Generalized constant ratio submanifolds of semi-Euclidean spaces

One of the most basic objects studied to understand geometrical properties of a submanifold of a Euclidean space or semi-Euclidean space is its position vector. In this direction, the notion of constant ratio (CR) submanifolds in Euclidean spaces introduced by B. Y. Chen. For each Riemannian manifold M isometrically immersed in E^m , there is a natural orthogonal decomposition of the position vector x at each point on M; namely x = x^T + x? where x^T and x? denote the tangential (tangent) and normal components of x, respectively. If the ratio of length of these two vectors is constant, then M is said to be a CR submanifold or an equiangular submanifold. In the particular case of the codimension= 1, if M is an CR hypersurface, then the tangential part x^T of x is a principal direction of M. However, the converse of this statement does not hold in general. For example, it is proved that all of rotational surfaces in E3 have this property. But a rotational surface is not a CR surface unless its profile curve is chosen specifically. Therefore, the definition of generalized constant ratio (GCR) surface has been recently given: If x^T is a principal direction of a surface, then the surface is said to be a GCR surface. In Euclidean and semi-Euclidean spaces, generalized constant ratio (GCR) submanifolds have been studied and obtained some new important results by many researchers interested in geometry. There are still many open problems in this regard. As also mentioned in the thesis, GCR surfaces can also be thought as a generalization of the surface endowed with canonical principal direction (CPD) obtained by taking a constant vector k rather than the position vector x. Given the above definitions, it is easy to see that these definitions can be made the same for hypersurfaces. However, when the GCR submanifold is defined, the definition of the A-class of the S^nxR and H^nxR spaces is used in the case of codimension being larger than one. By using this definition given by R. Tojiero and B. Mendoza, a submanifold of Euclidean and semi-Euclidean spaces is called a GCR submanifold if the tangential component of the position vector of that submanifold is the principal direction of all shape operators. In this thesis, the generalized constant ratio (GCR) submanifolds in Euclidean and semi-Euclidean spaces are discussed and this thesis is organized into six sections. In the first two chapters of the thesis, some general information about the topic of the thesis is given; In the third, fourth and fifth sections, the main results are presented. The organization of thesis is as follows: In the first chapter; papers studied by considering the geometric properties of generalized constant ratio (GCR) submanifolds and the submanifolds are summarized. In Chapter 2; basic definitions and theorems to be used throughout the thesis are given. In Chapter 3; we have classified constant angle (CAS) surfaces, surfaces endowed with constant canonical direction (CPD) surfaces, constant slope (CSS) surfaces and finally generalized constant ratio (GCR) surfaces with codimension= 2 in E^4 Euclidean space, respectively. In Chapter 4; firstly we give some important theorems and results to be used in this section by investigating some geometrical properties for hypersurfaces endowed with canonical principal direction (CPD) in Minkowski space E^n+1_1. In addition, a classification was made for surfaces endowed with a canonical principal direction (CPD) with codimension= 1 in Minkowski space E^3_1. Subsequently, minimal (maximal) cases of these surfaces have been investigated and have been given some important characterizations. Finally, we give a new classification for constant angle (CAS) surfaces with a light-like constant direction. In Chapter 5; firstly we give some important theorems and results to be used in this section by investigating some geometrical properties for generalized constant ratio (GCR) hypersurfaces in Minkowski space E^n+1_1. In addition, a classification was made for generalized constant ratio (GCR) surfaces with codimension= 1 in Minkowski space E^4_1. In the last chapter; first the obtained results are evaluated and then some open problems are revealed.