Is the Hamiltonian geometrical criterion for chaos always reliable?

It is found that the application of a newly developed geometrical criterion, in which negative eigenvalues of the associated matrix determined by the dynamical curvature of a conformal metric for a Hamiltonian system are used to identify the onset of local instability or chaos, is somewhat problematic in some circumstances. In fact, this criterion is neither necessary nor sufficient for the prediction of instability of orbits on a same energy hypersurface because it is not in good agreement with information on unstable or chaotic behavior given by the maximal Lyapunov exponent in general.