A parametrization of isometric immersions between anti-de Sitter space-times
The anti-de Sitter space forms of constant curvature \(-1\) are considered. If \(\widetilde H^n_1\) denotes the universal pseudo-Riemannian covering manifold of the \(n\)-dimensional anti-de Sitter spacetime \(H^n_1\), a complete description of the space of isometric immersions of \(\widetilde H^n_1\) into \(\widetilde H^{n+1}_1\) is given. The main result essentially states that for each isometric immersion \(f: \widetilde H^n_1\to\widetilde H^{n+1}_1\) there exists a family of countable \(n\)-tuples of real-valued \(C^\infty\) functions which completely describe the second fundamental form of the immersion and hence the immersion itself. Conversely, there exists an isometric immersion \(f:\widetilde H^n_1\to\widetilde H^{n+1}_1\) whose relative nullity foliation is completely determined by functions in a certain family of \(C^\infty\) functions.
universal pseudo-Riemannian covering manifold
scientific article; zbMATH DE number 1180764
scientific article