The power of the normal bundle associated to an immersion of $\RP^n$, its complexification and extendibility

Abstract

The purpose of this paper is to establish the formulas on the power of the normal bundle associated to an immersion of the real projective space $\RP^n$ in the Euclidean space $\R^{n+k}$, and apply them to the problem of extendibility and stable extendibility. Furthermore, we give an example of a $2$-dimensional $\R$-vector bundle over $\RP^2$ that is stably extendible to $\RP^3$ but is not extendible to $\RP^3$.