  For a two-dimensional surface in the four-dimensional Euclidean space we
introduce an invariant linear map of Weingarten type in the tangent space of
the surface, which generates two invariants k and kappa.
  The condition k = kappa = 0 characterizes the surfaces consisting of flat
points. The minimal surfaces are characterized by the equality kappa^2=k. The
class of the surfaces with flat normal connection is characterized by the
condition kappa = 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, which prove to
be surfaces with flat normal connection, and describe the rotational surfaces
with constant invariants.
