For \(\Omega \subseteq \mathbb {C}\) a connected open set, and \({\mathcal {U}}\) a unital \(C^*\)-algebra, let \({\mathcal {I}} ({\mathcal {U}})\) and \({\mathcal {P}}({\mathcal {U}})\) denote the sets of all idempotents and projections in \({\mathcal {U}}\) respectively. A set \({\mathcal {P}}({\mathcal {U}})\) is called the Grassmann manifold of \(\mathcal {U}\) and \({\mathcal {I}} ({\mathcal {U}})\) is called the extended Grassmann manifold. If \(P:\Omega \rightarrow {\mathcal {P}}({\mathcal {U}})\) is a real-analytic \({\mathcal {U}}\)-valued map which satisfies \(\overline{\partial } PP=0\), then P is called a holomorphic curve on \({\mathcal {P}}({\mathcal {U}})\). In this note, we will define the formulas of curvature and itâs covariant derivatives for holomorphic curves on \(C^*\)-algebras. It can be regarded as a generalization of curvature and itâs covariant derivatives from complex geometry. By using the curvature formula, we give the unitarily and similarity classification theorems for the holomorphic curves and extended holomorphic curves on \(C^*\)-algebras respectively. Furthermore, we also give a description of the trace of the covariant derivatives of curvature for any Hermitian holomorphic vector bundles. Applications include the similarity of holomorphic Hermitian vector bundles and CowenâDouglas operators.