Characterization of projective spaces and ${\mathbb{P}}^r$-bundles as ample divisors

Let $X$ be a projective manifold of dimension $n$. Suppose that $T_X$ contains an ample subsheaf. We show that $X$ is isomorphic to ${\mathbb{P}}^n$. As an application, we derive the classification of projective manifolds containing a ${\mathbb{P}}^r$-bundle as an ample divisor by the recent work of D.~Litt.