The method of moving frames applied to Kähler submanifolds of complex space forms (Q2640163)

In this article the author studies Kähler submanifolds of complex space forms by the method of moving frames, which was used originally by E. Cartan. He introduces the \(S_ c\)-structure (P,\(\omega\)) over a connected complex manifold M, where P is a principal fiber bundle over M, \(\omega\) is a \(g_ c(N)\)-valued 1-form on P and \(g_ c(N)\) is the Lie algebra of the group \(G_ c(N)\) of holomorphic isometries for the N- dimensional complex space form \(S_ c(N)\) of holomorphic sectional curvature 4c. The author proves the uniqueness of \(S_ c\)-structures for the Kähler metric on M and the rigidity theorem of Calabi. It is shown that a Kähler immersion of M into \(S_ c(N)\) induces an \(S_ c\)- structure over M and conversely if M is simply connected, every \(S_ c\)- structure over M is obtained in this way. The author also considers necessary and sufficient conditions that a connected complete Kähler submanifold M of \(S_ c(N)\) is homogeneous.