Rights First published in [Publication] in [volume and number, or year],published by the American Mathematical Society Recognizing Certain Rational Homogeneous Manifolds of Picard Number 1 from their Varieties of Minimal Rational Tangents

Let X be a uniruled projective manifold, i.e., a projective manifold that can be filled up by rational curves. Fixing a polarization and minimizing degrees among free rational curves we have the notion of minimal rational curves. Collecting the set of all tangents to minimal rational curves passing through a given general point we obtain a variety of minimal rational tangents. In a series of articles with Jun-Muk Hwang we have been studying uniruled projective manifolds, especially Fano manifolds of Picard number 1, in terms of the geometry of varieties of minimal rational tangents. Here Fano manifolds admit rational curves by the pioneering work of Mori [18], and they are uniruled by Miyaoka-Mori [16]. In our geometric study of uniruled projective manifolds the general philosophy is that one should be able to recover the complex structure and algebro-geometric properties of a pro-jective uniruled manifold, especially in the case of Fano manifolds of Picard number 1, from the geometry of its varieties of minimal rational tangents. In this direction, when varieties of minimal rational tangents of a Fano manifold X of Picard number 1 are positive-dimensional, under very mild conditions we have shown in Hwang-Mok [8] that X is determined up to bi-holomorphism by the germ of the fiber space of varieties of minimal rational tangents at