  The variety of minimal rational tangents associated to Hecke curves was used
by J.-M.Hwang [8] to prove the simplicity of the tangent bundle on the moduli
of vector bundles over a curve. In this paper, we use the tangent maps of the
symplectic and orthogonal Hecke curves to prove an analogous result for
symplectic and orthogonal bundles. In particular, we show the nondegeneracy of
the associated variety of minimal rational tangents, which implies the
simplicity of the tangent bundle on the moduli spaces of symplectic and
orthogonal bundles over a curve. We also show that for large enough genus, the
tangent map is an embedding for a general symplectic or orthogonal bundle.
