  It is shown that the beta functions for four dimensional N=2 supersymmetric
Yang-Mills theory without matter give integral curves on the moduli space some
of which are geodesics of the natural metric on the moduli space. In particular
the flow lines which cross-over from from the weak coupling limit
(asymptotically free theory) to the singular points, representing the strong
coupling limit, are geodesics. A possible connection with irreversibility is
discussed.
