On the existence of spherically bent submanifolds, an analogue of a theorem of E. Cartan
type: Item
instance of: scholarly article
title: On the existence of spherically bent submanifolds, an analogue of a theorem of E. Cartan
zbMATH Open document ID: 1053.53014
author: Knut Pawel
author: Helmut Reckziegel
published in: Kodai Mathematical Journal
publication date: 2004-06-18T00:00:00Z
review text: In E. Cartan's book \textit{Leçons sur la géométrie des espaces de Riemann}, Gauthier-Villars, Paris (1951; Zbl 0044.18401) on can find a result providing a condition sufficient for a linear subspace \(V\subset T_pM\) to become the tangent space of a totally geodesic submanifold of \(M\). Here, the authors provide an analogous condition for so called \textit{spherical submanifolds}: a submanifold \(N\) of \(M\) is said to be \textit{spherical} whenever it is totally umbilical and its mean curvature vector is parallel in the normal bundle of \(N\). The condition in question is expressed in terms of the parallel transport and the curvature tensor of \(M\); it seems to be natural but complicated enough not to be provided here in details.
Mathematics Subject Classification ID: 53B25
Mathematics Subject Classification ID: 53C40
Mathematics Subject Classification ID: 53B20
zbMATH DE Number: 2076058
zbMATH Keywords: Riemannian manifold
zbMATH Keywords: mean curvature
zbMATH Keywords: spherical submanifold
reviewed by: Paweł Walczak
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.2996/kmj/1061901062
OpenAlex ID: W1985432104
DOI: 10.2996/KMJ/1061901062
description: scientific article; zbMATH DE number 2076058