Extrinsic spheres in a symmetric space
type: Item
instance of: scholarly article
title: Extrinsic spheres in a symmetric space
zbMATH Open document ID: 0798.53051
author: Y. Nikolayevsky
published in: Mathematical Notes
publication date: 1994-02-08T00:00:00Z
review text: A submanifold \(N^ \ell \subset M^ n\) is called completely umbilic if \(h(X,Y) = Hg(X,Y)\) for any vector fields \(X\), \(Y\) tangent to \(N^ \ell\) where \(g\) and \(h\) are the first and second quadratic forms of \(N^ \ell\). If the mean curvature vector \(H\) is parallel to the normal connection and does not equal zero then the completely umbilic \(N^ \ell\) is called an extrinsic sphere iff it is an extrinsic hypersphere in a completely geodesic submanifold \(\widetilde{N}^{\ell + 1} \subset M^ n\) of constant curvature. The author gives the example of an extrinsic sphere in a Riemannian manifold which is not an extrinsic hypersphere in some completely geodesic submanifold.
Mathematics Subject Classification ID: 53C40
Mathematics Subject Classification ID: 53C35
zbMATH DE Number: 484242
zbMATH Keywords: completely umbilic
zbMATH Keywords: mean curvature vector
zbMATH Keywords: extrinsic sphere
reviewed by: V. T. Fomenko
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 484242