  Enlightened by the similar equation form between the GCHS \footnote{GCHS:
Generalized Covariant Hamilton System\\GSPB:Generalized structural Poisson
bracket} defined by the GSPB and the geodesic equation expressed by geospin
variable, we find a deep connection between the geospin matrix and S-dynamics.
In this contrastive way, we actually proves that the GCHS is a compatible
theory suitable for the curved spacetime as primitively stated. By contrast,
geospin matrix in Riemannian geometry has the same physical nature as
S-dynamics in GCHS. We obtain a fact that geodesic equation can be naturally
derived by the GCHS in terms of the velocity field.
  We strictly prove that the geometrio $\hat{S}{{\left( {{x}_{k}},{{p}_{i}},H
\right)}^{T}}={{\left( {{b}_{k}},{{A}_{i}},w \right)}^{T}}$ holds by using
structural operator $\hat{S}$ directly induced by structural derivative
${A}_{i}$ in terms of position ${x}_{k}$, momentum ${p}_{i}$ and Hamiltonian
$H$ respectively. It evidently proves that the GCHS on the Riemannian manifold
is certainly determined by the Christoffel symbols. As an application, we
consider the GCHS on Riemannian geometry.
