Conformal-twisted and warped-twisted product manifolds and submanifolds

The main purpose of this thesis is to research the geometry of conformal-twisted product and warped-twisted product submanifolds. These types of manifolds are studied as submanifolds of globally conformal Kaehler manifolds and Einstein-like warped-twisted product manifolds are investigated. This study consists of five main chapters. In the first chapter, the history of this study is given, which means that the previous studies in the literature that were studied by researchers related to this subject have been mentioned. The second chapter consists of five subsections. In the first subsection, the definitions of the Riemannian manifold, the main tensors defined on the Riemannian manifold and different classes of Einstein-like manifolds are given. In the second subsection, the definition of submanifold of Riemannian manifolds is given and the main properties of a submanifold are given. The equations of Gauss and Weingarten which establish relationship between submanifold and the ambient manifold are given as well. In the third subsection, definitions of almost Hermitian manifold, Kaehlerian manifold and the main properties of this kind of manifolds are given. In addition to this, some special submanifolds of almost Hermitian manifolds which are called holomorphic, totally real, semi-slant, hemi-slant etc. are introduced. In the fourth subsection, the definitions of locally and globally conformal Kaehler manifolds are given and main theorems and equations valid for these submanifolds are reminded. The fifth subsection includes the definition of doubly twisted product submanifolds and the theorem which characterizes some special cases of such manifolds. Besides, the covariant derivative formulas of doubly warped product manifolds are given. \par In the third chapter, tools and methods that are used through the thesis are mentioned. The fourth chapter is the main part of the thesis. This chapter consists of two subsections. In these subsections, the notion of conformal-twisted product submanifolds of the form $_{f_2}M^{T}\times_{f_1}M^{\theta}$ and $_{f_2}M^{\theta}\times_{f_1}M^{T}$ is introduced, where $M^T$ is a holomorphic submanifold and $M^\theta$ is a proper slant submanifold of $M$ in a globally conformal Kaehler manifold and $f_2$ and $f_1$ are conformal factor and twisting function, respectively. Necessary and sufficient conditions for proper semi-slant submanifold to be a locally conformal-twisted product for such submanifolds of the form $_{f_2}M^{T}\times_{f_1}M^{\theta}$ and $_{f_2}M^{\theta}\times_{f_1}M^{T}$ are given. A general inequality for the square norm of second fundamental form of these types of submanifolds is established. Moreover, the notion of warped-twisted product hemi-slant submanifolds of the form $_{f_2}M^{\bot}\times_{f_1}M^{\theta}$ with warping function $f_2$ on $M^\theta$ and twisting function $f_1$ is introduced, where $M^\bot$ is a totally real and $M^\theta$ is a slant submanifold of a globally conformal Kaehler manifold. A warped-twisted product hemi-slant submanifold of a globally conformal Kaehler manifold is proved to be a locally doubly warped product. Then, a general inequality for doubly warped product mixed geodesic hemi-slant submanifolds is established and some results for such submanifolds are gotten by using the equality sign of the general inequality. Additionally, necessary and sufficient conditions are given for the factor manifolds of Einstein-like warped-twisted manifolds to be a Einstein-like manifolds and different classes of Einstein-like warped-twisted product manifolds are characterized. Conversely, when the factor manifolds of warped-twisted product manifold are Einstein-like manifold, whether the warped-twisted product is Einstein-like is examined. In the fifth chapter, the study is reviewed. The contribution of thesis to the literature is mentioned and what's more, it is mentioned that this study will inspire to different research topics.