On hypersurfaces in a Riemannian vector bundle with prescribed Gaussian curvature and convexity
type: Item
instance of: scholarly article
title: On hypersurfaces in a Riemannian vector bundle with prescribed Gaussian curvature and convexity
zbMATH Open document ID: 1059.53035
author: Abdellah Hanani
published in: Mathematische Zeitschrift
publication date: 2004-07-01T00:00:00Z
review text: Let \(E\rightarrow M\) be a metric vector bundle over a Riemannian manifold \(M\) and \(\Sigma \subset E\) the unit sphere subbundle. The total space \(E\) is endowed with a compatible Riemannian metric such that the orthogonal complement to the vertical space is the horizontal space of a metric connection in the bundle \(E\rightarrow M\). The main results are: 1) there is no radial graph \(\Gamma \subset E\) over \(\Sigma\) of positive Gauss curvature, 2) the Gauss curvature of any convex radial graph is zero and the Gauss curvature of its fibres is constant along horizontal curves and 3) there exists a convex radial graph with fibres of prescribed positive Gauss curvature, under suitable assumptions on the prescribed function.
reviewed by: V. Cortés
Mathematics Subject Classification ID: 53C21
Mathematics Subject Classification ID: 35J60
Mathematics Subject Classification ID: 58J60
Mathematics Subject Classification ID: 53C42
zbMATH DE Number: 2078067
zbMATH Keywords: hypersurfaces
zbMATH Keywords: prescribed Gauss curvature
zbMATH Keywords: Sasaki metric
zbMATH Keywords: vector bundle
zbMATH Keywords: metrics on sphere bundles
Wikidata QID: Q115388866
MaRDI profile type: Publication
full work available at URL: s00209-003-0605-0
OpenAlex ID: W2035277309
DOI: 10.1007/S00209-003-0605-0
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description: scientific article; zbMATH DE number 2078067