Karamanoğlu Mehmetbey Üniversitesi Mühendislik ve Doğa Bilimleri Dergisi

Öz

Contact manifolds have many applications to medical science, technology, geometric optics, geometric quantization, control theory, thermodynamics, and classical mechanics. Therefore, studies on the Riemann geometry of contact manifolds are important. On the other hand, one of the important tools of Riemann geometry are curvature tensors. By using curvature tensors, some geometric properties and physical applications of contact manifolds can be examined. Especially, important results are obtained on symmetry of manifolds which is the one of the main study topics of Riemannian geometry. A special kind of curvature tensor is the (0,2) -type $\mathcal{Z}$-tensor which has some geometric properties different from Ricci curvature tensor. This type of tensor gives us important results on contact manifolds. Especially, semi-symmetry conditions which are related to $\mathcal{Z}$-tensor present nice results. In this study, we work on$N(k)$- contact metric manifolds which are a special kind of contact manifolds. We present some results on $N(k)$-contact metric manifolds by using$\mathcal{Z}$-tensor. We classify the manifolds by using some semi-symetry conditions such as $R(\xi ,W).\mathcal{Z}=0$, $\mathcal{P}(\xi ,W).\mathcal{Z}=0$, $\mathcal{L}(\xi ,W).\mathcal{Z}=0$ and \[{{\mathcal{W}}_{2}}(\xi ,W).\mathcal{Z}=0\], where R the Riemann curvature tensor, $\mathcal{P}$is the Projective curvature tensor, $\mathcal{L}$ is the concircular curvature tensor and \[{{\mathcal{W}}_{2}}\]is the $W_2$ curvature tensor.