Extendibility and stable extendibility of the square of the normal bundle associated to an immersion of the real projective space
type: Item
instance of: scholarly article
title: Extendibility and stable extendibility of the square of the normal bundle associated to an immersion of the real projective space
zbMATH Open document ID: 1020.55002
author: Teiichi Kobayashi
author: Kazushi Komatsu
published in: Hiroshima Mathematical Journal
publication date: 2003-10-06T00:00:00Z
review text: Let \(P^n\) denote the \(n\)-dimensional real projective space and let \(\nu\) denote the normal bundle associated to an immersion of \(P^n\) in \(\mathbb{R}^{n+k}\), where \(k> 0\). The purpose of this paper is to study the problem of determining the dimension \(n\) for which \(\nu^{\otimes 2}\) and \((\nu\otimes\mathbb{C})^{\otimes 2}\) are extendible (or stably extendible) to \(P^m\) for every \(m> n\). The main result of this paper consists of four theorems (Theorems 1-4). Two of them deal with the special case when \(k= n+1\). The results are: \(\nu^{\otimes 2}\) (resp. \((\nu\otimes\mathbb{C})^{\otimes 2}\)) is extendible to \(P^m\) for every \(m> n\) if and only if \(1\leq n\leq 17\) (resp. \(18\)) or \(n= 20,21\) (Theorems 2 and 4). The other two ones treat different situations, but the conditions for \(n\), \(k\) are rather too complicated to be repeated here. This paper may be considered in some respects as a continuation of two previous ones by the first author, \textit{H. Maki} and \textit{T. Yoshida} [Hiroshima Math. J. 5, 487-497 (1975; Zbl 0314.55028); ibid. 29, 631-638 (1999; Zbl 0946.55010)]. The proofs of the above theorems use some of the results obtained there.
reviewed by: Haruo Minami
Mathematics Subject Classification ID: 55R50
Mathematics Subject Classification ID: 55N15
zbMATH DE Number: 1911553
zbMATH Keywords: vector bundle
zbMATH Keywords: real projective space
zbMATH Keywords: normal bundle
zbMATH Keywords: immersion
zbMATH Keywords: extendible
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.32917/hmj/1151007487
OpenAlex ID: W2112807218
Wikidata QID: Q128872020
DOI: 10.32917/HMJ/1151007487
description: scientific article; zbMATH DE number 1911553