A Generalization of Bonnet's Theorem on Darboux Surfaces

The well-known Bonnet theorem claims that, on a Darboux surface in three-dimensional Euclidean space, along each line of curvature, the corresponding principal curvature is proportional to the cube of another principal curvature. In the present paper, this theorem is generalized (with respect to dimension) to $n$-dimensional hypersurfaces of Euclidean spaces.