Manuel García Fernández

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.

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.

Bosonic string theory in Lorentzian principal circle bundles over anti-de-Sitter space AdS3

Abstract In this note we show a bosonic conformal string theory on U (1)-bundles over AdS 3 . To this end, we first look for r -elastic helices in the space AdS 3 , because they generate solutions of the motion equation on backgrounds P which are principal circle-bundles over AdS 3 endowed with the standard metric or generalized Kaluza–Klein metrics. In fact, we reduce the search of U (1)-symmetric string configurations on P to the search of r -elastic curves in the orbit space AdS 3 .

REDUCTION OF VARIABLES FOR WILLMORE-CHEN SUBMANIFOLDS IN SEVEN SPHERES

We obtain a reduction of variables criterion for 4-dimensional Willmore-Chen submanifolds associated with the generalized Kaluza-Klein conformal structures on the 7-sphere. This argument connects the variational problem of Willmore-Chen with a variational problem for closed curves into 4-spheres. It involves an elastic energy functional with potential. The method is based on the extrinsic conformal invariance of the Willmore-Chen variational problem, and the principle of symmetric criticality. It also uses several techniques from the theory of pseudo-Riemannian submersions. Furthermore, we give some applications, in particular, a result of existence for constant mean curvature Willmore-Chen submanifolds which is essentially supported on the nice geometry of closed helices in the standard 3-sphere.