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.
@article{bwmeta1.element.doi-10_2478_s11533-010-0073-9,
author = {Georgi Ganchev and Velichka Milousheva},
title = {Invariants and Bonnet-type theorem for surfaces in $\mathbb{R}$4},
journal = {Open Mathematics},
volume = {8},
year = {2010},
pages = {993-1008},
zbl = {1213.53010},
language = {en},
url = {http://dml.mathdoc.fr/item/bwmeta1.element.doi-10_2478_s11533-010-0073-9}
}
Georgi Ganchev; Velichka Milousheva. Invariants and Bonnet-type theorem for surfaces in ℝ4. Open Mathematics, Tome 8 (2010) pp. 993-1008. http://gdmltest.u-ga.fr/item/bwmeta1.element.doi-10_2478_s11533-010-0073-9/
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