The equations of a holomorphic subspace in a complex Finsler space

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. 1. Holomorphic subspaces In this section we briefly recall the general settings obtained in [M-P], [Mu1] concerning the holomorphic subspaces of a complex Lagrange space, particularly a Finsler space, and complete them with Gauss andWeingarten formulas. This class of submanifolds seems to be fruitful in results because the tangent induced connection by the Chern-Lagrange connection of the ambient complex Lagrange space is exactly the Chern-Lagrange connection of the subspace. Particularly this assertion remains true for complex Finsler spaces. For this reason the theory of holomorphic subspaces becomes attractive and can cover a large area from the geometry of submanifolds. In this paper our aim is to obtain the equations of the subspace and then to study the induced holomorphic curvature. Let M be a complex manifold of dimension n, (z)k=1,n complex coordinates in a local chart, and M̃ an embedded complex submanifold of M , {ξ}α=1,m local complex coordinates in M̃ . Assume that the inclusion map i: M̃ →֒ M is holomorphic. It follows that M̃ has a local holomorphic representation of the form Mathematics subject classification number : 53B40, 53B25.