Abstract. In this paper, we show new results on slant submanifolds of an almost contact metric manifold. We study and characterize slant submanifolds of Kcontact and Sasakian manifolds. We also study the special class of three-dimensional slant submanifolds. We give several examples of slant submanifolds.1991 Mathematics Subject Classi®cation. 53C15, 53C40.0. Introduction. Slant immersions in complex geometry were de®ned by B.-Y. Chen as a natural generalization of both holomorphic immersions and totally real immersions [2]. Examples of slant immersions into complex Euclidean spaces C 2 and C 4 were given by Chen and Tazawa [2, 4, 5], while slant immersions of KaÈ hler Cspaces 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 a Riemannian manifold into an almost contact metric manifold and he has proved some properties of such immersions. A. Lotta and A. M. Pastore have obtained examples of slant submanifolds in the Sasakian-space-form R 2m 1 as the leaves of a harmonic Riemannian 3-dimensional foliation [8]. Finally, A. Lotta has also studied some properties about the intrinsic geometry of 3-dimensional non-anti-invariant slant submanifolds of K-contact manifolds [6].The purpose of the present paper is to study slant immersions in K-contact and Sasakian manifolds. We ®rst review, in Section 1, basic formulas and de®nitions for almost contact metric manifolds and their submanifolds, which we shall use later. In Section 2, we recall the de®nition of a slant submanifold of an almost contact metric manifold and we show a ®rst characterization theorem. In Section 3, we give many interesting examples of slant submanifolds in almost contact metric manifolds and in Sasakian manifolds. Then, we characterize slant submanifolds by means of the covariant derivative of the square of the tangent projection T over the submanifold of the almost contact structure of a K-contact manifold. Later, we study the ®rst interesting class of slant submanifolds: the three-dimensional slant submanifolds. We show some results concerning the tangent T and the normal N projections. We also use the given examples in order to remark some facts concerning the main theorems of the paper. We study slant submanifolds in K-contact manifolds and threedimensional slant submanifolds in Sections 4 and 5 respectively.

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.

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, LaTe

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 magnetics 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.

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 pseudoisotropy and then to study the rigidity of this class of immersion into the pseudoEuclidean 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.