  In this paper, the general disagreement of the geometrical lyapunov exponent
with lyapunov exponent from tangent dynamics is addressed. It is shown in a
quite general way that the vector field of geodesic spread $\xi^k_G$ is not
equivalent to the tangent dynamics vector $\xi^k_T$ if the parameterization is
not affine and that results regarding dynamical stability obtained in the
geometrical framework can differ qualitatively from those in the tangent
dynamics. It is also proved in a general way that in the case of Jacobi metric
-frequently used non affine parameterization-, $\xi^k_G$ satisfies differential
equations which differ from the equations of the tangent dynamics in terms that
produce parametric resonance, therefore, positive exponents for systems in
stable regimes.
