# Invariants and Bonnet-type theorem for surfaces in ℝ4

Georgi Ganchev; Velichka Milousheva

Open Mathematics (2010)

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

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topGeorgi 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 -

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