Let X be a Fano manifold of Picard number 1 with numerically effective tangent bundle. According to the principal case of a conjecture of Campana-Peternell's, X should be biholomorphic to a rational homogeneous manifold G/P, where G is a simple Lie group, and P subset of G is a maximal parabolic subgroup. In our opinion there is no overriding evidence for the Campana-Peternell Conjecture for the case of Picard number 1 to be valid in its full generality. As part of a general programme that the author has undertaken with Jun-Muk Hwang to study uniruled projective manifolds via their varieties of minimal rational tangents, a new geometric approach is adopted in the current article in a special case, consisting of (a) recovering the generic variety of minimal rational tangents C-x, and (b) recovering the structure of a rational homogeneous manifold from C-x. The author proves that, when b(4)(X) = 1 and the generic variety of minimal rational tangents is 1-dimensional, X is biholomorphic to the projective plane P-2, the 3-dimensional hyperquadric Q(3), or the 5-dimensional Fano homogeneous contact manifold of type G(2), to be denoted by K(G(2)). The principal difficulty is part (a) of the scheme. We prove that C-x subset of PTx(X) is a rational curve of degrees less than or equal to3, and show that d = 1 resp. 2 resp. 3 corresponds precisely to the cases of X = P-2 resp. Q(3) resp. K(G(2)). Let K be the normalization of a choice of a Chow component of minimal rational curves on X. Nefness of the tangent bundle implies that K is smooth. Furthermore, it implies that at any point x is an element of X, the normalization K-x of the corresponding Chow space of minimal rational curves marked at x is smooth. After proving that K-x is a rational curve, our principal object of study is the universal family U of K, giving a double fibration rho : U --> K,mu : U --> X, which gives P-1-bundles. There is a rank-2 holomorphic vector bundle V on K whose projectivization is isomorphic to rho : U --> K. We prove that V is stable, and deduce the inequality d less than or equal to 4 from the inequality c(1)(2) (V) less than or equal to 4c(2)(V) resulting from stability and the existence theorem on Hermitian-Einstein metrics. The case of d = 4 is ruled out by studying the structure of the curvature tensor of the Hermitian-Einstein metric on V in the special case where c(1)(2)(V) = 4c(2)(V). [References: 21]
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