Cubic form geometry for surfaces in \({\mathbf S}^3(1)\)
type: Item
instance of: scholarly article
title: Cubic form geometry for surfaces in \({\mathbf S}^3(1)\)
zbMATH Open document ID: 1010.53016
author: Tsasa Lusala
published in: Beiträge zur Algebra und Geometrie
publication date: 2002-08-20T00:00:00Z
full work available at URL: https://eudml.org/doc/229413
full work available at URL: 21.html
review text: Let \(M^2\) be a nondegenerate surface in a sphere \(S^3(1)\), and \(C\) the difference tensor field between the Levi-Civita connections \(\nabla^I\) and \(\nabla^{III}\) of its first and third fundamental forms, i.e. \(C:= {1\over 2}(\nabla^I- \nabla^{III})\). A local classification is given for nondegenerate surfaces \(M^2\) in \(S^3(1)\subset R^4\) with vanishing traceless part \(\widetilde C\) of \(C\). It is proved that such an \(M^2\) is locally either totally umbilical, or a part of a rotational surface whose profile curve is an arc of an ellipse or of a hyperbola, or a part of a quadratic surface of a special type, which is described in the paper by the explicit parametric equations in \(R^4\).
Mathematics Subject Classification ID: 53B25
Mathematics Subject Classification ID: 53A15
zbMATH DE Number: 1784608
zbMATH Keywords: nondegenerate surfaces in the 3-sphere
zbMATH Keywords: principal curvature functions
zbMATH Keywords: rotational surfaces
zbMATH Keywords: cubic form geometry
reviewed by: Ülo Lumiste
MaRDI profile type: Publication
description: scientific article; zbMATH DE number 1784608