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 -fold with the same condition and Picard number greater than is either a rational homogeneous manifold or the product of copies of and a Fano -fold constructed by G. Ottaviani. We also clarify that has a non-nef tangent bundle and in particular is not rational homogeneous.
"Extremal Rays and Nefness of Tangent Bundles." Michigan Math. J. 68 (2) 301 - 322, June 2019. https://doi.org/10.1307/mmj/1549681299