Totally geodesic holomorphic subspaces

In [G. Munteanu, Complex Spaces in Finsler, Lagrange and Hamilton Geometries, vol. 141, Kluwer Academic Publishers, Dordrecht, FTPH, 2004.] we underlined the motifs of a remarkable class of complex Finsler subspaces, namely the holomorphic subspaces. With respect to the Chern–Finsler complex connection (see [M. Abate, G. Patrizio, Finsler Metrics—A Global Approach, Lecture Notes in Mathematics, vol. 1591, Springer, Berlin, 1994.]) we studied in [G. Munteanu, The equations of a holomorphic subspace in a complex Finsler space, Publicationes Math. Debrecen, submitted for publication.] the Gauss, Codazzi and Ricci equations of a holomorphic subspace, the aim being to determine the interrelation between the holomorphic sectional curvature of the Chern–Finsler connection and that of its induced tangent connection.In the present paper, by means of the complex Berwald connection, we study totally geodesic holomorphic subspaces. With respect to complex Berwald connection the equations of the holomorphic subspace have simplified expressions. The totally geodesic subspace request is characterized by using the second fundamental form of complex Berwald connection.