On compositions of isometric immersions
type: Item
instance of: scholarly article
title: On compositions of isometric immersions
zbMATH Open document ID: 0768.53027
author: Marcos Dajczer
author: Ruy Tojeiro
published in: Journal of Differential Geometry
publication date: 1993-04-01T00:00:00Z
review text: The background of this paper is the problem of local and global isometric immersions with low codimension of the \(n\)-dimensional sphere \(S^ n\) into Euclidean space \(\mathbb{R}^{n+p}\); the case \(p = 2\) and \(n \geq 4\) was considered since 1965 [\textit{B. O'Neill}, Duke Math. J. 32, 149-159 (1965; Zbl 0134.395); \textit{J. Erbacher}, Nagoya Math. J. 45, 139-165 (1972; Zbl 0211.537); \textit{W. Henke}, Math. Ann. 219, 261-276 (1976; Zbl 0303.53024); \textit{L. Whitt}, J. Differ. Geom. 14, 295-302 (1979; Zbl 0427.53027)] and \textit{J. D. Moore}]. In the first section of the present paper, the authors give several conditions on isometric immersions \(f: M^ n \to Q_ c^{n+1}\) and \(g: M^ n \to Q_ c^{n+p}\), which imply, that \(g\) is a composition \(g = h\circ f\) with \(h: U \subset Q_ c^{n+1} \to Q_ c^{n+p}\) (\(M^ n\) is an \(n\)-dimensional Riemannian manifold and \(Q^ N_ c\) are complete, simply connected Riemannian manifolds of constant sectional curvature c). Section 2 gives a contribution to the question of realization of an \(n\)- dimensional Riemannian manifold \(M^ n\) as a hypersurface in two space forms of different sectional curvature. From a result of do Carmo-Dajczer it follows, that for \(n \geq 4\quad M^ n\) has to be conformally flat; the authors show, that for \(n = 3\) this is not necessary. Further results, concerning the composition of pseudo-umbilical isometric immersions complete this interesting paper.
Mathematics Subject Classification ID: 53C42
Mathematics Subject Classification ID: 53C20
zbMATH DE Number: 147454
zbMATH Keywords: constant sectional curvature
zbMATH Keywords: conformally flat
zbMATH Keywords: pseudo-umbilical isometric immersions
reviewed by: Friedrich Manhart
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.4310/jdg/1214448440
OpenAlex ID: W1606013585
Wikidata QID: Q115180745
DOI: 10.4310/JDG/1214448440
Recommended article: Q4890673
Recommended article: Compositions of isometric immersions in higher codimension
Recommended article: Q4892954
Recommended article: On isometric immersions into complex space forms
Recommended article: On Compositions of Conformal Immersions
Recommended article: Q4953677
Recommended article: Q3596603
Recommended article: Isometric immersions and compensated compactness
Recommended article: Q4427618
Recommended article: Isometric immersions in Minkowski spaces
description: scientific article; zbMATH DE number 147454