On Sectional Curvatures of (ε)-Sasakian Manifolds

The index of a metric plays significant roles in differential geometry as it generates variety of vector fields such as space-like, time-like, and light-like fileds. With the help of these vector fields, we establish interesting properties on ( )-Sasakian manifolds, which was introduced by Bejancu and Duggal [1] and further investigated by Xufeng and Xiaoli [2]. Since Sasakian manifolds with indefinite metrics play crucial roles in physics [3], hence the study of these manifolds becomes the central theme in present scenario. Here the next section is concerned with the basic results of Riemannian curvature tensor of ( )-Sasakian manifolds. In Section 3, these results will be used to obtain the equivalent relations among φ-sectional curvature, totally real sectional curvature, and totally real bisectional curvature. In [1], authors defined the ( )-Sasakian manifold as follows. LetM be a real (2n+1)-dimensional differentiable manifold endowed with an almost contact structure (φ,η,ξ), where φ is a tensor field of type (1,1), η is a 1-form, and ξ is a vector field onM satisfying