Circles and spheres in pseudo-Riemannian geometry
type: Item
instance of: scholarly article
title: Circles and spheres in pseudo-Riemannian geometry
zbMATH Open document ID: 0705.53026
DOI: 10.1007/BF01833144
author: Elsie Sterbin Gottlieb
published in: Aequationes Mathematicae
publication date: 1990-01-01T00:00:00Z
full work available at URL: https://eudml.org/doc/137345
review text: In the present paper pseudo-Riemannian submanifolds \(M_{\alpha}\) of a pseudo-Riemannian manifold \(\bar M_{\beta}\) are studied. \(\alpha\) (resp. \(\beta\)) is the index of the metric tensor of \(M_{\alpha}\) (resp. \(\tilde M_{\beta})\). When \(M_{\alpha}\) is totally umbilical and has parallel mean curvature vector field it is called an extrinsic sphere. A theorem of \textit{K. Nomizu} and \textit{K. Yano} in cases of submanifolds in a Riemannian manifold [Math. Ann. 210, 163-170 (1974; Zbl 0273.53039)] is generalized by the present authors to the following result. Let \(M_{\alpha}\) be a pseudo-Riemannian submanifold in a pseudo-Riemannian manifold \(M_{\beta}\) and \(\epsilon_ 0=+1\quad or\quad -1,\epsilon_ 1=+1,-1\quad or\quad 0\) \((2-2\alpha \leq \epsilon_ 0+\epsilon_ 1\leq 2n-2\alpha -2).\) For any positive constant k the following conditions (a) and (b) are equivalent, (a) every circle in \(M_{\alpha}\) with \(g(X,X)=\epsilon_ 0\) and \(g(\nabla_ XX,\nabla_ XX)=\epsilon_ 1k^ 2\) is a circle in \(\tilde M_{\beta}\), (b) \(M_{\alpha}\) is an extrinsic sphere. Here X is the unit tangent vector field of the circle and \(\nabla\) is the covariant differentiation of \(M_{\alpha}\). A theorem of \textit{K. Sakamoto} [Tohoku Math. J., II. Ser. 29, 25-56 (1977; Zbl 0357.53035)] is also generalized to pseudo-Riemannian cases.
Mathematics Subject Classification ID: 53C40
Mathematics Subject Classification ID: 53B30
Mathematics Subject Classification ID: 53B25
zbMATH DE Number: 4157422
zbMATH Keywords: constant isotropic
zbMATH Keywords: pseudo-Riemannian submanifolds
zbMATH Keywords: extrinsic sphere
zbMATH Keywords: circle
Wikidata QID: Q115393148
reviewed by: Yosio Mutō
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 4157422