On hypersurfaces in Banach manifolds
type: Item
instance of: scholarly article
title: On hypersurfaces in Banach manifolds
zbMATH Open document ID: 1067.58004
author: El-Said R. Lashin
published in: Journal of the Egyptian Mathematical Society
publication date: 2004-10-25T00:00:00Z
review text: The author establishes Gauss-Codazzi equations for hypersurfaces in a Banach manifold and a generalization of the Schur theorem to the case of Banach manifolds. Concepts of bending and equi-affinity are introduced for infinite-dimensional hypersurfaces in Hilbert manifolds. The following theorems are proven: 1) A hypersurface \(N\) of constant sectional curvature in a Hilbert manifold \(M\) can be bent into a hyperplane if \(\beta= 0\), and into a hypersphere if \(\beta>0\); 2) The necessary and sufficient condition for a Hilbert manifold to be locally isomorphic to a manifold of constant sectional curvature is that it has the same constant curvature. Finally is proven: 3) The locally equi-affine hypersurfaces of a Hilbert manifold \(M\) of constant sectional curvature are the only hyperspheres.
Mathematics Subject Classification ID: 58B20
Mathematics Subject Classification ID: 53C20
zbMATH DE Number: 2109771
zbMATH Keywords: Gauss-Codazzi equation
zbMATH Keywords: Banach manifold
zbMATH Keywords: Hilbert manifold
reviewed by: Vasil G. Angelov
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 2109771