Abstract
We introduce a novel family of two-dimensional surfaces in \({\mathbb {E}}^4\) which generalize the classical Dini surfaces in \({\mathbb {E}}^3\) by inheriting their geometric features concerning the degeneration of the Bianchi-Bäcklund transformation.
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Notes
The choice of segments in question is governed by a specific system of first order pde’s representing the Bianchi-Bäcklund analytical transformation of solutions of the sine-Gordon equations.
Instead of unit segments one can consider segments of an arbitrary fixed length. By applying dilatations in \(\mathbb {\mathbb {E}}^3\) this more general situation can be reduced to the case we study in this note.
To illustrate the impact of the value of r, consider \((v^1,v^2,v^3)\) as Cartesian coordinates in an auxiliary \({\mathbb {R}}^3\). Then solutions of (12)–(13) represent arcs of circles obtained by intersecting the unit sphere \((v^1)^2+(v^2)^2+(v^3)^2=1\) with hyperplanes \((v^2-\frac{1}{r}) / v^3 = const\). The hyperplanes share the same straight line \(v^2=\frac{1}{r}\), \(v^3=0\), whose placement with respect to the unit sphere strongly depends on r and affects the structure of solutions: if \(r>1\), then we have a family of disjoint circles with two different limit points; if \(r=1\), then we have a family of circles sharing the same points which is also the limit point of the family; if \(r<1\), then we have a periodic family of circles with two common points.
Without loss of generality, we can set \(c_3=0\) by shifting the arc length \(s\rightarrow s+\frac{c_3}{\lambda }\) of \({\tilde{\gamma }}\).
The same phenomenon of the degeneracy of Bianchi-Bäcklund transformations related to the lines of curvatures is well known in the classical theory of pseudo-spherical surfaces in \({\mathbb {E}}^3\), see [15].
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Acknowledgements
The first author would like to express his deepest gratitude to Yuri Nikolayevsky, Vlad Yaskin, Alla and Dmitry Shepelsky, Tatiana and Dmitry Bolotov, Sergey Bezuglyi, Eugen Petrov, Alexander Yampolsky for their support and assistance.
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All authors contributed to the study conception and design. The first draft of the manuscript was written by VG. All authors commented on previous versions of the manuscript. All authors read and approved the final manuscript.
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Gashurenko, D., Gorkavyy, V. Circular Dini Surfaces in \({\mathbb {E}}^4\). Results Math 79, 20 (2024). https://doi.org/10.1007/s00025-023-02044-9
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DOI: https://doi.org/10.1007/s00025-023-02044-9