A correspondence for isometric immersions into product spaces and its applications
Let \(\psi : M^n \to N^n \times P^r\) be an isometric immersion from an \(n\)-dimensional semi-Riemannian manifold \(M^n\) into a product manifold \(N^n \times P^r\) equipped with a semi-Riemannian warped product metric \(g_N \otimes_{\mathcal F} g_P\). Assume that \(\psi\) is a graph, that is, \(\psi\) is of the form \(\psi = (\pi,h \circ \pi)\), where \(\pi\) is a diffeomorphism from \(M^n\) into some open domain \(\Omega \subset N^n\), and \(h : \Omega \to P^r\) is smooth. Then \(\widehat{\psi} = (\pi^{-1},h)\) is an isometric immersion from \(\Omega \subset N^n\) into the product manifold \(M^n \times P^r\) equipped with the semi-Riemannian metric \(g^\prime = \frac{1}{{\mathcal F}^2}(g_M \otimes (-g_P))\). The authors investigate the correspondence between the two maps \(\psi\) and \(\widehat{\psi}\). As a consequence they obtain several existence and non-existence results for hypersurfaces in Riemannian or Lorentzian spaces. For example, if \(M^n\) is a complete Riemannian manifold with negative Ricci curvature whose scalar curvature is bounded from above by a negative constant, they conclude that the Euclidean space \({\mathbb R}^n\) cannot be isometrically immersed into the Lorentzian product space \(M^n \times {\mathbb L}\).
scientific article; zbMATH DE number 5791217
scientific article