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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.

The family of all the submanifolds of a given Riemannian or pseudo-Riemannian manifold is large enough to classify them into some interesting subfamilies such as minimal (maximal), totally geodesic, Einstein, etc. Most of these have been extensively studied by many authors, but as far as we know, no paper has hitherto been published on the class of isotropic submanifolds. The purpose of this paper is therefore to gain a better understanding of this interesting class of submanifolds that arise naturally in mathematics and physics by studying their relationships with other closely distinguished families.

We give a simple proof of the existence of an almost contact metric structure on any orientable 3-dimensional Riemannian manifold (M3, g) with the prescribed metric g as the adapted metric of the almost contact metric structure. By using the key formula for the structure tensor obtained in the proof this theorem, we give an application which allows us to completely determine the magnetic flow of the contact magnetic field in any 3-dimensional Sasakian manifold.

Several notions of isotropy of a (pseudo-)Riemannian manifold have been introduced in the literature, in particular, the concept of pseudo-isotropic immersion. The aim of this paper is to look more closely at this notion of pseudo-isotropy and then to study the rigidity of this class of immersion into the pseudo-Euclidean space. It is worth pointing out that we first obtain a characterization of the pseudo-isotropy condition by using tangent vectors of any causal character. Then, rigidity theorems for pseudo-isotropic immersions are proved, and in particular, some well known results for the Riemannian case arise. Later, we bring together the notions of pseudo-isotropy, intrinsically and extrinsically isotropic manifolds, and prove interesting relations among them. Finally, we pay special attention to the case of codimension two Lorentz surfaces.

In the present work, we briefly summarize our contributions to the study of CR submanifolds of (locally conformal almost) Kaehler manifolds and normal CR submanifolds of S-manifolds.

In this note we use the Hopf map … : S 3 ! S 2 to construct an in- teresting family of Riemannian metrics h f on the 3-sphere, which are parametrized on the space of positive smooth functions f on the 2-sphere. All these metrics make the Hopf map a Riemannian submersion. The Hopf tube over an immersed curve ∞ in S 2 is the complete lift … i1 (∞) of ∞, and we prove that any geodesic of this Hopf tube satisfles a Clairaut relation. A Hopf tube plays the role in S 3 of the surfaces of revolution in R 3 . Furthermore, we show a geometric integration method of the Frenet equations for curves in those non-standard S 3 . Finally, if we consider the sphere S 3 equipped with a family h f of Lorentzian metrics, then a new Clairaut relation is also obtained for timelike geodesics of the Lorentzian Hopf tube, and a geometric integration method for curves is still possible.

STRUCTURE ON A SLANT SUBMANIFOLD OF A CONTACT MANIFOLD

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 .

We obtain a reduction of the symmetry holographic principle for symmetric configurations of Nambu–Goto–Polyakov string theories in a semi-Riemannian space. The argument reduces the search of string configurations with a certain degree of symmetry to that for elastic curves in a corresponding orbit space. These solutions are solitons which are holographically related to particles that evolve along elastic worldlines in the orbit space. We also exhibit examples and applications to obtain soliton string shapes with cylindrical, rotational, toroidal etc. symmetry. In most of the cases we can determine the whole moduli space of symmetric solitons.

The principle of symmetric criticality allows one to reduce the search of symmetric critical points for the conformal total tension functional in a Kaluza‐Klein conformal universe, to the search of closed curves in its gravitatory component which are critical points for certain r-elastic energy functionals. The constancy of the mean curvature is preserved in this reduction of symmetry. In this framework we study ther-elasticity of fibres in a semi-Riemannian warped tube. We obtain a wide class of tubes which are foliated by r-elastic circles and give some applications including a characterization of the photon sphere in a Schwarzschild spacetime from the point of view of the r-elasticity. © 2000 Elsevier Science B.V. All rights reserved.

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.

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.

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.

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.

0. Introduction. Many authors have studied the geometry of sub-manifolds of Kaehlerian and Sasakian manifolds. On the other hand, DavidE. Blair has initiated the study of S-manifolds, which reduce, in particularcases, to Sasakian manifolds ([1, 2]).I. Mihai ([8]) and L. Ornea ([9]) have investigated CR-submanifolds ofS-manifolds. The purpose of the present paper is to study a special kind ofsuch submanifolds, namely the normal CR-submanifolds.In Sections 1 and 2, we review basic formulas and definitions for sub-manifolds in Riemannian manifolds and in S-manifolds, respectively, whichwe shall use later. In Section 3, we introduce normal CR-submanifolds ofS-manifolds and we study some properties of their geometry. Finally, in Sec-tion 4, we consider those submanifolds in the case of the ambient S-manifoldbeing an S-space form.1. Preliminaries. Let N be a Riemannian manifold of dimension nand M an m-dimensional submanifold of N. Let g be the metric tensor fieldon N as well as the induced metric on M. We denote by ∇ the covariantdifferentiation in N and by ∇ the covariant differentiation in M determinedby the induced metric. Let T(N) (resp. T(M)) be the Lie algebra of vectorfields in N (resp. in M) and T(M)

Hyperglycemia is a frequent observation in the acute coronary syndrome. We analyzed the relationship between hyperglycemia on admission and patients with acute coronary syndrome.Prospective study of 455 patients with acute coronary syndrome with and without elevation of ST segment with high risk according to ACA/AHA criteria. We divided the sample according to the median glycemia on admission into139 mg/dl and ≥ 139 mg/dl. We studied the analytic, electrocardiography, echocardiography and epidemiologic variables. Using the Cox Proportional Hazard Model, we analyzed their relationship with the mortality as principal variable during a six-month period after the acute coronary syndrome.Mean age was 64.3 ± 12.7 years, 80.4% were male and 21.8% had been diagnosed with diabetes. Mean glycemia on admission was 163.3 ± 71.8 mg/dl. Forty-seven patients died (10.3%), Mean glycemia of those who had died was 189.8 ± 78.8 mg/dl compared to 160.3 ± 70.4 mg/dl in the survival group (P = 0.003). Patients with hyperglycemia on admission ≥ 139 mg/dl had higher mortality, hazard ratio (HR) =2.98 (confidence interval [CI 95%]: 1.06-8.4; P = 0.039). Elderly patients, being a male, having ventricular dysfunction and initial decrease of blood pressure also showed an independent relationship with mortality.Hyperglycemia on admission ≥ 139 mg/dl in acute coronary syndrome patients is associated with a higher risk of death in the following six months, independently of diabetes or other risk factors known.

Hyperammonemia causes several alterations, mainly in the central nervous system. If hepatic failure is not its etiology, other less frequent causes must be investigated in the search for a definitive diagnosis.We report the case of a 16 year old patient admitted for acute encephalopathy and hyperammonemia. After analysis, brain CT, ultrasound and abdominal Doppler, we began empirical treatment of hyperammonemia secondary to disorders of the urea cycle. We treated the brain edema and eliminated ammonia but we did not obtain favourable results and the patient died four days later.The complex management of hyperammonemia and the high morbidity and mortality involved require a multidisciplinary approach. Only early treatment and identification of the hyperammonemia's etiology can avoid high morbidity and mortality in these patients.

In this paper we define the contact number of a pseudo-Riemannian submanifold into the pseudo-Euclidean space, and prove that this contact number is closely related to the notion of pseudo-isotropic submanifold. We give a classification of hypersurfaces into the pseudo-Euclidean space with contact number at least 3. A classification of the complete spacelike codimension-2 submanifolds of the Lorentz-Minkowski space with contact number at least 3 is also obtained.

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. Ministerio de Educación y Ciencia Fondo Europeo de Desarrollo Regional Junta de Andalucía

In this note we use the Hopf map to construct a family of metrics in the 3-sphere parametrized on the space of positive smooth functions in the 2-sphere. All these metrics make the Hopf map a Riemannian submersion. Also, the fibres are all geodesics if and only if the metric comes from a constant function and so, we have a Berger 3-sphere. Every geodesic in a 3-dimensional Riemannian manifold is a minimum for each elastic energy functional. Therefore, we characterize those func- tions on the 2-sphere that locally give metrics which have all the fibres being elastica, i.e., critical points of those functionals. Some applications are given including one to the Willmore-Chen variational problem.

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.

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.

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 Schrödinger equation showing the solitonic nature of those. Ministerio de Educación y Ciencia Fondo Europeo de Desarrollo Regional Junta de Andalucía

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 correspond with a limit case obtained when the force of the Gauss-Landau-Hall increases arbitrarily. We also obtain new properties related with the completeness of flowlines for a general magnetic fields. The paper also contains new 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., Comment: 20 pages, LaTeX

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