An analogue of Bäcklund's theorem in affine geometry

Abstract

It is well-known that there is a correspondence between solutions of the Sine-Gordon equation (SGE) -~—,~Hr= sin q> dx dt Y and the surfaces of constant curvature -1 in R (see below). The classical Backlund transformation of such surfaces furnishes a way to generate new solutions of the SGE from a given solution. This has received much attention in recent studies of the soliton solutions of the SGE, and the technique has been used successfully in the study of other non-linear evolution equations. In the first section of this paper we present a simple derivation of the classical Backlund theorem and its applications by using the method of moving frames. Our main result concerns affine minimal surfaces. They arise as the solution of the variation problem for affine area. The corresponding Euler-Lagrange equation is a fourth order partial differential equation. In §2,we develop the basic properties of affine minimal surfaces.In §3 we study the transformation of affine surfaces by realizing them as the focal surfaces of a line congruence. The natural conditions that the congruence be a W-congruence and that the affine normals at corresponding points be parallel lead to the conclusion that both surfaces are affine minimal. This is the content of Theorem 4, the main result of our paper. As in the classical case, the Theorem leads to the construction of new affine minimal surfaces from a given one by the solution of a completely integrable system of first order partial differential equations. 1. The classical Backlund theorem and its consequences. Let M be a surface in R. We choose a local field of orthonormal frames v\, v O3 be the dual coframe of Vi, v2, #3. We can write dx 2^0ava a s Here and throughout this paper we shall agree on the index ranges (1.2) 1 £ij,k£2, 1 £a,s,r H 3. The structure equations of R are *Work done under partial support of NSF grant MCS74-23180. **Work done under partial support of NSF Grant MCS76-01692. Received by the editors on April 13, 1978. Copyright © 1980 Rocky Mountain Mathematics Consortium 105 106 S. S. CHERN AND C. L. TERNG dda = 2] Os A Osa, 6as + Osa = 0 ddas = 2 0«r A 0ri8. Restricting these forms to the frames defined above, we have (1.4) 03 = 0 and hence (1.5) 0 = </03 = L 0,-A 0,3. By Cartan's lemma we may write (1.6) o« = S M y A/y = V The first equation of (1.3) gives (1.7) d0< = S 0 y A0„y (where 012 is the Levi-Civita connection form on M which is uniquely determined by these two equations). The Gauss equation is (1.8) <*0,7 = E 0» A 0*y + e „ where ©12 = flu» A 023 = (det(/j,7))0i A 02, (1.9) K^LL det(Af-y) = the Gaussian curvature of M. The Codazzi equations are (1.10) rf0f-3 = £ 0l7 A 0y3. y The first fundamental form of the surface is (1.11) I = (0i) + (02) , and the second fundamental form is (1.12) II = £ 0,0,3 = E h fi fi j . i ij Near a non-umbilical point M can be parametrized by its lines of curvature, i.e., there are coordinates utin which I and II are both diagonaHzed. Explicitly we write I = (*i)(<fai) + (a2f(du2f II = bfaWutf + b2(a2)\du2)