Stable extendibility of vector bundles over and the stable splitting problem

Let F be the real number field R or the complex number field C, and let RP n denote the real projective n-space. In this paper, we study the conditions for a given F-vector bundle over RP n to be stably extendible to RP m for every m > n , and establish the formulas on the power ζ r = ζ ⊗ ⋯ ⊗ ζ (r-fold) of an F-vector bundle ζ over RP n . 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 RP n , its complexification and extendibility, Hiroshima Math. J. 37 (2007) 101–109]. Furthermore, we answer the stable splitting problem for F-vector bundles over RP n by means of arithmetic conditions.