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 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...

Recently there has been a lot of interest in geometrically motivated approaches to data analysis in high dimensional spaces. We consider the case where data is drawn from sampling a probability distribution that has support on or near a submanifold of Euclidean space. We show how to “learn” the homology of the submanifold with high confidence. We discuss an algorithm to do this and provide lear...

It is well known that diffeomorphism groups play an important role in certain areas of mathematical physics, such as quantitative fluid dynamics and plasma physics, gauge theories and general Hamiltonian theory. Let (M, g) be a compact oriented Riemannian manifold and μ the corresponding volume element. Denote by D = Diff(M) the diffeomorphism group of M , completed in the r-Sobolev topology, a...

We prove that distortion of a knotted curve in R is great than 4.76. This improves a result obtained by John M. Sullivan and Elizabeth Denne in [DS]. 1 The definition and results M. Gromov introduced the notion of distortion for submanifolds of R in [G], p.113-114. In the case of a simple closed curve K (closed 1-dimensional submanifold of R) distortion is U(K) = max P,Q∈K |PQ|K |PQ| , where |P...

A link is a smooth, oriented submanifold L = {Kx, . . . , Km} of S which is the ordered disjoint union of m manifolds each piecewise-linearly homeomorphic to the «-sphere (if m = 1, L is called a knot). Knots and links play an essential role in the classification of manifolds and, in this regard, perhaps the most important equivalence relation on links is that of link concordance. LQ and L{ are...

Special Lagrangian submanifolds of complex Euclidean space C have been studied widely over the last few years. These submanifolds are volume minimizing and, in particular, they are minimal submanifolds. When n = 2, special Lagrangian surfaces of C are exactly complex surfaces with respect to another orthogonal complex structure on R ≡ C. A very important problem here, and even a good starting p...