  In this paper, a new criterion for the similarity of commuting tuples of
operators on Hilbert spaces is introduced. As an application, we obtain a
geometric similarity invariant of tuples in the Cowen-Douglas class which gives
a partial answer to a question raised by R.G. Douglas about the similarity of
quasi-free Hilbert modules. Moreover, a new subclass of commuting tuples of
Cowen-Douglas class is obtained.
