Hypersurfaces of two space forms and conformally flat hypersurfaces

We address the problem of determining the hypersurfaces f : M-n -> Q(s)(n+1)(c) with dimension n >= 3 of a pseudo-Riemannian space form of dimension n+1 constant curvature c and index s is an element of{0,1} for which there exists another isometric immersion (f) over tilde : M-n -> Q((s) over tilde) (n+1) with (c) over tilde not equal c. For n >= we provide a complete solution by extending results for s = 0 (s) over tilde by do Carmo and Dajczer (Proc Am Math Soc 86:115-119, 1982) and by Dajczer and Tojeiro (J Differ Geom 36:1-18, 1992). Our main results are for the most interesting case n = 3 and these are new even in the Riemannian case s = 0 (s) over tilde In particular, we characterize the solutions that have dimension n = 3 and three distinct principal curvatures. We show that these are closely related to conformally flat hypersurfaces of Q(s)(4)(c) with three distinct principal curvatures, and we obtain a similar characterization of the latter that improves a theorem by Hertrich-Jeromin (Beitr Algebra Geom 35:315-331, 1994).