Extendible and stably extendible vector bundles over real projective spaces

Abstract

The purpose of this paper is to study extendibility and stable extendibility of vector bundles over real projective spaces. We determine a necessary and sufficient condition that a vector bundle $\zeta$ over the real projective $n$-space $R{P}^n$ is extendible (or stably extendible) to $R{P}^m$ for every $m>n$ in the case where $\zeta$ is the complexification of the tangent bundle of $R{P}^n$ and in the case where $\zeta$ is the normal bundle associated to an immersion of $R{P}^n$ in the Euclidean $(n+k)$-space $R^{n+k}$ or its complexification, and give examples of the normal bundle which is extendible to $R{P}^N$ but is not stably extendible to $R{P}^{N+1}$.