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

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 x. The condition k = x = 0 characterizes the surfaces consisting of flat points. The minimal surfaces are characterized by the equality x(2) - k = 0. The class of the surfaces with flat normal connection is characterized by the condition x = 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.