Abstract
We focus on investigating the differential geometric properties of cuspidal edge in the three-sphere from a viewpoint of duality. Using Legendrian duality, we study a special kind of flat surface along cuspidal edge in three-dimensional sphere space. This kind of surface is dual to the singular set of the cuspidal edge surface. Thus, we call it the flat \(\Delta \)-dual surface. Flatness of a surface can be defined by the degeneracy of the dual surface. It is similar to the case for the Gauss map of a flat surface in Euclidean space. Moreover, classifications of singularities of the flat \(\Delta \)-dual surface are shown. We also investigate the dual relationships of singularities between flat \(\Delta \)-dual surface and flat approximations of the original cuspidal edge surface. At last, we consider a global geometry of the singular set of a cuspidal edge surface using the flat \(\Delta \)-dual surface.
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Acknowledgements
The authors will express their really appreciate for the valuable suggestions from the anonymous reviewers. The second author would like to thank Professor Shyuichi Izumiya for his meaningful discussion about this topic when the second author visited at Research Institute for Mathematical Sciences (RIMS). The work is supported by the National Nature Science Foundation of China (No. 12271086).
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Haibo Yu and Liang Chen wrote the main manuscript text and Yong Wang prepared figures 1 and 2. Haibo Yu also prepared tables and charts. All authors reviewed the manuscript and share the same contributions to this work.
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Yu, H., Chen, L. & Wang, Y. Singularities of Flat Dual Surfaces of Cuspidal Edges in the Three-Sphere from Duality Viewpoint. Mediterr. J. Math. 21, 103 (2024). https://doi.org/10.1007/s00009-024-02645-w
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DOI: https://doi.org/10.1007/s00009-024-02645-w