Shugailo O. The structure of submanifolds in affine space with degenerate affine fundamental form

Properties and structure of multidimensional affine submanifolds with degenerate affine fundamental form with the additional conditions on the connection, cubic form and the Weingarten map are studied. Sufficient conditions for parallelism of null distribution of affine immersion are derived. The structure of affine submanifold of rank two is described in dependence on its point type. An affine analogues of Chern-Kuiper estimates are proved. Parallel affine immersions of codimension two with flat connection are classified in dependence on dimension of the Weingarten mapping.The structure of affine umbilical immersions with flat connection, locally symmetric connection and with nilpotent curvature operator is described.