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 projective space of Rn has a natural open subset: the set of lines spanned by vectors with all coordinates > 0. Such a subset can be defined more generally for any partial flag manifold of a split semisimple real algebraic group. The main result of the paper is that this subset can be defined by algebraic equalities and inequalities. Let G be a simply connected semisimple algebraic group ov...

In this paper we prove that there do not exist warped product GCR-lightlike submanifolds in the form M = N⊥ ×λ NT such that N⊥ is an anti-invariant submanifold tangent to V and NT an invariant submanifold of M̄ , other than GCR-lightlike product in an indefinite cosymplectic manifold. We also obtain some characterizations for a GCR-lightlike submanifold to be locally a GCR-lightlike warped produ...

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 generalize a Bernstein-type result due to Albujer and Alı́as, for maximal surfaces in a curved Lorentzian product 3-manifold of the form Σ1 ×R, to higher dimension and codimension. We consider M a complete spacelike graphic submanifold with parallel mean curvature, defined by a map f : Σ1 → Σ2 between two Riemannian manifolds (Σ1 ,g1) and (Σ n 2,g2) of sectional curvatures K1 and K2, respecti...

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.

In this paper, we study lightlike submanifolds of indefinite Kaehler manifolds. We introduce a class of lightlike submanifold called semi-invariant lightlike submanifold. We consider lightlike submanifold with respect to a quarter-symmetric non metric connection which is determined by the complex structure. We give some equivalent conditions for integrability of distributions with respect to th...

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.

Let $M$ be an orientable hypersurface in the Euclidean space $R^{2n}$ with induced metric $g$ and $TM$ be its tangent bundle. It is known that the tangent bundle $TM$ has induced metric $overline{g}$ as submanifold of the Euclidean space $R^{4n}$ which is not a natural metric in the sense that the submersion $pi :(TM,overline{g})rightarrow (M,g)$ is not the Riemannian submersion. In this paper...