Finite type submanifolds and Gauss maps

In the late 1970's, the theory of finite type submanifolds in Euclidean space was initiated by B.-Y. Chen. Later, the notion of finite type was extended to differentiable maps on Riemannian manifolds, especially, to Gauss maps of Euclidean submanifolds. Since then, many geometers have obtained substantial results about characterizing or classifying the submanifoldsof Euclidean space or pseudo--Euclidean space in terms of finite type. The generalization to more general maps attracted the interest of people working in analysis to finite type theory, and it is now considered an active research field with several open problems. In this thesis, there are two main problems which are studied. One of them is related to different types of rotational surfaces in the pseudo--Euclidan space with pointwise 1--type Gauss map and the other one is pseudo--spherical submanifolds with finite type pseudo--spherical Gauss map. A short overview of the results obtained in this thesis is given as follows. In the first chapter, main topics of thesis are discussed briefly and a review of literature about finite type theory and submanifolds with pointwise 1--type Gauss map is mentioned. In the second chapter, a short survey of all the theory that will be needed in the following chapters is given. The aim of the third chapter is to obtain some classification and characterization theorems of some special classes of rotational surfaces in the pseudo--Euclidean space with pointwise 1--type Gauss map of the first kind and the second kind. The scheme of problems in the third chapter is as following: First, timelike rotational surface in the Minkowski 4--space with profile curve lying in 2--dimensional timelike plane, which is called double rotational surface, is studied. All such surfaces with pointwise 1--type Gauss map of the first kind are classified and then timelike double rotational surfaces with pointwise 1--type Gauss map of the second kind are focused. Then, it is proved that there exists no non--planar timelike rotational surface with flat normal bundle and pointwise 1--type Gauss map of the second kind. Secondly, a different class of rotational surfaces in Minkowski space E^4_1 with 2--dimensional axis is considered. There are three types of rotational surfaces with 2--dimensional axis, called rotational surfaces of elliptic, hyperbolic or parabolic type which are invariant under spacelike rotation, hyperbolic rotation and screw rotation, respectively. All flat spacelike rotational surfaces of elliptic and hyperbolic types with pointwise 1--type Gauss map of the first and second kind are determined. Then, flat spacelike rotational surfaces of parabolic type with harmonic Gauss map are characterized. Also, it is shown that there exists no flat spacelike rotational surface of parabolic type in E^4_1 with pointwise 1--type Gauss map of the second kind. Similarly, the classification and characterization of flat timelike rotational surfaces of elliptic, hyperbolic and parabolic types in Minkowski space with 2--dimensional axis and having pointwise 1--type Gauss map are done. Also, it is shown that there exists no flat timelike rotational surface of parabolic type in E^4_1 with pointwise 1-type Gauss map. Finally, some special classes of non--planar rotational surfaces in the pseudo--Euclidean space E^4_2 with profile curves lying in 2--planes are considered. The differential equation that characterizes such rotational surfaces in E_2^4 with zero mean curvature is solved and examples of maximal surfaces and Lorentz surfaces with zero mean curvature in E^4_2 are given. Then, such rotational surfaces in the pseudo--Euclidean space E^4_2 with zero mean curvature vector and having pointwise 1--type Gauss map of the second kind are obtained. Furthermore, all such pseudo--umbilical rotational surfaces in the pseudo--Euclidean E^4_2 are determined, and they are examined in terms of having pointwise 1--type Gauss map. In the fourth chapter, pseudo--spherical submanifolds with finite type pseudo--spherical Gauss map which is obtained from Obata's map for pseudo--spherical immersions are studied. The scheme of problems in the fourth chapter is as following: First of all, the pseudo--Riemannian submanifolds in the pseudo--sphere with harmonic pseudo--spherical Gauss map are considered. Then, Riemannian and Lorentzian surfaces of arbitrary codimension in a pseudo--sphere whose pseudo--spherical Gauss map is harmonic are completely classified. In some cases, a global classification is obtained, while in the other cases the surfaces are described by an explicit system of partial differential equations. Secondly, the characterization of pseudo--Riemannian submanifolds in the pseudo--sphere S^{m-1}_s\subset E^m_s with 1--type pseudo--spherical Gauss map is obtained. This is divided into two cases depending on whether the spectral decomposition of the pseudo--spherical Gauss map contains a non--zero constant component or not. It is proven that a pseudo--Riemannian submanifold of S^{m-1}_s\subset E^m_s has 1--type pseudo--spherical Gauss map with zero constant component in its spectral decomposition if and only if it has zero mean curvature vector in S^{m-1}_s\subset E^m_s, constant scalar curvature and flat normal connection. Also, a complete classification for spacelike surfaces and Lorentzian surfaces in the pseudo--sphere S^{m-1}_s\subset E^m_s is determined. Then, the pseudo--Riemannian submanifolds of a pseudo--sphere with 1--type pseudo--spherical Gauss map having a non--zero constant component in its spectral decomposition are considered. According to the casual character of the mean curvature, two different classification theorems are obtained. Moreover, it is showed that an $n$--dimensional pseudo horosphere in S^{n+1}_s\subset E^{n+2}_s, that is, a flat and totally umbilical hypersphere of S^{n+1}_s, has biharmonic pseudo--spherical Gauss map. Finally, pseudo--Riemannian submanifolds in a pseudo--sphere with 2--type pseudo--spherical Gauss map are considered. It is shown that a Lorentzian surface in a pseudo--sphere S^4_2\subset E^5_2 with zero mean curvature vector in S^4_2 has 2--type pseudo--spherical Gauss map if and only if it has Gaussian curvature K=1/3 and normal curvature K^D with |K^D|=2/3. As an example for above statement, Lorentzian Veronese surface is mentioned and the geometric quantities of this surface are given. It is known that Veronese surface in S^4 is the only minimal surface with 2--type spherical Gauss map. However, there are surfaces in S^4_2\subset E^5_2 with 2--type pseudo--spherical Gauss map apart from Lorentzian Veronese surface. Moreover, pseudo--Riemann hypersurfaces in a pseudo--sphere are studied. The necessary and sufficient condition having 2--type pseudo--spherical Gauss map for pseudo--Riemannian hypersurfaces with a diagonalizable shape operator, which is called proper hypersurfaces, is obtained. That is, a non--totally umbilical proper hypersurface in a pseudo--sphere S^{n+1}_s\subset E^{n+2}_s with non--zero constant mean curvature has 2--type pseudo--spherical Gauss map if and only if it has constant scalar curvature. In particular, a classification of proper surfaces in the de Sitter space S^3_1\subset E^4_1 with 2--type pseudo--spherical Gauss map is given. It is well--known that the shape operator of a pseudo--Riemannian surface does not need to be diagonalizable. Because of this fact, there are significant differences between Riemannian and pseudo--Riemannian cases. There are some families of surfaces in the pseudo--Riemannian manifolds which have no Riemannian counterparts such as the B--scrolls and the complex circles. Then, Lorentzian surfaces in the de Sitter space S^3_1\subset E^4_1 with non--diagonalizable shape operator are examined. It is given that the B--scroll over a null curve in S^3_1\subset E^4_1 has null 2--type pseudo--spherical Gauss map. As a consequence, it is proved that a Lorentzian surface in S^3_1 with non--zero constant mean curvature and a non--diagonalizable shape operator at any point has null 2--type pseudo--spherical Gauss map if and only if it is an open part of a non--flat B--scroll over a null curve.