Golden structures on manifolds and their submanifolds

This study prepared as a philosophy doctoral thesis consists of six sections. The first section is the part of of the introduction which is divided into a summary of general literature and the information related to the contents of the philosophy doctoral thesis. The second section includes some basic concepts which will be used throughout the philosophy doctoral thesis. The other sections form the original part of the philosophy doctoral thesis. The third section consists of an investigation related to the concepts of parallelism, half parallelism, anti-half-parallelism, integrability and geodesicity on golden manifolds. These concepts with respect to an arbitrary fixed linear connection and Schouten and Vrănceanu connections, defined by it, are scrutinized. The fourth section is devoted to the construction of a golden structure on locally product manifolds. Accordingly, the concepts of a locally product golden Riemannian manifold and a locally decomposable golden Riemannian manifold are introduced. The decomposability of locally product golden Riemannian manifolds is addressed. The condition for each of the components of a locally decomposable golden Riemannian manifold to be an Einstein manifold (or a constant curvature manifold) is examined. Besides, a golden Riemannian structure on the Riemannian product of two Riemannian manifolds is defined and it is shown that the Riemannian product manifold is a locally decomposable golden Riemannian manifold. The fifth chapter deals with the geometry of the submanifolds of a golden Riemannian manifold and constitutes a large part of this study. Some properties of the canonical structures on the tangent and normal bundles of any isometrically immersed submanifold of a golden Riemannian manifold, induced by the golden structure of the ambient manifold, are given. Isometrically immersed non-invariant, invariant and anti-invariant submanifolds of a golden Riemannian manifold with the help of induced structures on them, by the golden structure of the ambient manifold, are investigated. The conditions for any isometrically immersed non-invariant submanifold to be totally geodesic (or minimal) are determined. In addition, some important results are found in cases where the tangent vector fields of the induced structure on the isometrically immersed non-invariant submanifold are linearly dependent. It is shown that any isometrically immersed invariant submanifold of a locally decomposable golden Riemannian manifold is a locally decomposable golden Riemannian manifold. It is tried to find the equivalent expressions to the invariance of isometrically immersed submanifolds. Totally geodesicity of isometrically immersed invariant submanifolds is dwelt on. Some interesting results regarding isometrically immersed invariant submanifolds are reached. Some properties of isometrically immersed anti-invariant submanifolds are obtained and the conditions under which for them to be totally geodesic are discussed. Under some assumptions, a local orthonormal frame is established for the normal bundle of any isometrically immersed anti-invariant submanifold. Besides, it is shown that the second fundamental tensors corresponding to normal vector fields determined by a local orthonormal frame of the tangent bundle of the isometrically immersed anti-invariant submanifold are zero. Furthermore, the geometric properties of semi-invariant submanifolds of a golden Riemannian manifold are examined and some examples are given on them. A wide investigation is made for any semi-invariant submanifold on characterizations, parallelism of canonical structures on its tangent and normal bundles, integrability and parallelism of the distributions which are defined on it, some classifications such as totally geodesic, mixed totally geodesic and totally umbilical and de Rham cohomology groups. The sixth section is separated into a new structure called an f(3,-2,-1)-structure or a para f(3,2,1)-structure on differentiable manifolds. Some examples that show the relationship between para f(3,2,1)-structures and golden structures are obtained. The conditions for the induced structure on any isometrically immersed submanifold of a golden Riemannian manifold to be a para f(3,2,1)-structure are addressed. The existence of para f(3,2,1)-structures on the tangent and normal bundles of semi-invariant submanifolds of a golden Riemannian manifold is researched. The notions of partially integrability and integrability of any para f(3,2,1)-structure are described. Fundamental properties and integrability of para f(3,2,1)-structures and their distributions, which are naturally defined, are investigated.