On analogues of B\"acklund theorem in affine differential geometry of surfaces

TL;DR

This paper explores affine Bäcklund theorems for surfaces, extending classical results by examining conditions for local symmetry and proportionality of affine fundamental forms in affine differential geometry.

Contribution

It formulates new affine Bäcklund theorems with non-parallel transversal fields and identifies conditions for local symmetry and proportionality of affine fundamental forms.

Findings

Formulated affine Bäcklund theorems with non-parallel transversal fields

Identified geometric conditions for local symmetry of induced connections

Provided necessary and sufficient conditions for proportional affine fundamental forms

Abstract

We recall the well-known Chern-Terng theorem concerning affine minimal surfaces. Next we formulate some complementary (with transversal fields necessarily not parallel) affine B\"acklund theorem. We describe some geometrical conditions which imply the local symmetry of both induced connections. We give also some necessary and sufficient conditions under which the affine fundamental forms are proportional.