Stable extendibility of vector bundles over real projective spaces

Let F be either the real number field IS or the complex number field C and RPn 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 RPn to be stably extendible to RPn for every m >= n. In this paper, we simplify the theorems and apply them to the tangent bundle of RPn, its complexification, the normal bundle associated to an immersion of RPn in Rn+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]. (C) 2013 Elsevier B.V. All rights reserved.