Para-Sasaki like Riemann manifolds

In this thesis, para-Sasaki-like Riemannian manifolds, which are a special class of almost paracontact almost paracomplex Riemannian manifolds, have been investigated. Almost paracontact almost paracomplex Riemann manifolds equipped with generalized symmetric metric connection have been studied. Specifically, the relationship between the Levi-Civita connection and the generalized symmetric metric connection for para-Sasaki-like Riemann manifolds is given. Relationships bet- ween curvature tensor, Ricci tensor and scalar curvature tensor are obtained. Para- Einstein manifolds are examined and para-Ricci-like solitons with generalized symmetric metric connection on para-Sasaki-like Riemannian manifolds are studied. 3 and 5 dimensional examples supporting the obtained theorems are given. In the last section, D−homothetic deformation is defined on apapR manifolds, and curva- ture tensors are calculated specifically for para-Sasaki-like Riemannian manifolds. It is shown that the D−homothetic deformation of the para-Einstein-like manifold is also a para-Einstein-like manifold. It has been proven that if the para-Sasaki-like Riemannian manifold admits para-Ricci-like soliton, the D−homothetic deformation of the para-Sasaki-like Riemannian manifold also admits para-Ricci-like soliton. Some differential operators corresponding to D−homotetic deformation have been calcu- lated. A 5-dimensional example supporting the obtained theorems is given.