The Schwarzian derivative in several complex variables (III)

In the point view of Lie group, the cross ratio and Schwarzian derivative in Cⁿ are defined and discussed, especially the Schwarzian derivative of holomorphic mappings on the domains in matrix space C^{m x n} is defined and discussed. It is proved that it is invariant up to similarity under the group of holomorphic automorphism of the Grassmann manifold CG(m, n). And it is also proved that the Schwarzian derivative equals zero if and only if the mapping is liaearly fractional. Project supported by the National Natural Science Foundation of China.