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.     

A note on hypersurfaces in a sphere

We study the influence of a unit Killing vector field on the geometry of a hypersurface in the unit sphere. The combination of the Killing vector field on the hypersurface and the conformal vector field on the ambient sphere triggers the presence of four specific smooth functions on the hypersurface, we use these four functions to derive different sufficient conditions for a hypersurface to be the totally geodesic sphere and for a minimal hypersurface to be the totally geodesic sphere, Clifford minimal hypersurface respectively. In particular we classify compact minimal hypersurfaces with a unit Killing vector field in the unit sphere.     

The energy of Hopf vector fields

The 3-dimensional Hopf vector field is shown to be a stable harmonic section of the unit tangent bundle. In contrast, higher dimensional Hopf vector fields are unstable harmonic sections; indeed, there is a natural variation through smooth unit vector fields which is locally energy-decreasing, and whose asymptotic limit is a singular vector field of finite energy. This energy is explicitly calculated, and conjectured to be the infimum of the energy functional over all smooth unit vector fields. Received: 17 March 1999     

Geodesic vector fields and Eikonal equation on a Riemannian manifold

In this paper, we study the impact of geodesic vector fields (vector fields whose trajectories are geodesics) on the geometry of a Riemannian manifold. Since, Killing vector fields of constant lengths on a Riemannian manifold are geodesic vector fields, leads to the question of finding sufficient conditions for a geodesic vector field to be Killing. In this paper, we show that a lower bound on the Ricci curvature of the Riemannian manifold in the direction of geodesic vector field gives a sufficient condition for the geodesic vector field to be Killing. Also, we use a geodesic vector field on a 3-dimensional complete simply connected Riemannian manifold to find sufficient conditions to be isometric to a 3-sphere. We find a characterization of an Einstein manifold using a Killing vector field. Finally, it has been observed that a major source of geodesic vector fields is provided by solutions of Eikonal equations on a Riemannian manifold and we obtain a characterization of the Euclidean space using an Eikonal equation.     

A critical radius for unit Hopf vector fields on spheres

The volume of a unit vector field V of the sphere (n odd) is the volume of its image V() in the unit tangent bundle. Unit Hopf vector fields, that is, unit vector fields that are tangent to the fibre of a Hopf fibration are well known to be critical for the volume functional. Moreover, Gluck and Ziller proved that these fields achieve the minimum of the volume if n = 3 and they opened the question of whether this result would be true for all odd dimensional spheres. It was shown to be inaccurate on spheres of radius one. Indeed, Pedersen exhibited smooth vector fields on the unit sphere with less volume than Hopf vector fields for a dimension greater than five. In this article, we consider the situation for any odd dimensional spheres, but not necessarily of radius one. We show that the stability of the Hopf field actually depends on radius, instability occurs precisely if and only if In particular, the Hopf field cannot be minimum in this range. On the contrary, for r small, a computation shows that the volume of vector fields built by Pedersen is greater than the volume of the Hopf one thus, in this case, the Hopf vector field remains a candidate to be a minimizer. We then study the asymptotic behaviour of the volume; for small r it is ruled by the first term of the Taylor expansion of the volume. We call this term the twisting of the vector field. The lower this term is, the lower the volume of the vector field is for small r. It turns out that unit Hopf vector fields are absolute minima of the twisting. This fact, together with the stability result, gives two positive arguments in favour of the Gluck and Ziller conjecture for small r.     

Energy of Radial Vector Fields on Compact Rank One Symmetric Spaces

We consider the energy (or the total bending) of unit vector fields oncompact Riemannian manifolds for which the set of its singularitiesconsists of a finite number of isolated points and a finite number ofpairwise disjoint closed submanifolds. We determine lower bounds for theenergy of such vector fields on general compact Riemannian manifolds andin particular on compact rank one symmetric spaces. For this last classof spaces, we compute explicit expressions for the total bending whenthe unit vector field is the gradient field of the distance function toa point or to special totally geodesic submanifolds (i.e., for radialunit vector fields around this point or these submanifolds).     

Conformal vector fields on Finsler spaces

Here, it is shown that every vector field on a Finsler space which keeps geodesic circles invariant is conformal. A necessary and sufficient condition for a vector field to keep geodesic circles invariant, known as concircular vector fields, is obtained. This leads to a significant definition of concircular vector fields on a Finsler space. Finally, complete Finsler spaces admitting a special conformal vector field are classified.     

Almost contact metric manifolds whose Reeb vector field is a harmonic section

We investigate almost contact metric manifolds whose Reeb vector field is a harmonic unit vector field, equivalently a harmonic section. We first consider an arbitrary Riemannian manifold and characterize the harmonicity of a unit vector field ??, when ??? is symmetric, in terms of Ricci curvature. Then, we show that for the class of locally conformal almost cosymplectic manifolds whose Reeb vector field ?? is geodesic, ?? is a harmonic section if and only if it is an eigenvector of the Ricci operator. Moreover, we build a large class of locally conformal almost cosymplectic manifolds whose Reeb vector field is a harmonic section. Finally, we exhibit several classes of almost contact metric manifolds where the associated almost contact metric structures ?? are harmonic sections, in the sense of Vergara-Diaz and Wood?[25], and in some cases they are also harmonic maps.     

局部对称流形上的数量曲率

本文讨论了无共轭点测地线上的Ｊａｃｏｂｉ声，证明了具非负数量曲率的局部对称的无共轭点流形及具非负数量曲率的具极点的局部对称的流形之数量曲率只能是零。部分解决了Ｅ．Ｈｏｐｆ猜想。     

Hopf hypersurfaces in pseudo-Riemannian complex and para-complex space forms

The study of real hypersurfaces in pseudo-Riemannian complex space forms and para-complex space forms, which are the pseudo-Riemannian generalizations of the complex space forms, is addressed. It is proved that there are no umbilic hypersurfaces, nor real hypersurfaces with parallel shape operator in such spaces. Denoting by J be the complex or para-complex structure of a pseudo-complex or para-complex space form respectively, a non-degenerate hypersurface of such space with unit normal vector field N is said to be Hopf if the tangent vector field JN is a principal direction. It is proved that if a hypersurface is Hopf, then the corresponding principal curvature (the Hopf curvature) is constant. It is also observed that in some cases a Hopf hypersurface must be, locally, a tube over a complex (or para-complex) submanifold, thus generalizing previous results of Cecil, Ryan and Montiel.     

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. Supported by funds of the University of Lecce and M.I.U.R.(PRIN).     

Almost Contact Lagrangian Submanifolds of Nearly Kaehler 6-Sphere

For a Lagrangian submanifold M of S ⁶ with nearly Kaehler structure, we provide conditions for a canonically induced almost contact metric structure on M by a unit vector field, to be Sasakian. Assuming M contact metric, we show that it is Sasakian if and only if the second fundamental form annihilates the Reeb vector field ξ, furthermore, if the Sasakian submanifold M is parallel along ξ, then it is the totally geodesic 3-sphere. We conclude with a condition that reduces the normal canonical almost contact metric structure on M to Sasakian or cosymplectic structure.     

Special isoparametric orbits in Riemannian symmetric spaces

In the present paper orbits of isotropy subgroups in Riemannian symmetric spaces are discussed. Principal orbits of an isotropy subgroup are isoparametric in the sense of Palais and Terng (seeCritical Point Theory and Submanifold Geometry, Springer-Verlag, Berlin, 1988). We show that excepting some special cases, the shape operator with respect to the radial unit vector field determines a totally geodesic foliation on a given principal orbit. Furthermore, we prove that the shape operators and the curvature endomorphisms with respect to the normal vectors commute on these isoparametric submanifolds.     

Line integration of Ricci curvature and conjugate points in Lorentzian and Riemannian manifolds

Generalizing results of Cohn-Vossen and Gromoll, Meyer for Riemannian manifolds and Hawking and Penrose for Lorentzian manifolds, we use Morse index theory techniques to show that if the integral of the Ricci curvature of the tangent vector field of a complete geodesic in a Riemannian manifold or of a complete nonspacelike geodesic in a Lorentzian manifold is positive, then the geodesic contains a pair of conjugate points. Applications are given to geodesic incompleteness theorems for Lorentzian manifolds, the end structure of complete noncompact Riemannian manifolds, and the geodesic flow of compact Riemannian manifolds.Partially supported by NSF grant MCS77-18723(02).     

Indice local pour des ensembles symétriques par arcs

Let M be a complete Riemannian manifold with sectional curvature and dimension . Given a unit vector and a point we prove the existence of a complete geodesic through x whose tangent vector never comes close to v. As a consequence we show the existence of a bounded geodesic through every point in a complete negatively pinched manifold with finite volume and dimension . Received April 13, 1998; in final form July 23, 1999 / Published online October 11, 2000     

Forms of certain hopf algebras

Let Ψ be a field, G a finite group of automorphisms of Ψ, and Φ the fixed field of G. Let H be a Hopf algebra over Ψ. For g ∈ G we define a Hopf algebra H_g which has the same underlying vector space as H and modified operations and show that the tensor product (over Ψ) ?_{g ∈ G} H_g has a Φ-form. As a consequence we see that if n>0 is an integer and Φ is a field of characteristic zero or p>0 with (n,p)=1, then there is a finite dimensional Hopf algebra over Φ with antipode of order 2n.     

A new discrete Hopf–Rinow theorem

We prove a version of the Hopf–Rinow theorem with respect to path metrics on discrete spaces. The novel aspect is that we do not a priori assume local finiteness but isolate a local finiteness type condition, called essentially locally finite, that is indeed necessary. As a side product we identify the maximal weight, called the geodesic weight, generating the path metric in the situation when the space is complete with respect to any of the equivalent notions of completeness proven in the Hopf–Rinow theorem. As an application we characterize the graphs for which the resistance metric is a path metric induced by the graph structure.     

Vector bundles over projective spaces. The case $${\mathbb F_1}$$

Over the field of one element, vector bundles over n-dimensional projective spaces are considered. It is shown that all line bundles are tensor powers of the Hopf bundle and all vector bundles are direct sums of line bundles. This is in complete analogy to the case of the projective line over an arbitrary classical field, but drastically simpler in comparison with projective spaces of higher dimensions.