  We consider a subclass of the Cowen-Douglas class in which the problem of
deciding whether two operators are similar becomes more manageable. A
similarity criterion for Cowen-Douglas operators is known to be dependent on
the trace of the curvatures of the corresponding eigenvector bundles. Unless
the given eignvector bundle is a line bundle, the computation of the
curvatures, in general, is not so simple as one might hope. By using a
structure theorem given in \cite{JW}, we reduce the problem of finding the
trace of the curvatures to looking at the curvatures of the associated line
bundles. Moreover, several questions related to the similarity problem are also
taken into account.
