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

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

2011-05-17

arxiv.org/abs/1105.3367v1

Cent. Eur. J. Math., 8 (6) (2010), 993-1008

Mathematics

Differential Geometry

18 pages

Scientific paper

10.2478/s11533-010-0073-9

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.

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