Stable extendibility of vector bundles over real projective spaces

Let F be either the real number field R or the complex number field C and RP n the real projective space of dimension n. Theorems A and C in Hemmi and Kobayashi (2008) [2] give necessary and sufficient conditions for a given F -vector bundle over RP n to be stably extendible to RP m for every m ⩾ n . In this paper, we simplify the theorems and apply them to the tangent bundle of RP n , its complexification, the normal bundle associated to an immersion of RP n in R n + r ( r > 0 ) , and its complexification. Our result for the normal bundle is a generalization of Theorem A in Kobayashi et al. (2000) [8] and that for its complexification is a generalization of Theorem 1 in Kobayashi and Yoshida (2003) [5].