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.