We consider similarity and quasi-affinity problems for Hilbert modules in the Cowen–Douglas class associated with the complex geometric objects, the hermitian anti-holomorphic vector bundles and curvatures. Given a “simple” rank one Cowen–Douglas Hilbert module $$\mathscr {M}$$ , we find necessary and sufficient conditions for a class of Cowen–Douglas Hilbert modules satisfying some positivity conditions to be similar to We also show that under certain uniform bound condition on the anti-holomorphic frame, a Cowen–Douglas Hilbert module is quasi-affinity to a submodule of the free module