In [Mu1] we underlined the motifs of holomorphic subspaces in a complex Finsler space: induced nonlinear connection, coupling
connections, and the induced tangent and normal connections. In the present paper we investigate the equations of Gauss, H-and A-Codazzi, and Ricci equations of a holomorphic subspace. We deduce the link between the holomorphic curvatures of the Chern-Finsler
connection and its induced tangent connection. Conditions for totally geodesic holomorphic subspaces are obtained.