Table of Contents
Geometry
Volume 2014, Article ID 715679, 11 pages
http://dx.doi.org/10.1155/2014/715679
Research Article

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

Department of Mathematics, University of Texas, Edinburg, TX 78540-2999, USA

Received 27 May 2014; Accepted 14 August 2014; Published 31 August 2014

Academic Editor: Chandrashekar Devchand

Copyright © 2014 Paul Bracken. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

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.