  In \cite{SH}, A. L. Shields proved a well-known theorem for the similarity of
unilateral weighted shift operators. By using the generalization of this
theorem for multivariable weighted shifts and the curvature of holomorphic
bundles, we give a necessary and sufficient condition for the similarity of
$m$-tuples in Cowen-Douglas class. We also present a necessary condition for
commuting $m$-tuples of backward weighted shift operators to be
$n$-hypercontractive in terms of the weight sequences.
