  Smooth Schubert varieties in rational homogeneous manifolds of Picard number
1 are horospherical varieties. We characterize standard embeddings of smooth
Schubert varieties in rational homogeneous manifolds of Picard number 1 by
means of varieties of minimal rational tangents. In particular, we mainly
consider nonhomogeneous smooth Schubert varieties in symplectic Grassmannians
and in the 20-dimensional $F_4$-homogeneous manifold associated to a short
simple root.
