On the Theory of Surfaces in the Four-dimensional Euclidean Space

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