Hopf Vector Fields Research Articles

Gauss maps of harmonic and minimal great circle fibrations

We investigate Gauss maps associated to great circle fibrations of the euclidean unit 3-sphere mathbb {S}^3. We show that the associated Gauss map to such a fibration is harmonic, respectively minimal, if and only if the unit vector field generating the great circle foliation is harmonic, respectively minimal. These results can be viewed as analogues of the classical theorem of Ruh and Vilms about the harmonicity of the Gauss map of a minimal submanifold in the euclidean space. Moreover, we prove that a harmonic or minimal unit vector field on mathbb {S}^3, whose integral curves are great circles, is a Hopf vector field.

Area-minimizing properties of Pansu spheres in the sub-Riemannian 3-sphere

Abstract We consider the sub-Riemannian 3-sphere ( 𝕊 3 , g h ) {(\mathbb{S}^{3},g_{h})} obtained by restriction of the Riemannian metric of constant curvature 1 to the planar distribution orthogonal to the vertical Hopf vector field. It was shown in [A. Hurtado and C. Rosales, Area-stationary surfaces inside the sub-Riemannian three-sphere, Math. Ann. 340 2008, 3, 675–708] that ( 𝕊 3 , g h ) {(\mathbb{S}^{3},g_{h})} contains a family of spherical surfaces { 𝒮 λ } λ ⩾ 0 {\{\mathcal{S}_{\lambda}\}_{\lambda\geqslant 0}} with constant mean curvature λ. In this work, we first prove that the two closed half-spheres of 𝒮 0 {\mathcal{S}_{0}} with boundary C 0 = { 0 } × 𝕊 1 {C_{0}=\{0\}\times\mathbb{S}^{1}} minimize the sub-Riemannian area among compact C 1 {C^{1}} surfaces with the same boundary. We also see that the only C 2 {C^{2}} solutions to this Plateau problem are vertical translations of such half-spheres. Second, we establish that the closed 3-ball enclosed by a sphere 𝒮 λ {\mathcal{S}_{\lambda}} with λ > 0 {\lambda>0} uniquely solves the isoperimetric problem in ( 𝕊 3 , g h ) {(\mathbb{S}^{3},g_{h})} for C 1 {C^{1}} sets inside a vertical solid tube and containing a horizontal section of the tube. The proofs mainly rely on calibration arguments.

Volume minimising unit vector fields on three dimensional space forms of positive curvature

We prove that on a closed 3-manifold of constant positive curvature the unit vector fields of minimum volume are exactly the Hopf vector fields.

Extended Fuller index, sky catastrophes and the Seifert conjecture

We extend the classical Fuller index, and use this to prove that for a certain general class of vector fields [Formula: see text] on a compact smooth manifold, if a homotopy of smooth non-singular vector fields starting at [Formula: see text] has no sky catastrophes as defined by the paper, then the time 1 limit of the homotopy has periodic orbits. This class of vector fields includes the Hopf vector field on [Formula: see text]. A sky catastrophe is a kind of bifurcation originally discovered by Fuller. This answers a natural question that existed since the time of Fuller’s foundational papers. We also put strong constraints on the kind of sky-catastrophes that may appear for homotopies of Reeb vector fields.

Hopf magnetic curves in the anti-de Sitter space ℍ13

We consider the anti-de Sitter space ℍ13 and the hyperbolic Hopf fibration h : ℍ13(1)→ℍ2(1/2). Using their description in terms of paraquaternions, we study the magnetic curves of the hyperbolic Hopf vector field. A complete classification is obtained for light-like magnetic curves, showing in particular the existence of periodic examples, and emphasizing their relationship with the hyperbolic Hopf fibration. Finally, we give a new interpretation of magnetic curves in ℍ13 using some techniques of Lie groups and Lie algebras.

Biconservative surfaces in BCV‐spaces

AbstractBiconservative hypersurfaces are hypersurfaces with conservative stress‐energy tensor with respect to the bienergy functional, and form a geometrically interesting family which includes that of biharmonic hypersurfaces. In this paper we study biconservative surfaces in the 3‐dimensional Bianchi–Cartan–Vranceanu spaces, obtaining their characterization in the following cases: when they form a constant angle with the Hopf vector field; when they are SO(2)‐invariant.

A theorem about energy and volume of vector fields with the “proportional volume property”

In this paper, we define a certain proportional volume property for a unit vector field on a spherical domain in S 3 . We prove that the volume of these vector fields has an absolute minimum, and that this value is equal to the volume of the Hopf vector field. Some examples of such vector fields are given. We also study the minimum energy of solenoidal vector fields which coincides with a Hopf flow along the boundary of a spherical domain of S 2 k + 1 .

Unit vector fields of minimum energy on quotients of spheres and stability of the Reeb vector field

In this paper we show that, in contrast with the situation for the sphere S2n+1(κ), on a quotient S2n+1(κ)/Γ, Γ≠{Id}, the unit Hopf vector fields are the unique unit vector fields which minimize the energy. Moreover, we show that such vector fields are stable critical points with respect to the volume functional. Then, we study the stability question, with respect to energy and volume, of the Reeb vector field for H-contact, K-contact manifolds, Sasakian space forms and 3-Sasakian manifolds.

Helix surfaces in the Berger sphere

We characterize helix surfaces in the Berger sphere, that is surfaces which form a constant angle with the Hopf vector field. In particular, we show that, locally, a helix surface is determined by a suitable 1-parameter family of isometries of the Berger sphere and by a geodesic of a 2-torus in the 3-dimensional sphere.

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˜).

Lipschitz minimality of Hopf fibrations and Hopf vector fields

Given a Hopf fibration of a round sphere by parallel great subspheres, we prove that the projection map to the base space is, up to isometries of domain and range, the unique Lipschitz constant minimizer in its homotopy class. Similarly, given a Hopf fibration of a round sphere by parallel great circles, we view a unit vector field tangent to the fibres as a cross-section of the unit tangent bundle of the sphere, and prove that it is, up to isometries of domain and range, the unique Lipschitz constant minimizer in its homotopy class. Previous attempts to find a mathematical sense in which Hopf fibrations and Hopf vector fields are optimal have met with limited success.

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.

Real hypersurfaces with constant principal curvatures in complex space forms

We present the motivation and current state of the classification problem of real hypersurfaces with constant principal curvatures in complex space forms. In particular, we explain the classification result of real hypersurfaces with constant principal curvatures in nonflat complex space forms and whose Hopf vector field has nontrivial projection onto two eigenspaces of the shape operator. This constitutes the following natural step after Kimura and Berndtʼs classifications of Hopf real hypersurfaces with constant principal curvatures in complex space forms.

Non-Hopf real hypersurfaces with constant principal curvatures in complex space forms

We classify real hypersurfaces in complex space forms with constant principal curvatures and whose Hopf vector field has two nontrivial projections onto the principal curvature spaces. In complex projective spaces such real hypersurfaces do not exist. In complex hyperbolic spaces these are holomorphically congruent to open parts of tubes around the ruled minimal submanifolds with totally real normal bundle introduced by Berndt and Bruck. In particular, they are open parts of homogenous ones. © Indiana University Mathematics Journal.

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.

Instability of Hopf vector fields on Lorentzian Berger spheres

In this work, we study the stability of Hopf vector fields on Lorentzian Berger spheres as critical points of the energy, the volume and the generalized energy. In order to do so, we construct a family of vector fields using the simultaneous eigenfunctions of the Laplacian and of the vertical Laplacian of the sphere. The Hessians of the functionals are negative when they act on these particular vector fields and then Hopf vector fields are unstable. Moreover, we use this technique to study some of the open problems in the Riemannian case.

Stability of unit Hopf vector fields on quotients of spheres

The volume of a unit vector field V of a Riemannian manifold ( M , g ) is the volume of its image V ( M ) in the unit tangent bundle endowed with the Sasaki metric. Unit Hopf vector fields, that is, unit vector fields that are tangent to the fiber of a Hopf fibration S n → C P n − 1 2 ( n odd) are well known to be critical for the volume functional on the round n-dimensional sphere S n ( r ) for every radius r > 1 . Regarding the Hessian, it turns out that its positivity actually depends on the radius. Indeed, in Borrelli and Gil-Medrano (2006) [2], it is proven that for n ⩾ 5 there is a critical radius r c = 1 n − 4 such that Hopf vector fields are stable if and only if r ⩽ r c . In this paper we consider the question of the existence of a critical radius for space forms M n ( c ) ( n odd) of positive curvature c. These space forms are isometric quotients S n ( r ) / Γ of round spheres and naturally carry a unit Hopf vector field which is critical for the volume functional. We prove that r c = + ∞ , unless Γ is trivial. So, in contrast with the situation for the sphere, the Hopf field is stable on S n ( r ) / Γ , Γ ≠ { Id } , whatever the radius.

HARMONIC SECTIONS OF TANGENT BUNDLES EQUIPPED WITH RIEMANNIAN g-NATURAL METRICS

Let $(M,g)$ be a Riemannian manifold. When $M$ is compact and the tangent bundle $TM$ is equipped with the Sasaki metric $g^s$, the only vector fields which define harmonic maps from $(M,g)$ to $(TM,g^s)$, are the parallel ones. The Sasaki metric, and other well known Riemannian metrics on $TM$, are particular examples of $g$-natural metrics. We equip $TM$ with an arbitrary Riemannian $g$-natural metric $G$, and investigate the harmonicity of a vector field $V$ of $M$, thought as a map from $(M,g)$ to $(TM,G)$. We then apply this study to the Reeb vector field and, in particular, to Hopf vector fields on odd-dimensional spheres.

LEFT-INVARIANT MINIMAL UNIT VECTOR FIELDS ON A LIE GROUP OF CONSTANT NEGATIVE SECTIONAL CURVATURE

Abstract. We find all left-invariant minimal unit vector fields and stro-ngly normal unit vector fields on a Lie group which is isometric to thehyperbolic space. 1. IntroductionA smooth unit vector field on a Riemannian manifold ( M,g ) is a cross sectionof its unit sphere bundle T 1 ( M ) and hence can be viewed as a submanifold of T 1 ( M ). If the manifold M is compact and T 1 ( M ) is equipped with a naturalRiemannian metric g s called the Sasaki metric, then the volume of the unitvector field is defined as the volume of this submanifold.For the problem of determining unit vector fields which have minimal vol-ume, Gluck and Ziller showed that the unit vector fields of minimal volume on S 3 are precisely the Hopf vector fields and no others ([7]). But in the higherdimensional spheres, S 2 n +1 ,k ≥ 2, this is not the case ([4], [8], [10]).The problem of finding unit vector fields of the minimum volume seems tobe very difficult, so it is natural to consider the problem of finding the criticalvalues or critical points of the volume functional.Gil-Medriano and Llinares-Fuster proved that a unit vector field is a criticalpoint of the volume functional if and only if the corresponding immersion in(

Stability of the Reeb vector field of H-contact manifolds

It is well known that a Hopf vector field on the unit sphere S 2n+1 is the Reeb vector field of a natural Sasakian structure on S 2n+1. A contact metric manifold whose Reeb vector field ξ is a harmonic vector field is called an H-contact manifold. Sasakian and K-contact manifolds, generalized (k, μ)-spaces and contact metric three-manifolds with ξ strongly normal, are H-contact manifolds. In this paper we study, in dimension three, the stability with respect to the energy of the Reeb vector field ξ for such special classes of H-contact manifolds (and with respect to the volume when ξ is also minimal) in terms of Webster scalar curvature. Finally, we extend for the Reeb vector field of a compact K-contact (2n+1)-manifold the obtained results for the Hopf vector fields to minimize the energy functional with mean curvature correction.