  We show that if the variety of minimal rational tangents (VMRT) of a uniruled
projective manifold at a general point is projectively equivalent to that of a
symplectic or an odd-symplectic Grassmannian, the germ of a general minimal
rational curve is biholomorphic to the germ of a general line in a
presymplectic Grassmannian. As an application, we characterize symplectic and
odd-symplectic Grassmannians, among Fano manifolds of Picard number 1, by their
VMRT at a general point and prove their rigidity under global K\"ahler
deformation. Analogous results for $G/P$ associated with a long root were
obtained by Mok and Hong-Hwang a decade ago by using Tanaka theory for
parabolic geometries. When $G/P$ is associated with a short root, for which
symplectic Grassmannians are most prominent examples, the associated local
differential geometric structure is no longer a parabolic geometry and standard
machinery of Tanaka theory cannot be applied because of several degenerate
features. To overcome the difficulty, we show that Tanaka's method can be
generalized to a setting much broader than parabolic geometries, by assuming a
pseudo-concavity type condition that certain vector bundles arising from
Spencer complexes have no nonzero sections. The pseudo-concavity type condition
is checked by exploiting geometry of minimal rational curves.
