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Abstract: Let be either the real number field or the complex number field and 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 -vector bundle over to be stably extendible to for every . In this paper, we simplify the theorems and apply them to the tangent bundle of , its complexification, the normal bundle associated to an immersion of in , 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]. [Copyright &y& Elsevier]

Abstract: Let F be the real number field R or the complex number field C, and let denote the real projective n-space. In this paper, we study the conditions for a given F-vector bundle over to be stably extendible to for every , and establish the formulas on the power (r-fold) of an F-vector bundle ζ over . Our results are improvements of the previous papers [T. Kobayashi, H. Yamasaki, T. Yoshida, The power of the tangent bundle of the real projective space, its complexification and extendibility, Proc. Amer. Math. Soc. 134 (2005) 303–310] and [Y. Hemmi, T. Kobayashi, Min Lwin Oo, The power of the normal bundle associated to an immersion of , its complexification and extendibility, Hiroshima Math. J. 37 (2007) 101–109]. Furthermore, we answer the stable splitting problem for F-vector bundles over by means of arithmetic conditions. [Copyright &y& Elsevier]