Submanifolds in Hyperbolic, Space

微分几何中的一个基本问题是刻画和确定空间形式中的所有子流形。如上所述，对问题的全面解决似乎完全超出了当前数学的能力范围。从历史上看，施加各种条件是为了使这个问题在某种程度上更可行，甚至更可行。这样的条件之一是将子流形限制为余维为一且与环境空间具有相同的常曲率。这个问题在这种相当受限的状态下得到了相当大的关注；事实上，它已经取得了很大的进展。例如，对于非负曲率的黎曼空间型，这个问题早就得到了解决，在双曲情况下，只存在一些部分解，直到最近得到了关于该空间的更详细的描述，在不确定的情况下，Graves给出了关于平坦洛伦兹空间型的问题的答案。Abe处理了涉及de Sitter空间形式的情况，在这篇报告中，我们讨论了常曲率-1的反de Sitter空间形式。给出了H^-1^n到H^-1^&lt；n+1&gt；的等距浸入空间的完全刻画。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.