  In view of Mori theory, rational homogenous manifolds satisfy a recursive
condition: every elementary contraction is a rational homogeneous fibration and
the image of any elementary contraction also satisfies the same property. In
this paper, we show that a smooth Fano $n$-fold with the same condition and
Picard number greater than $n-6$ is either a rational homogeneous manifold or
the product of $n-7$ copies of $\mathbb{P}^1$ and a Fano $7$-fold $X_0$
constructed by G. Ottaviani. We also clarify that $X_0$ has non-nef tangent
bundle and in particular is not rational homogeneous.
