Ricci operator in a Hopf real Hypersurfaces of complex space form

We initiate the study of real hypersurface of a complex space form and it is proved that Lie parallelism of the Ricci operator in the direction of ξ, locally symmetric Ricci operator in a real hypersurface of a complex space form reduce the hypersurface to Hopf hypersurface.


Introduction
Let M n (c) denote an n-dimensional Kaeherian manifold of constant holomorphic sectional curvature c is called a complex space form. An M be a real hypersurface in M n (c). In the event that Aξ = αξ is contented, then the structure vector field ξ of M could be a principal vector and M is named a Hopf hypersurface of M n (c), wherever A means the shape operator of M and α = g(Aξ, ξ). Complex space form could be a projective space P n (c), a Euclidean space c n or a hyperbolic space H n (c), according to c > 0, c = 0 or c < 0. Real hypersufaces of complex space forms are studied widely several authors like [1] - [14] and numerous others. In this paper, we tend to prove conditions in terms of symmetry, Lie parallelism of the Ricci operator for a hypersurface to be a Hopf hypersurface. The paper is organized as follows: The section 2, contains preliminaries of real hypersurface of a complex space form. In section 3, we tend to considered Lie ξ-parallel, locally symmetric and obtained conditions for real hypersurface to be a Hopf hypersurface. Further we tend to derived expressions for constant holomorphic sectional curvature c.

Preliminaries
Let M be a real hypersurface complex n-dimensional complex space form M n (c). We give basic formulas, The Gauss and Weingarten formulas the followings: From equations of Gauss and Codazzi: where the Ricci operator Q is characterized by Let Π be the open subset of M characterized by where α = η(Aξ). We put where W is a unit vector orthogonal to ξ and µ does not vanish on Π.

Hopf real hypersurface of a complex space form
In here consider Lie ξ-parallel, locally symmetric and obtained conditions for M to be a Hopf hypersurface. Further we derived expressions for c.
The hypersurface M is a Hopf hypersurface for constant h = traceA or holomorphic curvature c satisfies (23).

Conclusions
Consider ξ-Ricci-semi-symmetric, cyclic parallel, Lie-ξ-parallel and locally symmetric hypersurface of a complex space form and obtained conditions for real hypersurface to be a Hopf hypersurface. The constant holomorphic sectional curvature c is given explicitly interms of trace of shape operator.