Hypersurfaces of two space forms and conformally flat hypersurfaces

Author(s)

Canevari, S. ^[1] ; Tojeiro, R. ^[2]

Affiliation

^[1] Univ Fed Sergipe, Av Vereador Olimpio Grande S-N, Itabaiana - Brazil

^[2] Univ Fed Sao Carlos, Via Washington Luiz Km 235, BR-13565905 Sao Carlos - Brazil

• Document type: Journal article
• Source: Annali di Matematica Pura ed Applicata; v. 197, n. 1, p. 1-20, FEB 2018.
• Web of Science Citations: 6

Abstract

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). (AU)

• FAPESP's process: 11/21362-2 - Group actions, submanifold theory and global analysis in Riemannian and pseudo-Riemannian geometry
• Grantee: Paolo Piccione
• Support Opportunities: Research Projects - Thematic Grants