  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. ${\mathcal P}({\mathcal U})$ is called as the
Grassmann manifold of $\mathcal U$ and ${\mathcal I} ({\mathcal U})$ is called
as 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 formulaes of curvature and it's covariant
derivatives for holomorphic curves on $C^*$-algebras. It can be regarded as the
generalization of curvature and it's covariant derivatives of the classical
holomorphic curves. By using the curvature formulae, we give the unitarily and
similarity classifications for the holomorphic curves and extended holomorphic
curves on $C^*$-algebras respectively. And we also give a description of the
trace of the covariant derivatives of curvature for any Hermitian holomorphic
vector bundles. As applications, we also discuss the relationship between
holomorphic curves, extended holomorphic curves, similarity of holomorphic
Hermitian vector bundles and similarity of Cowen-Douglas operators.
