Submanifolds of hyperbolic and pseudo-hyperbolic spaces with finite type generalized Gauss map

The notion of finite type submanifolds of a Euclidean space was introduced by B.-Y. Chen in late 1970's. Since then the finite type submanifolds of Euclidean spaces or pseudo-Euclidean spaces have been studied extensively, and many important results have been obtained. A Riemannian submanifold of a Euclidean space or pseudo-Euclidean space is said to be of finite type if its position vector can be written as a sum of finitely many eigenvectors of the Laplace operator. If these eigenvectors are corresponding to $k$ distinct eigenvalues of the Laplace operator, then the submanifold is said to be of $k$-type. For an isometric immersion ${\bf x}: M^n \longrightarrow \mathbb{E}^m$ of a compact Riemannian manifold $ M^n$ into an Euclidean space $\mathbb{E}^m$, the constant vector ${\bf x_0}$ in the spectral decomposition is exactly the center of mass of $ M^n$ in $\mathbb{E}^m$, where $\bf{x_0}$ is the eigenfunction of the Laplacian with eigenvalue $\lambda_0=0$. A spherical finite type map ${\bf x}$ of a Riemannian manifold $M^n$ into the unit sphere $\mathbb S^{m-1}(1)$ centered at the origin of $\mathbb{E}^m$ is called mass-symmetric if the vector $\bf{x_0}$ in its spectral decomposition is the center of $\mathbb S^{m-1}(1)$. Finite type non-compact submanifolds of Euclidean spaces and pseudo-Euclidean spaces are studied by many researcher. When $ M^n$ is compact, the component $\bf{x_0}$ in the spectral decomposition is a constant vector. However, when $ M^n$ is non-compact the component $\bf{x_0}$ is not necessary a constant vector. If $ M^n$ is not compact, we can not make the spectral decomposition of a map on $ M^n$ in general. But, it is possible to define the notion of a map of finite type on a non-compact manifold. Finite type submanifolds of Euclidean spaces and pseudo-Euclidean spaces have been studied by many geometers, and also many classifications and characterizations of finite type submanifolds have been obtained. Later, the notion of finite type was extended to differentiable maps on compact manifolds, in particular to Gauss map of submanifolds. And many results have been obtained on the classification of submanifolds of Euclidean spaces and pseudo-Euclidean spaces with finite type Gauss map. Since a spherical submanifold can be viewed as a submanifold of a Euclidean space the Gauss map of the spherical submanifold can be determined in the ordinary sense. For the Gauss map to reflect the properties of submanifolds in sphere instead of Euclidean space, Obata modified the definition of Gauss map appropriately as follows: Let $\textbf{x}: M^n \rightarrow \widetilde M^m$ be an oriented isometric immersion from a Riemannian $n$-manifold $M^n$ into a space form $\widetilde M^m$ of constant curvature. The generalized Gauss map in the Obata's sense is a map which assigns to each $p\in M^n$ the totally geodesic $n$-space of $\widetilde M^m$ tangent to $\textbf{x}(M^n)$ at $\textbf{x}(p)$. In the case, $\widetilde M^m = \mathbb S^{m}(1)$\; (or resp. $\widetilde M^m = \mathbb H^{m}(-1)$ the hyperbolic space) the generalized Gauss map is also called the spherical Gauss map\;(or resp. the hyperbolic Gauss map). Recently, spherical (or hyperbolic) submanifolds with finite type spherical (or hyperbolic) Gauss map have been characterized and classified in some papers. It is known that the geometric behavior of classical and spherical Gauss map are different. For example, the classical Gauss map of every compact Euclidean submanifold is mass-symmetric; but the spherical Gauss map of a spherical compact submanifold is not mass-symmetric in general. In this thesis, submanifolds of hyperbolic spaces and pseudo-hyperbolic spaces with finite type generalized Gauss maps are studied. Firstly, the definition of pseudo-hyperbolic Gauss map of pseudo-hyperbolic submanifolds in the Obata's sense is given, and then characterization and classification of submanifolds of hyperbolic space and pseudo-hyperbolic space with finite type generalized Gauss map are obtained. In the first chapter, it is mentioned about a review of literature. And then, results obtained this thesis are summarized. In the second chapter, it is introduced some fundamental definitions and equations of pseudo-Riemannian submanifolds of pseudo-Riemannian manifolds. It is mentioned about the classical Gauss map from an oriented pseudo-Riemannian manifold into a pseudo-Euclidean space. In the third chapter, firstly, by using the definition of generalized Gauss map in Obata's sense, the definition of hyperbolic Gauss map and pseudo-hyperbolic Gauss map are given. Laplacian of pseudo-hyperbolic Gauss map of the pseudo-Riemannian $n$-submanifold $M^n_t$ with index $t$ in a pseudo-hyperbolic $(m-1)$-space $\mathbb{H}^{m-1}_s(-1)\subset \mathbb{E}_{s+1}^m$ with index $s$ is obtained. Then, the minimal polynomial criteria is given for a finite type differentiable map from the pseudo-Riemannian $n$-submanifold $M_t^n$ with index $t$ to $\mathbb{E}^m_s$ pseudo-Euclidean space. In the fourth chapter, submanifolds of hyperbolic spaces with finite type hyperbolic Gauss map are investigated. This chapter contains two sections. In the first section, the characterization and classification of submanifolds of hyperbolic space with 1-type hyperbolic Gauss map are obtained. It is concluded that a totally geodesic hyperbolic space $\mathbb{H}^{n}(-1)$ is the only minimal isoparametric hypersurface in $\mathbb{H}^{n+1}(-1)$ with 1-type hyperbolic Gauss map. Also, it is proved that an oriented surface $M$ in $\mathbb{H}^{m-1}(-1)\subset \mathbb{E}_1^m$ has 1-type hyperbolic Gauss map if and only if $M$ is an open part of a totally geodesic hyperbolic 2-space $\mathbb{H}^{2}(-1)$ in $\mathbb{H}^{m-1}(-1)$. Moreover, a classification theorem is given for a submanifold of $\mathbb{H}^{m}(-1)\subset \mathbb{E}_1^m$ with 1-type hyperbolic Gauss map such that its spectral decomposition contains a non-zero constant component. In the second section, the necessary and sufficient condition for hypersurfaces with non-zero constant mean curvature of hyperbolic space having 2-type hyperbolic Gauss map is obtained. A horohypersphere in $\mathbb{H}^{n+1}(-1)$ is introduced and then, it is showed that the horohypersphere in $\mathbb{H}^{n+1}(-1)$ has biharmonic hyperbolic Gauss map. It is also obtained that the standard product $\mathbb{H}^k(-\frac{1}{1+r^2})\times\mathbb{S}^{n-k}(\frac{1}{r^2})$ in $\mathbb{H}^{n+1}(-1)$ is the only isoparametric hypersurface in $\mathbb{H}^{n+1}(-1)$ with 2-type hyperbolic Gauss map. Finally, it is proved that a non-totally umbilical surface with non-zero constant mean curvature in $\mathbb{H}^{3}(-1)\subset \mathbb{E}_1^{4}$ has 2-type hyperbolic Gauss map if and only if it is an open portion of the product surface $\mathbb{S}^1(a^{-2})\times\mathbb{H}^1(-b^ {-2})$ in $\mathbb{H}^3(-1)$. In the last chapter, the pseudo-Riemannian submanifolds of pseudo-hyperbolic spaces with finite type pseudo-hyperbolic Gauss map are studied. It contains two sections. In the first section, a characterization of pseudo-Riemannian submanifold of a pseudo-hyperbolic space $\mathbb H^{m-1}_s (-1)$ with 1-type pseudo-hyperbolic Gauss map is obtained. It is given some examples of surfaces in $\mathbb{H}_1^3(-1)\subset \mathbb{E}^4_2$ and in $\mathbb{H}_1^4(-1)\subset \mathbb{E}^5_2$ such that they have finite type pseudo-hyperbolic Gauss map. Then, the classification of maximal surfaces in $\mathbb H^{m-1}_2 (-1) \subset \mathbb E^m_{3}$ with 1-type pseudo-hyperbolic Gauss map is obtained. A classification theorem on space-like oriented surfaces in the anti-de Sitter 4-space with null mean curvature vector and 1-type pseudo-hyperbolic Gauss map is given. Finally, pseudo-Riemannian submanifolds in $\mathbb H^{m-1}_s (-1) \subset \mathbb E^m_{s+1}$ with 1-type pseudo-hyperbolic Gauss map having non-zero constant component in its spectral decomposition are classified. In the second section, a characterization of space-like hypersurfaces with 2-type pseudo-hyperbolic Gauss map lying in $\mathbb{H}^{m-1}_1(-1)\subset \mathbb{E}^m_{2}$ is obtained. It is proved that a non-totally umbilical, space-like, oriented hypersurface $M^n$ with non-zero constant mean curvature in $\mathbb{H}^{n+1}_1(-1)\subset \mathbb{E}_2^{n+2}$ has 2-type pseudo-hyperbolic Gauss map if and only if it has constant scalar curvature. A classification is obtained for space-like oriented surfaces with constant mean curvature in $\mathbb{H}^{3}_1(-1)\subset \mathbb{E}_2^{4}$ having 2-type pseudo-hyperbolic Gauss map. Moreover, it is proved that hyperbolic Veronese surface in $\mathbb{H}_2^{4}(-1)$ is the only maximal surface fully lying in $\mathbb{H}_2^{4}(-1)\subset \mathbb{H}_2^{m-1}(-1) $,\;$m \geq 5$ with 2-type pseudo-hyperbolic Gauss map. Finally, it is concluded that there is no minimal space-like surface lying fully in $\mathbb{H}^4(-1)\subset \mathbb{E}^5_1$ with 2-type hyperbolic Gauss map and there is no maximal space-like surface lying fully in $\mathbb{H}^4_1(-1)\subset \mathbb{E}^5_2$ with 2-type pseudo-hyperbolic Gauss map.