Tohoku Mathematical Journal

Lie sphere geometry and integrable systems

Eugene V. Ferapontov

Full-text: Open access

Abstract

Two basic Lie-invariant forms uniquely defining a generic (hyper)surface in Lie sphere geometry are introduced. Particularly interesting classes of surfaces associated with these invariants are considered. These are the diagonally cyclidic surfaces and the Lie-minimal surfaces, the latter being the extremals of the simplest Lie-invariant functional generalizing the Willmore functional in conformal geometry. Equations of motion of a special Lie sphere frame are derived, providing a convenient unified treatment of surfaces in Lie sphere geometry. In particular, for diagonally cyclidic surfaces this approach immediately implies the stationary modified Veselov-Novikov equation, while the case of Lie-minimal surfaces reduces in a certain limit to the integrable coupled Tzitzeica system. In the framework of the canonical correspondence between Hamiltonian systms of hydrodynamic type and hypersurfaces in Lie sphere geometry, it is pointed out that invariants of Lie-geometric hypersurfaces coincide with the reciprocal invariants of hydrodynamic type systems. Integrable evolutions of surfaces in Lie sphere geometry are introduced. This provides an interpretation of the simplest Lie-invariant functional as the first local conservation law of the (2+1)-dimensional modified Veselov-Novikov hierarchy. Parallels between Lie sphere geometry and projective differential geometry of surfaces are drawn in the conclusion.

Article information

Source
Tohoku Math. J. (2), Volume 52, Number 2 (2000), 199-233.

Dates
First available in Project Euclid: 3 May 2007

Permanent link to this document
https://projecteuclid.org/euclid.tmj/1178224607

Digital Object Identifier
doi:10.2748/tmj/1178224607

Mathematical Reviews number (MathSciNet)
MR1756094

Zentralblatt MATH identifier
1058.53012

Subjects
Primary: 53A10: Minimal surfaces, surfaces with prescribed mean curvature [See also 49Q05, 49Q10, 53C42]
Secondary: 37K10: Completely integrable systems, integrability tests, bi-Hamiltonian structures, hierarchies (KdV, KP, Toda, etc.) 37K25: Relations with differential geometry 53A05: Surfaces in Euclidean space

Citation

Ferapontov, Eugene V. Lie sphere geometry and integrable systems. Tohoku Math. J. (2) 52 (2000), no. 2, 199--233. doi:10.2748/tmj/1178224607. https://projecteuclid.org/euclid.tmj/1178224607


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