José L. Cabrerizo

SLANT SUBMANIFOLDS IN SASAKIAN MANIFOLDS

Abstract. In this paper, we show new results on slant submanifolds of analmost contact metric manifold. We study and characterize slant submanifolds of K-contact and Sasakian manifolds. We also study the special class of three-dimen-sional slant submanifolds. We give several examples of slant submanifolds.1991 Mathematics Subject Classification. 53C15, 53C40.0. Introduction. Slant immersions in complex geometry were defined by B.-Y.Chen as a natural generalization of both holomorphic immersions and totally realimmersions [2]. Examples of slant immersions into complex Euclidean spaces C 2 andC 4 were given by Chen and Tazawa [2, 4, 5], while slant immersions of Ka¨hler C-spaces into complex projective spaces were given by Maeda, Ohnita and Udagawa[9].In a recent paper [7], A. Lotta has introduced the notion of slant immersion of aRiemannian manifold into an almost contact metric manifold and he has provedsome properties of such immersions. A. Lotta and A. M. Pastore have obtainedexamples of slant submanifolds in the Sasakian-space-form R

Semi-Slant Submanifolds of a Sasakian Manifold

We define and study both bi-slant and semi-slant submanifolds of an almost contact metric manifold and, in particular, of a Sasakian manifold. We prove a characterization theorem for semi-slant submanifolds and we obtain integrability conditions for the distributions which are involved in the definition of such submanifolds. We also study an interesting particular class of semi-slant submanifolds.

Magnetic vortex filament flows

We exhibit a variational approach to study the magnetic flow associated with a Killing magnetic field in dimension 3. In this context, the solutions of the Lorentz force equation are viewed as Kirchhoff elastic rods and conversely. This provides an amazing connection between two apparently unrelated physical models and, in particular, it ties the classical elastic theory with the Hall effect. Then, these magnetic flows can be regarded as vortex filament flows within the localized induction approximation. The Hasimoto transformation can be used to see the magnetic trajectories as solutions of the cubic nonlinear Schrodinger equation showing the solitonic nature of those.

The Gauss-Landau-Hall problem on Riemannian surfaces

We introduce the notion of Gauss-Landau-Hall magnetic field on a Riemannian surface. The corresponding Landau-Hall problem is shown to be equivalent to the dynamics of a massive boson. This allows one to view that problem as a globally stated, variational one. In this framework, flowlines appear as critical points of an action with density depending on the proper acceleration. Moreover, we can study global stability of flowlines. In this equivalence, the massless particle model corresponds with a limit case obtained when the force of the Gauss-Landau-Hall magnetic field increases arbitrarily. We also obtain properties related with the completeness of flowlines for general magnetic fields. The paper also contains results relative to the Landau-Hall problem associated with a uniform magnetic field. For example, we characterize those revolution surfaces whose parallels are all normal flowlines of a uniform magnetic field.

The Structure of a Class of K-contact Manifolds

In this paper we study a class of K-contact manifolds, namely φ-conformally flat K-contact manifolds and we show that a compact φ-conformally flat K-contact manifold with regular contact vector field is a principal S1-bundle over an almost Kaehler space of constant holomorphic sectional curvature.

The contact magnetic flow in 3D Sasakian manifolds

We first present a geometrical approach to magnetic fields in three-dimensional Riemannian manifolds, because this particular dimension allows one to easily tie vector fields and 2-forms. When the vector field is divergence free, it defines a magnetic field on the manifold whose Lorentz force equation presents a simple and useful form. In particular, for any three-dimensional Sasakian manifold the contact magnetic field is studied and the normal magnetic trajectories are determined. As an application, we consider the three-dimensional unit sphere, where we prove the existence of closed magnetic trajectories of the contact magnetic field, and that this magnetic flow is quantized in the set of rational numbers.

The curvature of submanifolds of anS-space form

Many authors have studied the geometry of submanifolds of Kaehlerian and Sasakian manifolds. On the other hand, David E. Blair has initiated the study of S-manifolds, which reduce, in particular cases, to Sasakian manifolds [1]. I. Mihai [7] and Ornea [8] have studied CR-submanifolds of S-manifolds. The purpose of the present paper is to investigate some properties ofinvariant and anti-invariant submanifolds of an S-manifold whose invariant f-sectional curvature is constant, that is, of an S-space form. Specifically, those ones related with the curvature tensor fields and with the scalar curvature on the submanifold. In Section 1 we review basic formulas for submanifolds in Riemannian manifolds and, in Section 2, for S-manifolds. In Sections 3 and 4 we study anti-invariant and invariant submanifolds, respectively, of an S-space form. Finally, in the last section we give some examples.

Magnetic fields in 2D and 3D sphere

In this note we study the Landau–Hall problem in the 2D and 3D unit sphere, that is, the motion of a charged particle in the presence of a static magnetic field. The magnetic flow is completely determined for any Riemannian surface of constant Gauss curvature, in particular, the unit 2D sphere. For the 3D case we consider Killing magnetic fields on the unit sphere, and we show that the magnetic flowlines are helices with the given Killing vector field as its axis.

Isotropy and Marginally Trapped Surfaces in a Spacetime

In this paper we shall study the notions of an isotropic and marginally trapped surface in a spacetime by using a differential geometric approach. We first consider spacelike isotropic surfaces in a Lorentzian manifold and, in particular, in a four-dimensional spacetime, where the isotropy function appears to be determined by the mean curvature vector field of the surface. Explicit examples of isotropic marginally outer trapped surfaces are given in the standard four-dimensional space forms: Minkowski, de Sitter and anti-de Sitter spaces. Then we prove rigidity theorems for complete spacelike 0-isotropic surfaces without flat points in these standard space forms. As a consequence, we also obtain characterizations of complete spacelike isotropic marginally trapped surfaces in these backgrounds.

Massless particles in three-dimensional Lorentzian warped products

The model of a massless relativistic particle with curvature-dependent Lagrangian is well known in (d+1)-dimensional Minkowski space. For other gravitational fields less rigid than those with constant (zero) curvature only a few results are known. In this paper, we give a geometric approach in order to solve the field equations associated with that Lagrangian in the setting of an interesting three-dimensional background, namely, a three-dimensional warped product with Lorentzian fibers. When some rigidity conditions are imposed to the fiber (constant Gauss curvature), the trajectories can be totally described. Several examples help us clarify this.