Sasaki-Kenmotsu manifolds

In the present paper, we introduce a new class of structures on an even dimensional differentiable Riemannian manifold which combines, well known in literature, the Sasakian and Kenmotsu structures simultaneously. The structure will be called a Sasaki-Kenmotsu structure by us. Firstly, we discuss the normality of the Sasaki-Kenmotsu structure and give some basic properties. Secondly, we present some important results concerning with the curvatures of the Sasaki-Kenmotsu manifold. Finally, we show the existence of the Sasaki-Kenmotsu structure by giving some concrete examples.