Parallel submanifolds (Q2655724)

The congruence theorems of Christoffel and Minkowski say that two ovaloids are congruent if the sum or the product of the curvature radii are equal in points with parallel tangent planes. Since parallelity of tangent planes is a concept of affine geometry, one may ask for similar results in affine differential geometry. Here it is shown, among other results, that a hypersurface is parallel to a proper affine sphere if it has constant affine curvature. Also parallel submanifolds of higher codimension are studied, especially Lagrangian submanifolds in complex affine space.