Weingarten Surfaces In Euclidean 3-space

Differential geometry is a subject which has a long history. In recent years, it has a deeper influence on other subjects more and more. Curve theory and surface theory are two important elements in differential geometry. It is well-known that the surfaces whose the Gaussian curvature and the mean curvature satisfy some relationships are Weingarten surfaces. The properties of the surfaces can be related to the Gaussian curvature and the mean curvature, so it is very important to study the Weingarten surfaces.In Euclidean 3-space, many people have studied the Weingarten surfaces. H.Hopf studied the Weingarten surfaces in 1956. J.A.Galvez, A.Martinez and F.Milan studied the linear Weingarten surfaces in R3 in 2002. Besides Juan A.Aledo Sanchez and Jose M.Espinar studied the hyperbolic linear Weingarten surfaces in R3 in 2006.In this paper leading in the Cauchy-Riemann operator and considering the method of local surface isothermal coordinates. We studied structural equation and integrable condition of the surfaces. And we obtain a representation for a linear Weingarten surface whose Gaussian curvature and mean curvature satisfy the relationship 2aH+bK=c.