NTMSCI 4, No. 1, 58-64 (2016) 58 New Trends in Mathematical Sciences http://dx.doi.org/10.20852/ntmsci.2016115613 Real hypersurface of a complex space form H.G.Nagaraja and Savithri Shashidhar Department of Mathematics, Bangalore University, Central College Campus, Bengaluru, India Received: 25 May 2015, Revised: 17 June 2015, Accepted: 17 August 2015 Published online: 6 January 2016 Abstract: The purpose of the present paper is to give characterization of real hypersurface of a complex space form. We find conditions for these hypersurfaces to be φ -symmetric and to have η - parallel curvature tensor. Further we prove totally η -umbilical real hypersurfaces of complex space forms have ξ -parallel Ricci tensor and ξ -parallel structure Jacobi operator. Keywords: Complex space form, real hypersurface, η -parallel, η umbilical, ξ -parallel, structure Jacobi operator. 1 Introduction A complex n(≥ 2) -dimensional K ¨ ahlerian manifold of constant holomorphic sectional curvature c is called a complex space form, which is denoted by M n (c). The induced almost contact metric structure of a real hypersurface M of M n (c) is denoted by (φ , ξ , η , g). In [1]and [2] Berndt has called real hypersurface in M n (c) with the principal vector ξ as Hopf real hypersurface. It can be easily seen that there does not exist any real hypersurface in M n (c), c ̸= 0, which is locally symmetric that is ∇R = 0. This motivates the introduction of the notion of η -parallel curvature tensor (Lee J.G etal [9]). The notion of η -parallel curvature tensor is defined by g((∇ X R)( Y, Z) U, V )= 0 for any X , Y, Z, U and V in a distribution orthogonal to ξ . This notion is weaker to the notion of η -parallel second fundamental tensor g((∇ X A) Y, Z)= 0 for any X , Y and Z. 2 preliminaries Let M n (c) denote the complex space form of complex dimension n with constant holomorphic sectional curvature 4c. Let M be a real (2n − 1)-dimensional hypersurface immersed in M n (c) with parallel almost complex structure J and N be unit normal vector field on M. For any vector field X tangent to M, we define φ , η and ξ by JX = φ X + η (X )N, JN = −ξ , (1) where φ X is the tangential part of JX , φ is a tensor field of a type (1, 1), η is a 1-form, and ξ is the unit vector field on M. Then they satisfy φ 2 X = −X + η (X )ξ , φξ = 0, η (ξ )= 1, g(X , ξ )= η (X ), η (φ X )= 0, g(φ X , φ Y )= g(X , Y ) − η (X )η ( Y ), g(φ X , Y )= −g(X , φ Y ), g(φ X , X )= 0, c ⃝ 2016 BISKA Bilisim Technology ∗ Corresponding author e-mail: hgnraj@yahoo.com