Affine Factorable Surfaces In Euclidean 3-space

The research on the surfaces has always been an important field of differential geometry. In 3-dimensional Euclidean space and in 3-dimensional Minkowski space, the factorable surface is very special and has been studied by many experts and scholars. Because of the indefiniteness of the metric, there are three kinds of vectors in 3-dimensional Minkowski space:spacelike vectors, timelike vectors and lightlike vectors. So there are six kinds of factorable surfaces along different directions. In 3-dimensional Euclidean space, there is one type.It is well-known that the surfaces whose Gaussian curvature and mean curvature satisfy some relationships are Weingarten surfaces. The properties of the surfaces can be decided by Gaussian curvature and mean curvature, so it is very important to study the Weingarten surfaces. Thus, the research on Weingarten factorable surfaces is very valuable. In 2007, Professor Huili Liu and Yanhua Yu made the classification of factorable minimal surfaces respectively in 3-dimensional Euclidean space and in 3-dimensional Minkowski space. In 2008, the idea that Weingarten factorable surfaces in 3-dimensional Euclidean space was promoted into the study of 3-dimensional Minkowski space by Huihui Meng. At the same time, the existence and expressions of several factorable surfaces were discussed. However, these studies were basically carried out under the condition of the orthogonal transformation. This paper mainly studied on affine factorable surfaces which were based on positive definite metrics and under affine transformation.First of all, the general way of expressing parametric equation of affine factorable surface with parameter a which is a constant was found. This definition of surface made the research scope of factorable surface more widely in Euclidean space due to the changing a. Then, according to the basic knowledge of differential geometry, the first and second basic forms of affine factorable surface and Gaussian curvature and mean curvature were directly calculated in Euclidean space. Finally, concrete expressions of factorable surface were got by stipulating Gaussian curvature and mean curvature are zero, and these affine factorable surfaces were classified.