Subbundles of tangent bundles

In this note we study the existence of subbundles or quotient bundles of the tangent bundle of some non-singular algebraic varieties. It is proved that the tangent bundle of a general complete intersection surface has no subbundles. Then we prove that the tangent bundle $T_{\mathbb{P}^n}$ of the projective space $\mathbb{P}^n$ has no subbundle or quotient bundle of rank r with $2 \leq r \leq 7$ if $r \leq n-2$. Furthermore, if $n+1=p_{1}^kp_{2}^s$ with $p_1$ and $p_2$ odd primes, $T_{\mathbb{P}^n}$ has no sub bundles. For the proof, we use the theory of uniform vector bundles on $T_{\mathbb{P}^n}$.