Non-extendibility of tangent bundles of complex projective spaces and quaternion projective spaces (Q1309114)

The main results of this paper are: (1) The real tangent bundle of complex projective \(n\)-space \(\mathbb{C} P^ n\) is extendible to \(\mathbb{C} P^{n + 1}\), i.e. is equivalent to the restriction of a real vector bundle on \(\mathbb{C} P^{n + 1}\), iff \(n = 1\) or 3. (2) The real tangent bundle of quaternion projective \(n\)-space \(HP^ n\) is extendible to \(HP^{n+1}\) iff \(n = 1\).