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

On the $N$-hypercontractions and similarity of multivariable weighted shifts