Tangent Sphere Bundle Research Articles

Integrable Systems with Variable Dissipation on the Tangent Bundle of a Sphere

Many problems of multidimensional dynamics involve systems for which the spaces of states are spheres of finite dimension and the spaces of phases are the tangent bundles of such spheres. We study conservative systems and present nonconservative force fields such that the systems involving such forces possess a complete collection of first integrals that are expressed through a finite combination of elementary functions and, in general, are transcendental functions of their variables. Bibliography: 32 titles.

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.

Mean curvature flow and Riemannian submersions

We prove that a sufficient condition ensuring that the mean curvature flow commutes with a Riemannian submersion is that the submersion has minimal fibers. We then lift some results taken from the literature (i.e., Andrews and Baker in J Differ Geom 85:357–395, 2010; Baker in The mean curvature flow of submanifolds of high codimension, 2011; Huisken in J Differ Geom 20:237–266, 1984; Math Z 195:205–219, 1987; Pipoli and Sinestrari in Mean curvature flow of pinched submanifolds of \({\mathbb {CP}}^{n}\)) to create new examples of evolution by mean curvature flow. In particular we consider the evolution of pinched submanifolds of the sphere, of the complex projective space, of the Heisenberg group and of the tangent sphere bundle equipped with the Sasaki metric.

Ricci tensors on unit tangent sphere bundles over 4-dimensional Riemannian manifolds

Ricci tensors on unit tangent sphere bundles over 4-dimensional Riemannian manifolds

The bienergy of unit vector fields

The bienergy of a unit vector field on a Riemannian manifold \((M, g)\) is defined to be the bienergy of the corresponding map \((M, g)\mapsto (T_{1}M, g_{\mathrm{S}})\), where the unit tangent sphere bundle \(T_{1}M\) is equipped with the restriction of the Sasaki metric \(g_{\mathrm{S}}\). The constrained variational problem is studied, where variations are confined to unit vector fields, and the corresponding critical point condition characterizes biharmonic unit vector fields. Finally, we completely determine invariant biharmonic unit vector fields in three-dimensional unimodular Lie groups and the generalized Heisenberg groups \(H(1, r), r\ge 2\).

Paracontact metric structures on the unit tangent sphere bundle

Starting from $g$-natural pseudo-Riemannian metrics of suitable signature on the unit tangent sphere bundle $T_1 M$ of a Riemannian manifold $(M,\langle,\rangle)$, we construct a family of paracontact metric structures. We prove that this class of paracontact metric structures is invariant under $\mathcal D$-homothetic deformations, and classify paraSasakian and paracontact $(\kappa,\mu)$-spaces inside this class. We also present a way to build paracontact $(\kappa,\mu)$-spaces from corresponding contact metric structures on $T_1 M$.

Metrics of Kaluza–Klein type on the anti‐de Sitter space

We introduce and study a new family of pseudo‐Riemannian metrics on the anti‐de Sitter three‐space . These metrics will be called “of Kaluza‐Klein type” , as they are induced in a natural way by the corresponding metrics defined on the tangent sphere bundle . For any choice of three real parameters , the pseudo‐Riemannian manifold is homogeneous. Moreover, we shall introduce and study some natural almost contact and paracontact structures , compatible with , such that is a homogeneous almost contact (respectively, paracontact) metric structure. These structures will be then used to show the existence of a three‐parameter family of homogeneous metric mixed 3‐structures on the anti‐de Sitter three‐space.

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.

Minimal unit vector fields with respect to Riemannian natural metrics

Let (M,g) be a Riemannian manifold. We denote by G˜ an arbitrary Riemannian g-natural metric on the unit tangent sphere bundle T1M, such metric depends on four real parameters satisfying some inequalities. The Sasaki metric, the Cheeger–Gromoll metric and the Kaluza–Klein metrics are special Riemannian g-natural metrics. In literature, minimal unit vector fields have been already investigated, considering T1M equipped with the Sasaki metric G˜S[12]. In this paper we extend such characterization to an arbitrary Riemannian g-natural metric G˜. In particular, the minimality condition with respect to the Sasaki metric G˜S is invariant under a two-parameters deformation of the Sasaki metric. Moreover, we show that a minimal unit vector field (with respect to G˜) corresponds to a minimal submanifold. Then, we give examples of minimal unit vector fields (with respect to G˜). In particular, we get that the Hopf vector fields of the unit sphere, the Reeb vector field of a K-contact manifold, and the Hopf vector field of a quasi-umbilical hypersurface with constant principal curvatures in a Kähler manifold, are minimal unit vector fields (with respect to G˜).

ALMOST PARACONTACT STRUCTURES ON TANGENT SPHERE BUNDLE

Using the almost product structure given by Druta, we introduce a metrical framed f(3, -1)-structure on the tangent bundle of a Riemannian manifold. Then by restricting this metrical framed f(3, -1)-structure to the tangent sphere bundle, we obtain an almost metrical paracontact structure on the tangent sphere bundle.

Variations of gwistor space

We study natural variations of the \mathrm{G}_2 structure \sigma_0\in\Lambda^3_+ existing on the unit tangent sphere bundle SM of any oriented Riemannian 4-manifold M . We find a circle of structures for which the induced metric is the usual one, the so-called Sasaki metric, and prove how the original structure has a preferred role in the theory. We deduce the equations of calibration and cocalibration, as well as those of W_3 pure type and nearly-parallel type.

Sasaki–Einstein and paraSasaki–Einstein metrics from [formula omitted]-structures

We prove that every contact metric (κ,μ)-space admits a canonical η-Einstein Sasakian or η-Einstein paraSasakian metric. An explicit expression for the curvature tensor fields of those metrics is given and we find the values of κ and μ for which such metrics are Sasaki–Einstein and paraSasaki–Einstein. Conversely, we prove that, under some natural assumptions, a K-contact or K-paracontact manifold foliated by two mutually orthogonal, totally geodesic Legendre foliations admits a contact metric (κ,μ)-structure. Furthermore, we apply the above results to the geometry of tangent sphere bundles and we discuss some geometric properties of (κ,μ)-spaces related to the existence of Einstein–Weyl and Lorentzian–Sasaki–Einstein structures.

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.

Levi Harmonic Maps of Contact Riemannian Manifolds

We study Levi harmonic maps, i.e., C∞ solutions f:M→M′ to \(\tau_{\mathcal{H}} (f) \equiv \operatorname{trace}_{g} ( \varPi_{\mathcal{H}}\beta_{f} ) = 0\), where (M,η,g) is an (almost) contact (semi) Riemannian manifold, M′ is a (semi) Riemannian manifold, βf is the second fundamental form of f, and \(\varPi_{\mathcal{H}} \beta_{f}\) is the restriction of βf to the Levi distribution \({\mathcal{H}} = \operatorname{Ker}(\eta)\). Many examples are exhibited, e.g., the Hopf vector field on the unit sphere S2n+1, immersions of Brieskorn spheres, and the geodesic flow of the tangent sphere bundle over a Riemannian manifold of constant curvature 1 are Levi harmonic maps. A CR map f of contact (semi) Riemannian manifolds (with spacelike Reeb fields) is pseudoharmonic if and only if f is Levi harmonic. We give a variational interpretation of Levi harmonicity. Any Levi harmonic morphism is shown to be a Levi harmonic map.

Geometry of Kaluza–Klein metrics on the sphere $${\mathbb{S}^3}$$

In this paper, we introduce a new family of Riemannian metrics \({\tilde g_{\lambda\mu\nu}}\) on the three-sphere and study its geometric properties, starting from the description of their curvature. Such metrics, which include the standard metric g0 and Berger metrics on \({\mathbb{S}^3}\) as special cases, are called “of Kaluza–Klein type”, because they are induced in a natural way by the corresponding metrics defined on the tangent sphere bundle \({T_1 \mathbb{S}^2(\kappa)}\). Each sphere \({(\mathbb{S}^3, \tilde g_{\lambda\mu\nu})}\) is a homogeneous space, and we obtain a full classification of its homogeneous structures. Moreover, we introduce and study a natural almost contact structure \({(\varphi,\xi,\eta)}\), for which \({(\varphi,\xi,\eta,\tilde g_{\lambda\mu\nu})}\) is a (homogeneous) almost contact metric structure on the three-sphere. Finally, we see that for a suitable family of Kaluza–Klein type metrics \({\tilde g_{ac}}\) on \({\mathbb{S}^3}\), it is possible to construct a two-parameter family of harmonic morphisms from \({(\mathbb{S}^3, \tilde g_{ac})}\) to \({\mathbb{S}^2(\kappa)}\).

$g$-natural metrics of constant curvature on unit tangent sphere bundles

We completely classify Riemannian $g$-natural metrics of constant sectional curvature on the unit tangent sphere bundle $T_1 M$ of a Riemannian manifold $(M,g)$. Since the base manifold $M$ turns out to be necessarily two-dimensional, weaker curvature conditions are also investigated for a Riemannian $g$-natural metric on the unit tangent sphere bundle of a Riemannian surface.

Infinitesimal isometries on tangent sphere bundles over three-dimensional manifolds

In this article, we study the infinitesimal isometries on tangent sphere bundles over orientable three-dimensional Riemannian manifolds. Focusing on the vector fields which do not preserve fibers, we show the existence of lifts to the bundles of orthonormal frames. These lifts enable us to analyze the infinitesimal isometries by the symmetry of principal fiber bundles. We prove that the tangent sphere bundle admits a non-fiber-preserving infinitesimal isometry if and only if the base manifold has the same constant sectional curvatures as the fibers have. As an application, we classify the infinitesimal isometries on tangent sphere bundles for the three dimensional case.

H-CONTACT UNIT TANGENT SPHERE BUNDLES OF FOUR-DIMENSIONAL RIEMANNIAN MANIFOLDS

AbstractWe study the geometric properties of a base manifold whose unit tangent sphere bundle, equipped with the standard contact metric structure, is H-contact. We prove that a necessary and sufficient condition for the unit tangent sphere bundle of a four-dimensional Riemannian manifold to be H-contact is that the base manifold is 2-stein.

Geodesic restrictions for the Casimir operator

We consider the hyperbolic Casimir operator C defined on the tangent sphere bundle SY of a compact hyperbolic Riemann surface Y. We prove a non-trivial bound on the L 2 -norm of the restriction of eigenfunctions of C to certain natural hypersurfaces in SY. The result that we obtain goes beyond known (sharp) local bounds of L. Hörmander.

Tangent sphere bundles with constant trace of the Jacobi operator

We consider the unit tangent sphere bundle T1M of a Riemannian manifold (M, g) equipped with the canonical contact metric structure \({(\eta, \xi, \phi, \bar{g})}\) . We study the geometric properties of the base manifold M under the assumption that the trace of the Jacobi operator with respect to the characteristic vector field ξ is constant. Moreover, we give examples of Einstein Riemannian manifolds, other than rank one symmetric spaces or globally Ossermann manifolds, the unit tangent sphere bundles of which have constant trace of the Jacobi operator. Finally, we extend our main results to the tangent sphere bundle TrM of constant radius r > 0.