Let $n\ge 0$ be an integer, let $\mathbb{CP}^n$ be complex projective $n$-space, and let $T_{\mathbb R}\mathbb{CP}^n\to \mathbb{CP}^n$ denote its smooth real tangent bundle. Let $\mathcal O(1)\to \mathbb{CP}^n$ be the hyperplane line bundle, and write