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.
Highlights
Bonnet surfaces in three-dimensional Euclidean space have been of great interest for a number of reasons as a type of surface [1, 2] for a long time
Bonnet surfaces are of nonconstant mean curvature that admit infinitely many nontrivial and geometrically distinct isometries which preserve the mean curvature function
Nontrivial isometries are ones that do not extend to isometries of the whole space E3
Summary
Bonnet surfaces in three-dimensional Euclidean space have been of great interest for a number of reasons as a type of surface [1, 2] for a long time. Bonnet surfaces are of nonconstant mean curvature that admit infinitely many nontrivial and geometrically distinct isometries which preserve the mean curvature function. The approach first given by Chern [6] to Bonnet surfaces is considered. With many results the analysis is local and takes place under the assumptions that the surfaces contain no umbilic points and no critical points of the mean curvature function. To establish some information about what is known, consider an oriented, connected, smooth open surface M in E3 with nonconstant mean curvature function H. If the function by which a nontrivial isometry preserving the mean curvature rotates the principal frame is considered, as when there are infinitely many isometries, this function is a global function on M continuously defined [9–11]. The analysis will begin by formulating the structure equations for two-dimensional manifolds
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