Some affine version of the Bäcklund theorem (Q2628391)

Consider surfaces \(f,\hat{f}:M\rightarrow \mathbb{R}^{3}\) in Euclidean space \(\mathbb{R}^{3}\) with the following properties: (B) for every \(p\in M\), one has \(f(p)\neq \hat{f}(p)\), and the vector \(\hat{f}(p)-f(p)\) is tangent to \(f(M)\) at \(f(p)\) and to \(\hat{f}(M)\) at \(\hat{f}(p)\), (L) the length \(L:=|\hat{f}(p)-f(p)|\) of the vector \(\hat{f}(p)-f(p)\) is independent of \(p\), (A) the angle \(\sigma\) between the Euclidean normals \(\mathbf n\) and \(\mathbf {\hat n}\) at corresponding points of \(f(M)\) and \(\hat{f}(M)\) is constant with \(\sin\sigma\neq 0\). Then the classical Bäcklund theorem states: Both surfaces are of constant negative Gaussian curvature \(\kappa=\hat{\kappa}=-{{\sin^{2}{\sigma}}\over{L^{2}}}\) and the second fundamental forms of \(f\) and \(\hat{f}\) are proportional. The main result of the present paper is an affine version of the Bäcklund theorem, corresponding to \(\sigma = \pi/2\): \vskip 4pt Theorem 1.4: Let \(f,\hat{f}:M\rightarrow \mathbb{R}^{3}\) be non-degenerate immersions of a two-dimensional real manifold \(M\) into the affine space \(\mathbb{R}^{3}\), satisfying (B) from above and the following conditions: \vskip 4pt (T) for every \(p\in M\) the affine normal \({\mathcal N}_p\) of \(f\) at \(f(p)\) is tangent to \(\hat{f}(M)\) at \(\hat{f}(p)\) and the affine normal \(\hat{\mathcal N}_p\) of \(\hat{f}(M)\) at \(\hat{f}(p)\) is tangent to \(f(M)\) at \(f(p)\), (C) the affine fundamental forms of \(f\) and \(\hat{f}\) are conformal to each other, (D) for all equiaffine local sections \(\xi\) and \(\hat{\xi}\) of \(\mathcal N\) and \(\hat{\mathcal N}\) respectively, \(\det(\hat{f}-f,\xi,\hat{\xi})=\mathrm{const}\). Then the Blaschke connections \(\nabla\) and \(\hat{\nabla}\), of \(f\) and \(\hat{f}\) respectively, are locally symmetric. Under the assumptions of Theorem 1.4 it turns out, that \(\dim \operatorname{im} R = \dim \operatorname{im}\hat{R}\), where \(R\) and \(\hat R\) are the curvature tensors. Furthermore it is proved that, assuming the hypotheses of Theorem 1.4 and \(\dim \operatorname{im} R = 2\), the Blaschke connections of \(f\) and \(\hat{f}\) are metrizable. So exactly this case corresponds to the situation described in the classical Bäcklund theorem or in the Bäcklund theorem for surfaces in Minkowski space. The paper ends with an additional version of the Bäcklund theorem in Minkowski space.