Rational curves of minimal degree and characterizations of projective spaces

In this paper we investigate complex uniruled varieties X whose rational curves of minimal degree satisfy a special property. Namely, we assume that the tangent directions to such curves at a general point x. X form a linear subspace of TxX. As a first application of our main result, we give a unified geometric proof of Mori's, Wahl's, Campana-Peternell's and Andreatta-Wisniewski's characterizations of P-n. We also give a characterization of products of projective spaces in terms of the geometry of their families of rational curves of minimal degree.