Some affine version of the Bäcklund theorem

A particular case (∠(n,n̂)=π2) of the classical Bäcklund theorem is extended to surfaces in affine space. We consider a pair of non-degenerate surfaces f and f̂ such that for every p both tangent planes f∗(TpM) and f̂∗(TpM) contain the line R(f̂(p)−f(p)), moreover the affine normal vector field ξ of f is tangent to f̂(M), the affine normal vector field ξ̂ of f̂ is tangent to f(M), the volume of the parallelopipedon spanned by f̂−f, ξ, ξ̂ is constant and the affine fundamental forms are proportional. We prove that under the conditions stated above Blaschke connections of both f and f̂ are locally symmetric. The considered situation differs from that in the theorem of Chern and Terng, where the affine normals at corresponding points were supposed to be parallel. Depending on dimimR which is shown to be equal to dimimR̂, either we deal with such surfaces as in the classical theorems for an appropriate scalar or pseudoscalar product in R3, or the connections are non-metrizable, hence are not met with in the classical case. A particular metrizable case corresponds to some new result for a pair of surfaces in Minkowski space, when one surface is timelike and the other is spacelike.