In this paper we introduce a new type of folding called equi-Gaussian curvature folding of connected Riemannian 2-manifolds. We prove that the composition and the cartesian product of such foldings is again an equi-Gaussian curvature folding. In case of equi-Gaussian curvature foldings, f : M → P n , of an orientable surface M onto a polygon P n we prove thatand we generalize (iii) for #nT 2 .

In our previous work (2002), we proved that an essential second-order hypersurface in an infinite-dimensional locally affine Riemannian Banach manifold is a Riemannian manifold of constant nonzero curvature. In this note, we prove the converse, in other words, we prove that a hypersurface of constant nonzero Riemannian curvature in a locally affine (flat) semi-Riemannian Banach space is an essential hypersurface of second order.

We prove that an essential hypersurface of second order in an infinite-dimensional locally affine Riemannian Banach manifold is a Riemannian manifold of constant nonzero curvature.