The Plot Of A Class Of Weingarten Surfaces Can

This thesis is divided into four chapters. In the first chapter, we first review the classic Backlund transformation. This is a transformation on pseudo-spheres in three-dimensional Euclidean space, and it can be used to construct a lot of interesting pseudo-spheres; Then we introduce some generations of the classic Backlund transformation, such as the transformation on surfaces for which the Gaussian and mean curvatures are linearly related; Finally we state the problem studied in this thesis. In the second chapter, we point out that for surfaces in three-dimensional Euclidean space, its Gauss-Codazzi equations are the integrability conditions of its Gauss-Weingarten equations, therefore the Gauss-Weingarten equations are the Lax-pair of the Gauss-Codazzi equations. Moreover, since there exists a homomorphism from SO(3,R) into SL(2,C), the Lax-pair in2by2forms can be constructed. The third chapter is the main part of this thesis. In this chapter we study the integrability of surfaces for which the principal curvatures κ1and κ2satisfy a quadratic relationship. We classify Gauss-Codazzi equations for such surfaces, and give their Lax-pairs. Let κ1=k, κ2=f(k)=mk2+(n+1)k+l,, f(k)-k=mk2+nk+l=m(k+n/2m)2+4ml-n2/4m.Our discussion is divided into three cases:n2-4ml=0, n2-4ml>0, n2-4ml<0, and we obtain the following results:1. When n2-4ml=0, the fundamental equation for such a Weingarten surface is and its Lax pair is 2. When n2-4ml>0, the fundamental equation for such a Weingarten surface is and its Lax pair is3. When n2-4ml<0, the fundamental equation for such a Weingarten surface is and its Lax pair isIn the fourth chapter we verify the integrability of the above Lax pairs.