The present paper concerns local theory of operator tuples in the Cowen-Douglas class \(\mathcal {B}_n^m(\Omega) \). We start with point-wise localizations to introduce a kind of operator-valued invariants with which a Specht-type classification for unitary equivalence of \(\mathcal {B}_n^m(\Omega) \) is obtained. Further more, we investigate localization of \(\mathcal {B}_n^m(\Omega) \) on analytic sub-manifolds with a tensorial approach to its geometric classification theory where, among other things, the Specht's invariants are related to curvatures of the holomorphic vector bundles associated to \(\mathcal {B}_n^m(\Omega) \).