Invariants and Bonnet-type theorem for surfaces in ℝ4

Abstract

In the tangent plane at any point of a surface in the four-dimensional Euclidean space we consider an invariant linear map ofWeingarten-type and find a geometrically determined moving frame field. Writing derivative formulas of Frenet-type for this frame field, we obtain eight invariant functions. We prove a fundamental theorem of Bonnet-type, stating that these eight invariants under some natural conditions determine the surface up to a motion. We show that the basic geometric classes of surfaces in the four-dimensional Euclidean space, determined by conditions on their invariants, can be interpreted in terms of the properties of two geometric figures: the tangent indicatrix, which is a conic in the tangent plane, and the normal curvature ellipse. We construct a family of surfaces with flat normal connection.

Timelike surfaces with zero mean curvature in Minkowski 4-space

“…In Section 3 we proved that any timelike surface with H = 0 free of flat points is determined up to a motion in R 4 1 by two invariant functions µ and ν satisfying a system of two partial differential equations, namely system (9).…”

“…Using that x(u, v) and y(u, v) satisfy (10) we get that the integrability conditions z uv = z vu of system (14) The meaning of Theorem 3.5 is that any timelike surface with H = 0 free of flat points is determined up to a motion in R 4 1 by two invariant functions satisfying the system of natural partial differential equations (9).…”

“…In [3], [4], and [5] we called such points flat points of the surface. These points are analogous to flat points in the theory of surfaces in R 3 .…”

“…In the present paper we study the local theory of timelike surfaces with zero mean curvature in the four-dimensional Minkowski space R 4 1 . In Section 3 we introduce (Theorem 3.4) special isothermal parameters (canonical parameters) on any timelike surface with zero mean curvature free of flat points and clear the geometric nature of these parameters.…”

“…In the last section we make a survey of the background systems of partial differential equations describing surfaces with zero mean curvature in R 4 or R 4 1 . These systems written in canonical form are given below:…”