Restricted tangent bundle of rational curves on projective hypersurfaces

We determine all triples $(e,d,n)$ for which a general degree $d$ hypersurface $X\subset {\mathbb{P}}^n$ contains a degree $e$ rational curve $C$ with balanced restricted tangent bundle $T_X{\mathrm{\mid }}_C$. In addition, we show how to compute explicit examples of hypersurfaces with balanced $T_X{\mathrm{\mid }}_C$ when $C$ is a rational normal curve.