Submanifolds in Hyperbolic, Space

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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.