Invariants and Bonnet-type theorem for surfaces in \(\mathbb R^{4}\)
The article is a contribution to differential geometry of two-dimensional surfaces in four-dimensional Euclidean space and in particular to the theory of local differential invariants. After recalling some concepts from \textit{G. Ganchev} and \textit{V. Milousheva} [Kodai Math. J. 31, No.~2, 183--198 (2008; Zbl 1165.53003)], the authors interpret basic surface invariants in terms of properties of the tangent indicatrix and the normal curvature ellipse: Minimal surfaces have circular tangent indicatrices, surfaces with flat normal connection have right hyperbolas as tangent indicatrices or, equivalently, degenerate ellipses of normal curvature. Converse statements hold true as well. The authors prove a fundamental theorem for two-dimensional surfaces in four-space. The surface is determined, up to rigid motion, by eight invariants that appear as coefficients in Frenet-type formulas of a geometric moving frame field and satisfy certain integrability conditions. Finally, the authors construct examples of surfaces with a flat normal connection where either the Gaussian curvature, the mean curvature or a yet nameless differential invariant is constant.
scientific article