Numerical invariant associated with manifold

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.


Introduction
Let be a Riemannian manifold of dimension with the coordinate ( 1 , ⋯ , ). The distance on are given by first fundamental metrical tensor = 1 ,where will be assume to be analytic function of 1 , ⋯ , and let the distance element in this space be given by second fundamental quadratic form , where Ω will be assume to be analytic function of 1 , ⋯ , .
Definition 1.1. The properties of manifold which is invariant under homeomorphism is called as topological invariant of a manifold .

Definition 1.2. A numerical invariant of a manifold
is a number associated with which has the same value for any manifold homeomorphic to .
From the definition 1.2, numerical invariant of a manifold is also a topological property of . A complete set of invariant determines a manifold upto homeomorphism. Many mathematician studied the topological properties of a manifold. But till date no decent complete set of invariants for a manifold is known.
In [9], Oswald Veblen studied the topological properties of a manifold by using simplical complex. In [6] De Rham studied the topological properties of a manifold by using the theory of integration. In the investigation of the properties of a manifold by using the theory of integral,certain topological invariants such as the torsion group of a manifold can't be explain by using the theory of integration on a manifold. To study the certain topological invariant of a manifold, W.V.D.Hodge introduced the theory of harmonic integral by using first fundamental metrical tensor [2]. The role of second fundamental metrical tensor is absent in the theory of harmonic integral [7,10,11]. This statement motivate us to introduced the generalised theory of harmonic integral which consider first fundamental metrical tensor and second fundamental metrical tensor.
The paper is organised as follows: In section 2, we have studied the properties of cycle on a manifold. In section 3, we have studied the properties of differential form on a manifold . In section 4, we have studied the period of integral on a manifold . In section 5,we have studied harmonic integral on a manifold. In section 6, we have studied RP-harmonic integral on a manifold. In section 7, we have studied modified RP-harmonic integral on a manifold. In section 8, we have studied generalised harmonic integral on a manifold. In section 9, we have studied the period matrix of harmonic form, RPharmonic form, Generalized harmonic form. In section 10, we have studied the numerical invariant of a manifold.

Remark 1.3.
We have assumed that be a manifold of dimension with metric given by equation 1 and equation 2. Remark 1.4. denote the set of real numbers, denote the set of complex numbers, denote the set of rational numbers, denote the set of integers, denotes set of natural numbers.

Cycle on a manifold Let
= {( 1 , ⋯ , ): ∈ } be the Euclidean space of dimension. Let 0 , ⋯ , are + 1independent points in such that = ( 1 , ⋯ , ). The rectilinear p-simplex or p-simplex or a rectilinear simplex of p dimensions is denoted by ( 0 ⋯ ) and it is defined as follows: The closure of rectilinear simplex of p-dimensions ( 0 ⋯ ) is denoted by [ 0 ⋯ ] and it is defined as follows: The boundary of the p-simplex ( 0 ⋯ ) is the difference between ( 0 ⋯ ) and [ 0 ⋯ ]. Therefore, the boundary of p-simplex consists of ( p + 1 k + 1 ) simplexes of k dimension, where = 0, ⋯ , − 1. The polyhedral complex of dimension is denoted by K n and it is a finite collection of rectilinear simplexes in ξ of dimension 0, ⋯ , such that (i) no two simplexes of the set have a point in common; (ii) every simplex lying on the boundary of a simplex of the set belongs to the set. (ii) There are two classes of derangements which are obtain by even permutation of (0, ⋯ , ) and odd permutation of (0, ⋯ , ). (iii) An oriented simplex of ( 0 ⋯ ) is denoted by 0 ⋯ and it is obtained by associating one of the class of derangements of the suffixes to ( 0 ⋯ ) and for the other class, we denote it by − 0 ⋯ .
(iv) If we orient all the simplexes of a complex in an arbitrary manner, we get an oriented simplex. 3. An integral -chain whose boundary is zero is called as integral -cycle.

Definition 2.4.
A real -chain whose boundary is zero is called as real -cycle.

Definition 2.5.
A -chain on a field whose boundary is zero is called as -cycle on .
We know that every -cycle is the boundary of ( + 1)-chain not generally true. One can classify the cycle into two parts. If be a cycle which is a boundary of +1 chain then is called as bounding cycles and we say that is homologous to zero and we denote it as ~0.
If and ′ are two -chains such that − ′ is bounding cycle then − ′~0 . We denote it by ~′ and we call it as is homologous to ′ .

Betti Number and torsion coefficients
Let ( ) is an additive group of integral -chain of . Let ,1 ( ) is a subgroup of ( ) of bounding -cycle [3]. Then = ( ) ,1 ( ) form a group which is generated by free generators. The group is direct sum of and , where is infinite abelian group with free generator and is a finite abelian group generated by generator of the orders 1 , ⋯ , . The group is called as betti group of and the group is called as the ℎ torsion group of and 1 , ⋯ , are called as the ℎ torsion coefficient of .

Differential form on a manifold.
In this section,we have considered the basic theory of differential form in order to make this paper complete. For more detail (see [3]). If = 1 ! 1 ⋯ 1 ⋯ be any -form on a manifold of dimensions then + 1 form of is denoted by and it is called as the derived form or the exterior derivative of and it is obtain by using Stokes theorem which is expressed by using following equation: A -form is called as closed form on if and only if = 0.
Then the period of integral on the cycle Γ is defined as If the -cycle Γ is a bounding cycle then period of the integral of over Γ is zero.  If ϕ is a closed form on a manifold whose integral has all its periods equal to zero then ϕ is a null form.

Harmonic integral on a manifold
From the theorem(4.4), one can observe that ℎ betti number is equal to maximum number of closed form which are linearly independent. Hence by using theorem ,where will be assume to be analytic function of 1 , ⋯ , and let the distance element in this space be given by second fundamental quadratic form , where Ω will be assume to be analytic function of 1 , ⋯ , . Let = is determined uniquely by and the metrical tensor .Then form * is called the dual of the form . Definition 5.4. The tensor defined by the coefficient of a harmonic form is called as harmonic tensor.

RP-Harmonic integral on a manifold
From definition 5.2, we can observed that harmonic form on a manifold considered only first fundamental metrical tensor. There is no place given to second fundamental metrical tensor. To study the numerical invariant of a manifold, we have introduced new type of differential form, we called it as RP-harmonic form. ,where will be assume to be analytic function of 1 , ⋯ , and let the distance element in this space be given by second fundamental quadratic form , where Ω will be assume to be analytic function of 1 , ⋯ , . Let ,where will be assume to be analytic function of 1 , ⋯ , and let the distance element in this space be given by second fundamental quadratic form

Generalized Harmonic integral on a manifold
From section 6 and section 7, we can observed that harmonic form on a manifold considered only first fundamental metrical tensor and RP-harmonic form on a manifold considered only second fundamental metrical tensor. There is no place given to any tensor associated with a manifold . To study the numerical invariant of a manifold, we have introduced new type of differential form, we called it as Generalized Harmonic integral on a manifold. ,where will be assume to be analytic function of 1 , ⋯ , and let the distance element in this space be given by second fundamental quadratic form , where Ω will be assume to be analytic function of 1 , ⋯ , . Let be a tensor associated with . . By using the definition of homeomorphism, we must have rank of is equal to rank of W GH 2 .
Hence rank of period matrix = [ ], where = ∫ Γ is a numerical invariant of a manifold.

Geometric genus of a surface as a topological invariant by using harmonic integral
Let Hence, ∑ =1 = ∫ ϕ ϕ ′ = ∫ ϕ ϕ ′ = ∑ =1 . Therefore, we get = ′ , where ′ is a transpose of . . − ] is a positive definite form. Therefore is transformed to a diagonal matrix ′ , it has β negative terms. The number β is an invariant of the matrix . Since β = 2 + 1.
Hence is a topological invariant of a manifold. Hence is a topological invariant of a manifold.
Hence ℎ is a topological invariant of a manifold.

Conclusion
We know that there are no decent set of invariant to classify the manifold. 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