D-HOMOTHETIC DEFORMATION OF (κ,μ) MANIFOLD

unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. Abstract. In this paper, we study the invariance of certain curvature conditions in (κ,μ)-contact metric manifold under D-homothetic deformation. Finally we give an example to verify the results.


INTRODUCTION
The class of (κ, µ)-contact metric manifolds encases both Sasakian and non-Sasakian structures. This class of manifolds are invariant under D-homothetic transformation. It is noted that the class of spaces acquired through D-homothetic deformation [13] is a contact metric manifold whose curvature satisfies R(X,Y )ξ = 0. In [13], [14], the authors used D-homothetic deformation on Sasakian and K-contact structures to get results on the first Betti number , second Betti number and harmonic forms. A plane section in the tangent space T p (M) is called a φ -section if there exist a unit vector X in T p (M) orthogonal to ξ such that {X, φ X} is an orthonormal basis of the plane section. Then the sectional curvature K(X, φ X) = g(R(X, φ X)X, φ X) is called a φ -sectional curvature. A contact metric manifold M(φ , ξ , η, g) is said to be of constant φsectional curvature if at any point p ∈ M, the sectional curvature K(X, φ X) is independent of choice of non-zero X ∈ D p , where D denotes the contact distribution of the contact metric manifold defined by η = 0.
The Riemannian curvature tensor R of Sasakian manifold of constant φ -sectional curvature is determined by Ogiue [9]. The geometry of contact Riemannian manfolds of constant φsectional curvature is obtained by Tanno [15]. If the φ -sectional curvature H is constant on K-contact Riemannian manifold M(φ , ξ , η, g) then H can be deformed by a D-homothetic deformation of the structure tensors [16]. An extensive research about D-homothetic deformation on contact geometry is carried out in recent years. The D-homothetic deformation is related to the following tensor structures. In other words, it means that the changing of the tensor form where a is a positive constant. In particular, some authors (Carriazo et al [3]), (De et al [4]) studied D-homothetic deformations of certain structures . An almost contact metric manaifold is said to be η-Einstein if its Ricci tensor S is of the form where α and β are smooth functions on the manifold.
The notion of local symmetry of a Riemannian manifold has been studied by many authors in several ways to different structures. As a weaker version of local symmetry Takahashi [12] introduced the notion of a local φ -symmetry on a Sasakian manifold. Generalizing the notion of a local φ -symmetry of Takahashi [12]. De et al. [6] introduced the idea of φ -recurrent Sasakian manifolds. The notion of a generalized recurrent manifold has been introduced by Dubey [7] and studied by others. Again, the notion of a generalized Ricci recurrent manifold has been introduced and studied by De et. al. [5]. The properties of the extended generalized φ -recurrent β -Kenmotsu, Sasakian and (LCS) 2n+1 -manifolds have been studied in [11], [10] and [18] respectively. Motivated by the above studies, in this paper we characterize the (κ, µ)-contact metric manifolds under D-homothetic deformation. We study the invariance properties of extended generalized φ -recurrent, locally φ -Ricci symmetric (κ, µ) manifolds under D-homothetic deformation. Also η-parallel Ricci tensor is considered in (κ, µ)-contact metric manifolds. Finally, we give an example of such manifold.

PRELIMINARIES
Let M be (2n + 1)-dimensional almost contact metric manifold. Then it carries two fields φ and ξ and a 1-form η. The field φ represents the endomorphism of the tangent spaces, the field ξ is called characteristic vector field and η is a 1-form satisfying for any vector fields X,Y ∈ χ(M). In a contact metric manifold, we characterize a (1, 1) tensor holds. A contact metric manifold with ξ ∈ N(κ, µ) is called a (κ, µ)-contact metric manifold [1]. In a (κ, µ)-contact metric manifold M the following relations hold [1], [2]: where S and r are the Ricci tensor and scalar curvature respectively and Q is the Ricci opertor, i.e., g(QX,Y ) = S(X,Y ).
for any vector fields X, Y , Z on M.
Operatingφ = φ on both sides of above equation from the left, we have, Again, puttingφ X = φ X in (3.2) we have From (3.3) and (3.5) we get Hence the proof.
Hence the proof. Now, we deal with the study of η-parallel (κ, µ)-manifolds under D-homothetic deformation. Proof: Differentiating (3.1) covariantly with respect to W and then using (2.10) and (2.15) we have (4.13) Replacing the vector fields X by φ X and Y by φY in (4.13) and then by using (2.1) and (2.2) we obtain (4.14) Hence the Proof.

EXAMPLE
We consider 3-dimensional manifold M = {(x, y, z) ∈ R 3 }, where (x, y, z) are the standard coordinates in R 3 . Let {E 1 , E 2 , E 3 } be linearly independent global frame on M given by Let η be the 1-form defined by η(V ) = g(V, E 1 ) for any V ∈ χ(M). Let φ be the (1, 1)-tensor Using the linearity of φ and g, we have η( The Riemannian connection ∇ of the metric tensor g is given by Using Koszul's formula, we get the following, (5.1) From (5.1) it can be easily seen that (φ , ξ , η, g) is a (κ, µ) manifold. Next we find the curvature tensor as follows: In view of the expression of the curvature tensor we find the Ricci tensor as follows: Similarly we find S(E 2 , E 2 ) = −4 = S(E 3 , E 3 ). Hence r = −8.
It is well known that in a 3-dimensional manifold, the curvature tensor R satisfies the relation   In view of above relation we get Similarly we have K(E 3 , φ E 3 ) = 2µ + r 2 . Again from (3.1) it can be easily shown that −(1 − a)(3a − 1) Therefore (κ, µ)-manifold satisfies the relation (3.13) and hence Theorem

CONFLICT OF INTERESTS
The author(s) declare that there is no conflict of interests.