Extendibility and stable extendibility of the square of the normal bundle associated to an immersion of the real projective space
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.
scientific article; zbMATH DE number 1911553