Surfaces with $1$-type Gauss map

Introduction Submanifolds of finite type were introduced by B.-Y. Chen about thirteen years ago [2], Many works have been done in characterizing or classifying submanifolds in Euclidean space with this notion.On the other hand, several authors studied submanifolds with finite type Gauss map.B.-Y, Chen and P. Piccinni studied compact submanifolds with finite type Gauss map [3].And C. Baikoussis, B.-Y. Chen and L. Verstraelen classified ruled surfaces and tubes with finite-type Gauss map [1].Recently Y. H. Kim studied surfaces in 3-dimensional Euclidean space £3 with 1-type Gauss map and he proved that the only co-closed surfaces in E 3 with 1-type Gauss map are spheres and circular cylinders [6].In this paper we study surfaces in E z with 1-type Gauss map without the assumption of co-closedness and obtain the following theorem.THEOREM.Let M be an orientable, connected surface in E B .Then M has l-type Gauss map if and only if M is an open part of a sphere or an open part of a circular cylinder. PreliminariesLet M be an orientable, connected surface in E*.We now choose e ί and e 2 as principal normal vectors of M and let x and y the corresponding principal curvatures of the shape operator S associated with a unit normal vector e s .Let ω 1 , α> 2 , ω 3 be the dual 1-forms to e ίf e 2 and e 3 and α>5 the connection forms associated with ω 1 , ω 2 , ω 3 satisfying α>2+ α>^=0 and y=ωl(e 2 )=h(e 2) e 2 ), h(e l9 e 2 )= 1980 Mathematics Subject Classification (1985 Revision).53C40, 53A05.