An Intrinsic Characterization of Bonnet Surfaces Based on a Closed Differential Ideal

Geometry ◽

10.1155/2014/715679 ◽

2014 ◽

Vol 2014 ◽

pp. 1-11 ◽

Cited By ~ 1

Author(s):

Paul Bracken

Keyword(s):

Differential Equations ◽

Coordinate System ◽

Ordinary Differential Equations ◽

Dimensional Manifold ◽

Two Dimensional ◽

Differential Ideal ◽

Intrinsic Characterization ◽

Structure Equations ◽

Codazzi Equations

The structure equations for a two-dimensional manifold are introduced and two results based on the Codazzi equations pertinent to the study of isometric surfaces are obtained from them. Important theorems pertaining to isometric surfaces are stated and a theorem due to Bonnet is obtained. A transformation for the connection forms is developed. It is proved that the angle of deformation must be harmonic, and that the differentials of many of the important variables generate a closed differential ideal. This implies that a coordinate system exists in which many of the variables satisfy particular ordinary differential equations, and these results can be used to characterize Bonnet surfaces.

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An Intrinsic Characterization of Five Points in a CAT(0) Space

Analysis and Geometry in Metric Spaces ◽

10.1515/agms-2020-0111 ◽

2020 ◽

Vol 8 (1) ◽

pp. 114-165

Author(s):

Tetsu Toyoda

Keyword(s):

Metric Space ◽

Isometric Embedding ◽

Metric Spaces ◽

Necessary Conditions ◽

New Approach ◽

Intrinsic Characterization ◽

New Family

AbstractGromov (2001) and Sturm (2003) proved that any four points in a CAT(0) space satisfy a certain family of inequalities. We call those inequalities the ⊠-inequalities, following the notation used by Gromov. In this paper, we prove that a metric space X containing at most five points admits an isometric embedding into a CAT(0) space if and only if any four points in X satisfy the ⊠-inequalities. To prove this, we introduce a new family of necessary conditions for a metric space to admit an isometric embedding into a CAT(0) space by modifying and generalizing Gromov’s cycle conditions. Furthermore, we prove that if a metric space satisfies all those necessary conditions, then it admits an isometric embedding into a CAT(0) space. This work presents a new approach to characterizing those metric spaces that admit an isometric embedding into a CAT(0) space.

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On Homology and Cohomology Groups of Remainders

Georgian Mathematical Journal ◽

10.1515/gmj.2004.613 ◽

2004 ◽

Vol 11 (4) ◽

pp. 613-633

Author(s):

V. Baladze ◽

L. Turmanidze

Keyword(s):

Uniform Space ◽

Cohomology Groups ◽

Uniform Spaces ◽

Intrinsic Characterization

Abstract Border homology and cohomology groups of pairs of uniform spaces are defined and studied. These groups give an intrinsic characterization of Čech type homology and cohomology groups of the remainder of a uniform space.

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