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Invariants and Bonnet-type theorem for surfaces in \(\mathbb R^{4}\). (English) Zbl 1213.53010

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 G. Ganchev and 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.

MSC:

53A07 Higher-dimensional and -codimensional surfaces in Euclidean and related \(n\)-spaces
53A55 Differential invariants (local theory), geometric objects
53A10 Minimal surfaces in differential geometry, surfaces with prescribed mean curvature

Citations:

Zbl 1165.53003
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References:

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