On the theory of surfaces in the four-dimensional Euclidean space

Abstract

For a two-dimensional surface M2 in the four-dimensional Euclidean space E4 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 κ2 - 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 E4, which prove to be surfaces with flat normal connection, and describe the rotational surfaces with constant invariants.