Kähler manifolds of quasi-constant holomorphic sectional curvature and generalized Sasakian space forms

Abstract

The authors discuss the relation between the generalized Sasakian space forms and the notion of Kähler manifolds of quasi-constant holomorphic sectional curvature, that is, a Kähler manifold endowed with a unitary vector field \(\xi\), such that in each point all holomorphic planes making the same angle with \(\xi\) have the same curvature. On the other hand, a generalized Sasakian space form is an almost contact metric manifold \((M,\phi,\xi,\eta,g)\) sucht that the curvature tensor satisfies \begin{multline*} R(X,Y)Z=f_1\{g(Y,Z)X-g(x,Z)Y\}+f_2\{g(X,\phi Z)\phi Y - g(Y,\phi Z)\phi X +2g(X,\phi Y)\phi Z\}\\ +f_3\{\eta(X)\eta(Z)Y-\eta(Y)\eta(X)Z + g(X,Z)\eta(Y)\xi-g(Y,Z)\eta(X)\xi \}, \end{multline*} for all \(X,Y,Z \in\mathcal{X}(M)\), and for some real functions \(f_i\). In particular the authors show in Proposition \(1\) that a Kähler structure of quasi-constant holomorphic sectional curvature on a cylinder \(M\times\mathbb{R}\) induces a cosymplectic structure on \(M\), which is a generalized Sasakian space form. Also in Proposition 2 they show that a Kähler manifold of quasi-constant holomorphic sectional curvature can be extended (via a warped product) to a generalized Sasakian space form. Finally, they study Einstein hypersurfaces in a Kähler manifolds of quasi-constant holomorphic sectional curvature.