Let $X$ be a smooth complex projective variety such that the restriction of $TX$ on any curve $C$ in $X$ is ample. Is true in this case that $X$ is isomorphic to $\mathbb CP^n$?

I guess the above condition implies that that $TX$ is nef (i.e. $O(1)$ is nef on $\mathbb P(TX)$), but it is not clear for me that this condition implies that $TX$ is ample (which then implies that $X\cong \mathbb CP^n$ according a theorem of Mori).