On Complex Sasakian manifolds

TL;DR

This paper introduces complex Sasakian manifolds, exploring their defining properties, curvature relations, and flatness conditions, thereby expanding the understanding of complex contact geometry.

Contribution

It provides a formal definition of complex Sasakian manifolds, analyzes their properties, and investigates curvature relations and flatness conditions.

Findings

Defined complex Sasakian manifolds with complex contact forms

Derived curvature relations for these manifolds

Examined flatness conditions for the curvature tensor B

Abstract

In this article we study a class of normal{\theta}complex{\theta}contact{\theta}metric{\theta}manifold which is called a complex Sasakian manifold. This kind of manifold has a globally defined complex contact form and normal complex contact structure. We give the definition of a complex Sasakian manifold by consider the real case and we present general properties. Also we obtain some useful curvature relations. Finally we examine flatness conditions for general curvature tensor B.