Numerical invariant associated with manifold

Abstract

Let M be a Riemannian manifold of n dimension with the coordinate (x 1, …, xn ). The distance on M are given by first fundamental metrical tensor I = gijdxldxi , where gij will be assume to be analytic function of x1,…, xn and let the distance element in this space be given by second fundamental quadratic form II = Ω ijdxldxi , where Ω. will be assume to be analytic function of x 1, …, xn . In 1929, W.V.D. Hodge introduced the theory of harmonic integral. By using the theory of harmonic integral, he gave the topological definition of geometric genus Pg of a surface. But we have observed that in the theory of harmonic integral, there is no place for second fundamental form of a surface. This motivates us to introduce the new type of differential form by using second fundamental metrical tensor. In this paper, we have introduced the RP-harmonic integral, Modified RP-harmonic integral and Generalized harmonic integral. By using the period matrix corresponding to the RP-harmonic integral, Modified RP-harmonic integral and Generalized harmonic integral, we have studied the numerical invariant of a manifold M. As anologous to geometric genus of a surface, we have defined invariant of a surface, we called as RP-geometric genus Prp and Generalized geometric genus P gh.