Invariants and Bonnet-type theorem for surfaces in \(\mathbb R^{4}\)
type: Item
instance of: scholarly article
title: Invariants and Bonnet-type theorem for surfaces in \(\mathbb R^{4}\)
zbMATH Open document ID: 1213.53010
author: Georgi Ganchev
author: Velichka Milousheva
published in: Central European Journal of Mathematics
publication date: 2011-03-15T00:00:00Z
full work available at URL: https://arxiv.org/abs/1105.3367
review text: 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.
reviewed by: Hans-Peter Schröcker
Mathematics Subject Classification ID: 53A07
Mathematics Subject Classification ID: 53A55
Mathematics Subject Classification ID: 53A10
zbMATH DE Number: 5866001
zbMATH Keywords: Weingarten map
zbMATH Keywords: tangent indicatrix
zbMATH Keywords: normal curvature ellipse
zbMATH Keywords: fundamental theorem of Bonnet-type
MaRDI profile type: Publication
OpenAlex ID: W2069763491
arXiv ID: 1105.3367
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DOI: 10.2478/S11533-010-0073-9
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description: scientific article