# Invariants and Bonnet-type theorem for surfaces in ℝ4

Open Mathematics (2010)

• Volume: 8, Issue: 6, page 993-1008
• ISSN: 2391-5455

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## Abstract

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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.

## How to cite

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Georgi Ganchev, and Velichka Milousheva. "Invariants and Bonnet-type theorem for surfaces in ℝ4." Open Mathematics 8.6 (2010): 993-1008. <http://eudml.org/doc/269071>.

@article{GeorgiGanchev2010,
abstract = {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.},
author = {Georgi Ganchev, Velichka Milousheva},
journal = {Open Mathematics},
keywords = {Surfaces in the four-dimensional Euclidean space; Weingarten map; Tangent indicatrix; Normal curvature ellipse; Fundamental theorem of Bonnet-type; tangent indicatrix; normal curvature ellipse; fundamental theorem of Bonnet-type},
language = {eng},
number = {6},
pages = {993-1008},
title = {Invariants and Bonnet-type theorem for surfaces in ℝ4},
url = {http://eudml.org/doc/269071},
volume = {8},
year = {2010},
}

TY - JOUR
AU - Georgi Ganchev
AU - Velichka Milousheva
TI - Invariants and Bonnet-type theorem for surfaces in ℝ4
JO - Open Mathematics
PY - 2010
VL - 8
IS - 6
SP - 993
EP - 1008
AB - 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.
LA - eng
KW - Surfaces in the four-dimensional Euclidean space; Weingarten map; Tangent indicatrix; Normal curvature ellipse; Fundamental theorem of Bonnet-type; tangent indicatrix; normal curvature ellipse; fundamental theorem of Bonnet-type
UR - http://eudml.org/doc/269071
ER -

## References

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