Invariants and bonnet-type theorem for surfaces in $\r^4$

Abstract

In the tangent plane at any point of a surface in the four-dimensional Euclidean space we consider an invariant linear map of Weingarten-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.

18 pages

References in corpus (4)

• All superconformal surfaces in \R^4 in terms of minimal surfaces
• Conformal Geometry of Surfaces in the 4-Sphere and Quaternions
• Minimal Surfaces in the Four-Dimensional Euclidean Space
• Invariants of Lines on Surfaces in ${\mathbb{R}}^4$

Cited by in corpus (13)

• An invariant theory of marginally trapped surfaces in the four-dimensional Minkowski space
• Timelike surfaces with zero mean curvature in Minkowski 4-space
• An Invariant Theory of Spacelike Surfaces in the Four-dimensional Minkowski Space
• General Rotational Surfaces in ${\mathbb{R}}^4$ with Meridians Lying in Two-Dimensional Planes
• Marginally trapped meridian surfaces of parabolic type in the four-dimensional Minkowski space
• Marginally trapped surfaces with pointwise 1-type Gauss map in Minkowski 4-space
• Meridian Surfaces in E^4 with Pointwise 1-type Gauss Map
• Chen Rotational Surfaces of Hyperbolic or Elliptic Type in the Four-dimensional Minkowski Space
• Meridian Surfaces of Elliptic or Hyperbolic Type in the Four-dimensional Minkowski Space
• Meridian Surface of Weingarten Type in 4-dimensional Euclidean Space E^4
• Meridian Surfaces with Constant Mean Curvature in Pseudo-Euclidean 4-space with Neutral Metric
• Meridian Surfaces with Parallel Normalized Mean Curvature Vector Field in Pseudo-Euclidean 4-space with Neutral Metric
• On Generalized Spherical Surfaces in Euclidean Spaces