Submanifolds in Hyperbolic, Space

微分几何中的一个基本问题是描述和确定空间形式中的所有子流形。对上述问题的全面解决似乎超出了当前数学的能力范围。从历史上看，人们施加了各种条件，以便使问题在某种程度上更可行，如果不是更可行的话。其中一个条件是将子流形限制为余维为1且与环境空间具有相同的常数曲率。在这种相当有限的状态下，这个问题受到了相当大的关注；事实上，它已经取得了很大的进展。例如，非负曲率黎曼空间形式的问题早已解决，在双曲情况下，只有一些部分解存在，直到最近获得了一个冗长但更完整的空间描述，在不确定情况下，格雷夫斯给出了平坦洛伦兹空间形式的问题的答案。涉及德西特空间形式的案例由Abe处理。在本报告中，我们采用常曲率-1的反德西特空间形式。给出了H^^-_1^n等距浸入H^^-_1^<n+1>的空间的完整描述。这里我们用H^^-_1^n表示n维反dc Sitter时空H_1^n的泛伪黎曼覆盖流形。

A fundamental problem in differential geometry is to characterize and determine all the submanifolds in a space form. A complete solution to the problem in the generality as stated above simply seems beyond the reach of the current mathematics. Historically, various conditions were imposed upon so as to make the problem somewhat more feasible, if not more viable. One of such conditions is to restrict submanifolds to being of codimension one and of the same constant curvature as the ambient space. The problem has received considerable attention under this rather restricted state ; indeed, it has seen much progress.For example, the problem has long been settled for the Riemannian space forms of non-negative curvature, in the hyperbolic case, only some partial solutions existed until a lengthy but more complete description of the space was recently obtained, In the indefinite case, Graves gave the answer to the problem for the flat Lorentzian space forms. The case involving the de Sitter space forms was treated by Abe.In this report, we take up the anti-de Sitter space forms of constant curvature -1. We give a. complete description of the space of the isometric immersions of H^^-_1^n into H^^-_1^<n+1>. Here we denote by H^^-_1^n the universal pseudo-Riemannian covering manifold of the n-dimensional anti-dc Sitter space-time H_1^n.