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

At the 1958 International Congress of Mathematicians in Edinburgh, René Thom received one of two Field Medals for his development of cobordism. In his citation [11] Heinz Hopf described the definition of cobordism as one of those elementary and apparently trivial constructions which can hardly be expected to yield significant results. He compares this with Hurewicz’s definition of homotopy grou...

Part I introduced the notion of the projective linking number LinkP(Γ, Z) of a compact oriented real submanifold Γ of dimension 2p− 1 in complex projective n-space P with an algebraic subvariety Z ⊂ P−Γ of codimension p. It is shown here that a basic conjecture concerning the projective hull of real curves in P implies the following result: Γ is the boundary of a positive holomorphic p-chain of...

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.