The submanifold Dirac operator has been studied for this decade, which is closely related to Frenet-Serret and generalized Weierstrass relations. In this article, we will give a submanifold Dirac operator defined over a surface immersed in E with U(1)-gauge field as torsion in the sense of the Frenet-Serret relation, which also has data of immersion of the surface in E. MSC2000: 34L40, 53A05, 5...

The standard text-book Jacobi equation (equation of geodesic deviation) arises by linearizing the geodesic equation around some chosen geodesic, where the linearization is done with respect to the coordinates and the velocities. The generalized Jacobi equation, introduced by Hodgkinson in 1972 and further developed by Mashhoon and others, arises if the linearization is done only with respect to...

Using the submanifold quantum mechanical scheme, the restricted Dirac operator in a submanifold is defined. Then it is shown that the zero mode of the Dirac operator expresses the local properties of the submanifold, such as the Frenet-Serret and generalized Weierstrass relations. In other words this article gives a representation of a further generalized Weierstrass relations for a general k-s...

We examine some peculiarities of the subset of lattice preserving elements in a pseudo-Euclidean group, when the lattice under consideration contains a lightlike vector, or more generally, when the restriction of a pseudo-Euclidean metric to the real linear enveloppe of the lattice is not definite. For the case of a Lorentzian metric, it is shown in detail that the isometry group of the spaceti...

We introduce the notion of a weakly reflective submanifold, which is an austere submanifold with a certain global condition, and study its fundamental properties. Using these, we determine weakly reflective orbits and austere orbits of s-representations.

The lensing properties of superconducting cosmic strings endowed with a time dependent pulse of lightlike current are investigated. The metric outside the core of the string belongs to the pp–wave class, with a deficit angle. We study the field theoretic bosonic Witten model coupled to gravity, and we show that the full metric (both outside and inside the core) is a Taub-KerrShild generalizatio...

Abstract We construct the exponential map associated to a nonholonomic system that allows us define an exact discrete constraint submanifold. reproduce continuous flow as on this submanifold deriving version of equations. Finally, we derive general family integrators includes particular case trajectory.