An Invariant Theory of Surfaces in the Four-Dimensional Euclidean or Minkowski Space

2010 Mathematics Subject Classification: 53A07, 53A35, 53A10.

The present article is a survey of some of our recent results on the theory of two-dimensional surfaces in the four-dimensional Euclidean or Minkowski space. We present our approach to the theory of surfaces in Euclidean or Minkowski 4-space, which is based on the introduction of an invariant linear map of Weingarten-type in the tangent plane at any point of the surface under consideration. This invariant map allows us to introduce principal lines and an invariant moving frame field at each point of the surface. Writing derivative formulas of Frenet-type for this frame field, we obtain a system of invariant functions, which determine the surface up to a motion. We formulate the fundamental theorems for the general classes of surfaces in Euclidean or Minkowski 4-space in terms of the invariant functions. We show that the basic geometric classes of surfaces, determined by conditions on their invariants, can be interpreted in terms of the properties of two geometric figures: the tangent indicatrix and the normal curvature ellipse. We apply our theory to some special classes of surfaces in Euclidean or Minkowski 4-space.