On complex Sasakian manifolds
- PDF / 281,747 Bytes
- 10 Pages / 439.37 x 666.142 pts Page_size
- 78 Downloads / 205 Views
On complex Sasakian manifolds Aysel Turgut Vanlı1
· ˙Inan Ünal2
· Keziban Avcu1
Received: 17 November 2019 / Accepted: 19 September 2020 © African Mathematical Union and Springer-Verlag GmbH Deutschland, ein Teil von Springer Nature 2020
Abstract In this article, we study a class of normal complex contact metric manifold called a complex Sasakian manifolds. This kind of manifolds have a globally defined complex contact form and normal complex contact structure. We define a complex Sasakian manifold by considering 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. Keywords Complex Sasakian manifolds · Complex contact manifolds · Curvature tensors Mathematics Subject Classification 53C15 · 53C25 · 53D10
1 Introduction Complex contact manifolds still have many open problems. This subject’s importance is not only the complex version of real contact manifolds; also one can find many important informations about complex manifolds, Kähler manifolds. Besides, there are some applications in theoretical physics [19]. Although, complex contact manifolds have an old history like real contact manifolds, researchers could not give their attention to the subject. When we look at the 1980s, there are significant improvements in the Riemannian geometry of complex contact manifolds. Ishihara and Konishi constructed tensorial relations for a complex almost contact structure and presented normality [15–17]. The Riemann geometry of complex almost contact metric manifolds could be divided into three notions; 1. IK-normal complex contact metric manifolds : A complex contact manifold has a normal complex contact structure in the sense of Ishihara-Konishi. These type of manifolds were studied in [13–17,27].
B
Aysel Turgut Vanlı [email protected] ˙Inan Ünal [email protected]
1
Department of Mathematics, Gazi University, Ankara, Turkey
2
Department of Computer Engineering, Munzur University, Tunceli, Turkey
123
A. Turgut Vanli et al.
2. Normal complex contact metric manifolds: A complex contact structure in the sense of Korkmaz. These type of manifolds were studied in [2,3,5–7,20,21,23–26]. 3. Complex Sasakian manifolds: A complex contact manifold with a globally complex contact form and has a normal complex contact structure in the sense of Korkmaz. These type of manifolds were studied in [9,10,12]. In this work, we study on the third type of these manifolds. Firstly, we adopted a complex Sasakian manifold by considering the definition of a real Sasakian manifold. Later we give some fundamental equations and we obtain curvature properties. Finally, we examine some flatness conditions. We use a general tensor, which is defined in [22] and is called by B−tensor. We prove that a complex Sasakian manifold could not be B−flat.
2 Preliminaries In this section, we give some fundamental facts on complex contact manifolds. For details, the reader could be read [4,11,20]. Definition 1 Let N be a complex manifold
Data Loading...
