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In this paper we study quasi-homogeneous operators, which include thehomogeneous operators, in the Cowen-Douglas class. We give two separatetheorems describing canonical models (with respect to equivalence under unitaryand invertible operators, respectively) for these operators using techniquesfrom complex geometry. This considerably extends the similarity and unitaryclassification of homogeneous operators in the Cowen-Douglas class obtainedrecently by the last author and A. Kor\'{a}nyi. Specifically, the complexgeometric invariants used for our classification are the curvature and thesecond fundamental forms inherent in the definition of a quasi-homogeneousoperator. We show that these operators are irreducible and determine when theyare strongly irreducible. Applications include the equality of the topologicaland algebraic K-group of a quasi-homogeneous operator and an affirmative answerto a well-known question of Halmos on similarity for these operators.

Curvature and the Second fundamental form in classifying quasi-homogeneous holomorphic curves and operators in the Cowen-Douglas class