Metodología de la investigación en urología

Integral formulas in Riemannian geometry

: The paper deals with the motion of a curved and isolated vortex filament immersed in an infinite, incompressible, and inviscous fluid. The filament is not necessarily plane; Attention is focused on its curvature and its torsion. These quantities vary with the position along the filament and with the time.

General research on the curvature and torsion of an isolated vortex filament.

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.

/pdf/magnetic-vortex-filament-flows-3r9j4ubkrb.pdf

Magnetic vortex filament flows

* Basic Algebraic Notions* Basic Geometrical Notions* Walker Structures* Three-Dimensional Lorentzian Walker Manifolds* Four-Dimensional Walker Manifolds* The Spectral Geometry of the Curvature Tensor* Hermitian Geometry* Special Walker Manifolds

The Geometry of Walker Manifolds

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

Semi{slant submanifolds of a sasakian manifold

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 Gauss-Landau-Hall problem on Riemannian surfaces

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.

https://iopscience.iop.org/article/10.1088/1751-8113/42/19/195201/pdf

The contact magnetic flow in 3D Sasakian manifolds

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.

/pdf/on-the-existence-of-almost-contact-structure-and-the-contact-1jdothezwp.pdf

Manuel García Fernández

Papers

Semi{slant submanifolds of a sasakian manifold

Magnetic vortex filament flows

The Gauss-Landau-Hall problem on Riemannian surfaces

The contact magnetic flow in 3D Sasakian manifolds

On the existence of almost contact structure and the contact magnetic field