In the tangent plane at any point of a surface in the four-dimensional Euclidean space we consider an invariant linear map ofWeingarten-type and find a geometrically determined moving frame field. Writing derivative formulas of Frenet-type for this frame field, we obtain eight invariant functions. We prove a fundamental theorem of Bonnet-type, stating that these eight invariants under some natural conditions determine the surface up to a motion. We show that the basic geometric classes of surfaces in the four-dimensional Euclidean space, determined by conditions on their invariants, can be interpreted in terms of the properties of two geometric figures: the tangent indicatrix, which is a conic in the tangent plane, and the normal curvature ellipse. We construct a family of surfaces with flat normal connection.

Abstract. In the present article we study a special class of surfaces in the four-dimensional Euclidean space, which are one-parameter systems of meridians of the standard rotational hypersurface. They are called meridian surfaces. We show that a meridian surface has a harmonic Gauss map if and only if it is part of a plane. Further, we give necessary and sufficient conditions for a meridian surface to have pointwise 1-type Gauss map and find all meridian surfaces with pointwise 1-type Gauss map.

For a two-dimensional surface M 2 in the four-dimensional Euclidean space E 4 we introduce an invariant linear map of Weingarten type in the tangent space of the surface, which generates two invariants k and κ.The condition k = κ = 0 characterizes the surfaces consisting of flat points. The minimal surfaces are characterized by the equality κ 2 − k = 0. The class of the surfaces with flat normal connection is characterized by the condition κ = 0. For the surfaces of general type we obtain a geometrically determined orthonormal frame field at each point and derive Frenet-type derivative formulas.We apply our theory to the class of the rotational surfaces in E 4 , which prove to be surfaces with flat normal connection, and describe the rotational surfaces with constant invariants.2000 Mathematics Subject Classification. Primary 53A07, Secondary 53B25.