Weingarten Surfaces Of Main Curvature Satisfying A Class Of Rational Function Relations

This thesis is devoted to the study of Weingarten surfaces in three dimensional Euclidean space whose two principle curvatures satisfy a rational function. For such surfaces, we not only classify their fundamental equations, but also give their Lax pairs. This thesis is organized as follows.The first chapter is an introduction. We first introduce the Weingarten surface and its history. We then introduce some known results about Weingarten surface. Finally we state the problem studied in this thesis.The second chapter is preliminary. For a surface in three dimensional Euclidean space, the Gauss-Weingarten equations and Gauss-Codazzi equations can be obtained by differentiating its moving equations. The Gauss-Codazzi equations are the compatibility conditions of the Gauss-Weingarten equations. By use of a homomorphism from SO(3,R) into SL(2,C), one can obtain the Gauss-Weingarten equations in 2×2 matrix from. For a Weingarten surface, by choosing the lines of curvature as its parametric curves, one can solve the Codazzi equations by integration.The third chapter is the main part of this thesis. For a Weingarten surface whose two principal curvatures K1 and K2 satisfy the rational function K2＝aK1+b/K12+cK1+d,we classify its Gauss-Codazzi equations, and at the same time we obtain its Gauss-Weingarten equations in 2×2 matrix from. According to the factorization of numerator in K2-K1, we divide our discussions into four different cases. In each case, by solving the Codazzi equation, we obtain the Gauss equation and its Lax pair.In Chapter 4 we state some questions related to the results obtained in this thesis. It is well known that in three-dimensional Euclidean space, the fundamental equation of a pseudo-sphere is the sine-Gordon equation. The classical Backlund transformation on pseudo-sphere is defined by the Lax pair of the sine-Gordon equation. For Weingarten surfaces studied in this thesis, we obtain their fundamental equations and Lax pairs. In the next, one may consider if Backlund transformation can be constructed on these Weingarten surfaces.