For this quarter of century, quantum differential operators in a lower dimensional submanifold embedded or immersed in real n-dimensional euclidean space E n have been studied as physical models, which are realized as restriction of the operators in E n to the submanifold. For this decade, the Dirac operators in the submanifold have been investigated, which are identified with operators of the ...

In this paper we analyse briefly some properties of hemi-slant sub-manifold (LCS)n-manifold. Here discuss about necessary and sufficient conditions for distributions to be integrable obtain results in direction. We also study the geometry leaves submanifold At last give an example a

In this paper, we introduce the notion of probabilistic valued measures as a generalization of non-negative measures and construct the corresponding Lp spaces, for distributions p > "0. It is alsoshown that if the distribution p satises p "1 then, as in the classical case, these spaces are completeprobabilistic normed spaces.

Let M " be a smooth, closed, orientable 2«-manifold and suppose that Kx"~ is an orientable submanifold of M " dual to a cohomology class x . If d is a positive integer, the signatures of Kd"~ and K"~ are related by a numerical congruence. If n is odd, then any codimension 2 submanifold of CP" fixed by a diffeomorphism of odd prime order is dual to the generator of the cohomology algebra.

In this note we consider an arbitrary submanifold C of a Poisson manifold P and ask whether it can be embedded coisotropically in some bigger submanifold of P . We define the classes of submanifolds relevant to the question (coisotropic, Poisson-Dirac, pre-Poisson ones), present an answer to the above question and consider the corresponding picture at the level of Lie groupoids, making concrete...