  In 1978, M. J. Cowen and R.G. Douglas introduce a class of operators (known
as Cowen-Douglas class of operators) and associates a Hermitian holomorphic
vector bundle to such an operator in a very influential paper. They give a
complete set of unitary invariants in terms of involving the curvature and its
covariant partial derivatives. At the same time they ask: can one use geometric
ideas to characterize completely the similarity invariants of Cowen-Douglas
operators? We give a partial answer to this question. In this paper, we show
that the curvature and the second fundamental form completely characterize the
similarity invariants for a norm dense class of Cowen-Douglas operators.
