riemannian geometry; indeed, an intrinsic and invariant global system of differential forms of degree n associated to any given oriented Riemannian manifold M of dimension n + 1 . The framework is that of the tangent sphere bundle of M . We generalise to a Riemannian setting some results from the theory of hypersurfaces in flat Euclidean space. We give new applications and examples of the associated Euler–Lagrange differential systems.

Strongly pseudo-convex CR space forms

AbstractFor a contact manifold, we study a strongly pseudo-convex CR space form with constant holomorphic sectional curvature for the Tanaka-Webster connection. We prove that a strongly pseudo-convex CR space formMis weakly locally pseudo-Hermitian symmetric if and only if (i) dimM= 3, (ii)Mis a Sasakian space form, or (iii)Mis locally isometric to the unit tangent sphere bundleT1(𝔿n+1) of a hyperbolic space 𝔿n+1of constant curvature −1.

A Survey of Riemannian Contact Geometry

AbstractThis survey is a presentation of the five lectures on Riemannian contact geometry that the author gave at the conference “RIEMain in Contact”, 18-22 June 2018 in Cagliari, Sardinia. The author was particularly pleased to be asked to give this presentation and appreciated the organizers’ kindness in dedicating the conference to him. Georges Reeb once made the comment that the mere existence of a contact form on a manifold should in some sense “tighten up” the manifold. The statement seemed quite pertinent for a conference that brought together both geometers and topologists working on contact manifolds, whether in terms of “tight” vs. “overtwisted” or whether an associated metric should have some positive curvature. The first section will lay down the basic definitions and examples of the subject of contact metric manifolds. The second section will be a continuation of the first discussing tangent sphere bundles, contact structures on 3-dimensional Lie groups and a brief treatment of submanifolds. Section III will be devoted to the curvature of contact metric manifolds. Section IV will discuss complex contact manifolds and some older style topology. Section V treats curvature functionals and Ricci solitons. A sixth section has been added giving a discussion of the question of whether a Riemannian metric g can be an associated metric for more than one contact structure; at the conference this was an addendum to the third lecture.

Magnetic curves on tangent sphere bundles

In this paper we investigate contact magnetic curves on the unit tangent bundle UM of a Riemannian manifold M and we write the magnetic equations. In the case when M is a space form M(c), it follows that every contact normal magnetic curve is slant when $$c=1$$ , while for $$c\ne 1$$ a contact normal magnetic curve is slant if and only if it satisfies a conservation law. We perform a detailed study of contact magnetic curves in $$U{{\mathbb {S}}}^2$$ .

Pseudo-Einstein unit tangent sphere bundles

In the present paper, we study the pseudo-Hermitian almost CR structure of unit tangent sphere bundle $T_{1}M$ over a Riemannian manifold $M$. Then we prove that if the unit tangent sphere bundle $T_{1}M$ is pseudo-Einstein, that is, the pseudo- Hermitian Ricci tensor is proportional to the Levi form, then the base manifold $M$ is Einstein. Moreover, when $\dim M = 3$ or $4$, we prove that $T_{1}M$ is pseudo-Einstein if and only if $M$ is of constant curvature 1.

Totally geodesic property of the unit tangent sphere bundle with g-natural metrics

Abstract In this paper, we consider the tangent bundle of a Riemannian manifold (M, g) with g-natural metrics and among all of these metrics, we specify those with respect to which the unit tangent sphere bundle with induced g-natural metric is totally geodesic. Also, we equip the unit tangent sphere bundle T1M with g-natural contact (paracontact) metric structures, and we show that such structures are totally geodesic K-contact (K-paracontact) submanifolds of TM, if and only if the base manifold (M, g) has positive (negative) constant sectional curvature. Moreover, we establish a condition for g-natural almost contact B-metric structures on T1Msuch that these structures be totally geodesic submanifolds of TM.

Magnetic curves in tangent sphere bundles II

We study contact magnetic curves in the unit tangent sphere bundle over the Euclidean plane. In particular, we obtain all contact magnetic curves which are slant.

Kaluza–Klein type Ricci solitons on unit tangent sphere bundles

We consider g-natural pseudo-Riemannian metrics of Kaluza–Klein type on the unit tangent sphere bundle of a Riemannian manifold of constant sectional curvature and give necessary and sufficient conditions for these metrics to give rise to a Ricci soliton.On the one hand, we obtain a rigidity result in dimension three, showing that there are no nontrivial Ricci solitons among g-natural metrics of Kaluza–Klein type on the unit tangent sphere bundle of any Riemannian surface. On the other hand, while Ricci solitons determined by tangential lifts remain trivial in arbitrary dimension, horizontal lifts of vector fields related to the geometry of the base manifold (namely, homothetic vector fields) produce nontrivial Ricci solitons metrics of Kaluza–Klein type. Gradient Ricci solitons of Kaluza–Klein type are also completely characterized.

Unit tangent sphere bundles with the Reeb flow invariant Ricci operator

In this paper, we study unit tangent sphere bundles T1M whose Ricci operator $\bar{S}$ is Reeb flow invariant, that is, Lξ$\bar{S}$ = 0. We prove that for a 3-dimensional Riemannian manifold M, T1M satisfies Lξ$\bar{S}$ = 0 if and only if M is of constant curvature 1. Also, we prove that for a 4-dimensional Riemannian manifold M, T1M satisfies Lξ $\bar{S}$ = 0 and ℓ$\bar{S}$ξ = 0 if and only if M is of constant curvature 1 or 2, where ℓ = $\bar{R}$(·,ξ)ξ is the characteristic Jacobi operator.