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Using the tools of Differential Geometry, we define a new <> chaoticity indicator, able to detect dynamical instability of trajectories much more effectively, (i.e. "quickly") than the usual tools, like Lyapunov Characteristic Numbers (LCN's) or Poincare` Surface of Section. Moreover, at variance with other "fast" indicators proposed in the Literature, it gives informations about the asymptotic behaviour of trajectories, though being local in phase-space. Furthermore, it detects the chaotic or regular nature of geodesics without any reference to a given perturbation and it allows also to discriminate between different regimes (and possibly sources) of chaos in distinct regions of phase-space., Comment: 5 pages with EPS figures embedded. Short Version, to appear in the Proceedings of the 9th Marcel Grossmann Meeting. (World Scientific). E-mail: piero.cipriani@roma1.infn.it