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For a domain \Omega in \mathbb{C} and an operator T in \mathcal{B}_n(\Omega),Cowen and Douglas construct a Hermitian holomorphic vector bundle E_T over\Omega corresponding to T. The Hermitian holomorphic vector bundle E_T isobtained as a pull-back of the tautological bundle S(n,\mathcal{H}) definedover \mathcal{G}r(n,\mathcal{H}) by a nondegenerate holomorphic map z\mapsto{\rm{ker}}(T-z) for z\in\Omega. To find the answer to the converse, Cowen andDouglas studied the jet bundle in their foundational paper. The computations inthis paper for the curvature of the jet bundle are somewhat difficult tocomprehend. They have given a set of invariants to determine if two rank nHermitian holomorphic vector bundle are equivalent. These invariants arecomplicated and not easy to compute. It is natural to expect that theequivalence of Hermitian holomorphic jet bundles should be easier tocharacterize. In fact, in the case of the Hermitian holomorphic jet bundle\mathcal{J}_k(\mathcal{L}_f), we have shown that the curvature of the linebundle \mathcal{L}_f completely determines the class of\mathcal{J}_k(\mathcal{L}_f). In case of rank n Hermitian Holomorphic vectorbundle E_f, We have calculated the curvature of jet bundle \mathcal{J}_k(E_f)and also have generalized the trace formula for jet bundle \mathcal{J}_k(E_f).

Trace formulae for curvature of Jet Bundles over planar domain