Conformality on Semi-Riemannian Manifolds

• Published in: Mediterranean journal of mathematics Vol. 13; no. 4; pp. 2185 - 2198
• Main Authors: Bejan, Cornelia-Livia, Eken, Şemsi
• Format: Journal Article
• Language: English
• Published: Cham Springer International Publishing 01.08.2016
• Subjects: Mathematics; Mathematics and Statistics; 53C20; 53C50; 58E20; Conformal semi-Riemannian map; 53C43; Semi-Riemannian map; Harmonic map; Riemannian submersion
• ISSN: 1660-5446; 1660-5454
• DOI: 10.1007/s00009-015-0613-4

We introduce here the notion of conformal semi-Riemannian map between semi-Riemannian manifolds aiming to unify and generalize two geometric concepts. The first one is studied by García-Río and Kupeli (namely, semi-Riemannian map between semi-Riemannian manifolds). The second notion is defined by Şahin (namely, conformal Riemannian map between Riemannian manifolds) as an extension of the notion of Riemannian map introduced by Fischer. We support the main notion of this paper with several classes of examples, e.g. semi-Riemanninan submersions (see O’Neill’s book and Falcitelli, Ianus and Pastore’s book) and isometric immersions between semi-Riemannian manifolds. As a tool, we use the screen distributions (specific in semi-Riemannian geometry) of Duggal and Bejancu’s book to obtain some characterizations and to give a semi-Riemannian version of Fischer’s (resp. Şahin’s) results, using the new map introduced here. We study the generalized eikonal equation and at the end relate the main notion of the paper with harmonicity.