In this very famous paper:

Mori, Shigefumi, Projective manifolds with ample tangent bundles. Ann. of Math. (2) 110 (1979), no. 3, 593–606.

it is proven that over an algebraically closed field of chracteristic 0 $\mathbb P^n(K)$ is the only manifold $X$ with ample tangent bundle.

In the introduction the author points out that the statement is true under the weaker conditions: 1) $-K_X$ is ample; 2) for any non constant map $\mathbb P^1\to X$ the pull back of $T_X$ is ample on $\mathbb P^1$.

In this very famous paper:

Mori, Shigefumi, Projective manifolds with ample tangent bundles. Ann. of Math. (2) 110 (1979), no. 3, 593–606.

it is proven that over an algebraically closed field of characteristic 0 $\mathbb P^n(K)$ is the only manifold $X$ with ample tangent bundle.

In the introduction the author points out that the statement is true under the weaker conditions: 1) $-K_X$ is ample; 2) for any non constant map $\mathbb P^1\to X$ the pull back of $T_X$ is ample on $\mathbb P^1$.

In this very famous paper:

Mori, Shigefumi, Projective manifolds with ample tangent bundles. Ann. of Math. (2) 110 (1979), no. 3, 593–606.

it is proven that over an algebraically closed field of chracteristic 0 $\mathbb P^n(K)$ is the only manifold $X$ with ample tangent bundle.

In the introduction the author points out that the statement is true under the weaker conditions: 1) $K_X$ is ample; 2) for any non constant map $\mathbb P^1\to X$ the pull back of $T_X$ is ample on $\mathbb P^1$.

In this very famous paper:

Mori, Shigefumi, Projective manifolds with ample tangent bundles. Ann. of Math. (2) 110 (1979), no. 3, 593–606.

it is proven that over an algebraically closed field of chracteristic 0 $\mathbb P^n(K)$ is the only manifold $X$ with ample tangent bundle.

In the introduction the author points out that the statement is true under the weaker conditions: 1) $-K_X$ is ample; 2) for any non constant map $\mathbb P^1\to X$ the pull back of $T_X$ is ample on $\mathbb P^1$.