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Integrable systems in Möbius geometry. (English) Zbl 1127.53009

Mladenov, Ivaïlo (ed.) et al., Proceedings of the 8th international conference on geometry, integrability and quantization, Sts. Constantine and Elena, Bulgaria, June 9–14, 2006. Sofia: Bulgarian Academy of Sciences (ISBN 978-954-8495-37-0/pbk). 11-47 (2007).
This paper summarizes a series of lectures given by the author at the 8th International Conference on Geometry, Integrability and Quantization in Varna, Bulgaria in 2006. It is mainly concerned with two important families of immersions in conformal geometry, isothermic surfaces and conformally flat hypersurfaces. These families are studied under the light of curved flats, a simple type of integrable system, which arise as part of the solution to a classical problem already considered by Blaschke: When do the two envelopes of a sphere congruence in \(\mathbb{S}^3\) induce conformally equivalent metrics? (another part of the answer, dual Willmore surfaces, is not discussed here). The theory of isothermic surfaces and their transformation theory is well understood in this context and it is nicely explained in the paper. However, the relation between conformally flat hypersurfaces and curved flats needs a better understanding, and the emphasis is put on how the aspects of the theory discussed here can stimulate further research developments.
The paper is a good and well written introduction to some particularly interesting classical and recent results on the geometry of isothermic surfaces and conformally flat hypersurfaces, in connection with curved flats. The interested reader can find a more in-depth treatment of the topics included here in the author’s book: [U. Hertrich-Jeromin, Introduction to Möbius Differential Geometry. London Mathematical Society Lecture Note Series 300. Cambridge: Cambridge University Press (2003; Zbl 1040.53002)].
For the entire collection see [Zbl 1108.53003].

MSC:

53A30 Conformal differential geometry (MSC2010)
37K10 Completely integrable infinite-dimensional Hamiltonian and Lagrangian systems, integration methods, integrability tests, integrable hierarchies (KdV, KP, Toda, etc.)
37K25 Relations of infinite-dimensional Hamiltonian and Lagrangian dynamical systems with topology, geometry and differential geometry

Citations:

Zbl 1040.53002
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