  We show that deformations of a surjective morphism onto a Fano manifold of
Picard number 1 are unobstructed and rigid modulo the automorphisms of the
target, if the variety of minimal rational tangents of the Fano manifold is
non-linear or finite. The condition on the variety of minimal rational tangents
holds for practically all known examples of Fano manifolds of Picard number 1,
except the projective space. When the variety of minimal rational tangents is
non-linear, the proof is based on an earlier result of N. Mok and the author on
the birationality of the tangent map. When the varieties of minimal rational
tangents of the Fano manifold is finite, the key idea is to factorize the given
surjective morphism, after some transformation, through a universal morphism
associated to the minimal rational curves.
