Codazzi fields on surfaces immersed in Euclidean 4-space
type: Item
instance of: scholarly article
title: Codazzi fields on surfaces immersed in Euclidean 4-space
zbMATH Open document ID: 1173.53005
author: J. M. Gutiérrez Núñez
author: Carmen Romero-Fuster
author: Federico Sánchez-Bringas
published in: Osaka Journal of Mathematics
publication date: 2009-02-20T00:00:00Z
full work available at URL: https://projecteuclid.org/euclid.ojm/1227708823
review text: Consider a surface \(M\) immersed in \(\mathbb R^4\) and a vector field \(\nu\) in the normal bundle \(NM\). The authors deal with the question, under what conditions there exists an isometric immersion of \(M\) in \(\mathbb R^3\) so that there is a vector bundle isomorphism between the sub-bundle determined by \(\nu\) in \(NM\) and the normal bundle of the surface \(M'\) in \(\mathbb R^3\), taking the second fundamental form \(II_\nu\) of \(M\) to the second fundamental form of \(M'\). In the paper necessary and sufficient conditions for the normal field \(\nu\) are determined in order to get an affirmative answer to the above question. Furthermore geometric properties are deduced the surface \(M\) must fulfill in order to admit an isometric immersion in \(\mathbb R^3\) with prescribed second fundamental form.
reviewed by: Friedrich Manhart
Mathematics Subject Classification ID: 53A07
Mathematics Subject Classification ID: 57R25
zbMATH DE Number: 5509685
zbMATH Keywords: immersed surface
zbMATH Keywords: normal bundle
zbMATH Keywords: second fundamental form
MaRDI profile type: Publication
cites work: Über Nabelpunkte auf einer Eifläche
cites work: Q3999385
cites work: Local isometric immersions of R2 into R4.
cites work: Inflection points and topology of surfaces in 4-space
cites work: Closed principal lines of surfaces immersed in the Euclidean 4-space
cites work: On a Loewner umbilic-index conjecture for surfaces immersed in \({\mathbb{R}}^4\)
cites work: On G. Bol's proof of Carathéodory's conjecture
cites work: On singularities of submanifolds of higher dimensional Euclidean spaces
cites work: The complement of the bifurcation variety of a simple singularity
cites work: The geometry of surfaces in 4-space from a contact viewpoint
cites work: Q5428609
cites work: Lines of curvature near umbilical points on surfaces immersed in \(\mathbb{R}^ 4\)
cites work: Umbilicity of surfaces with orthogonal asymptotic lines in \(\mathbb R^ 4\).
cites work: A proof of a conjecture of Loewner and of the conjecture of Caratheodory on umbilic points
description: scientific article; zbMATH DE number 5509685