Lie sphere geometry and integrable systems

Two basic Lie-invariant forms uniquely defining a generic (hyper)surface in Lie sphere geometry are introduced. Particularly interesting classes of su rfaces associated with these invariants are considered. These are the diago nally cyclidic surfaces and the Lie-minimal surfaces, the latter being the extremals of the simplest Lie-invariant functional generalizing the Willmor e 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 particu lar, for diagonally cyclidic surfaces this approach immediately implies the stationary modified Veselov-Novikov equation, while the case of Lie-minima l surfaces reduces in a certain limit to the integrable coupled Tz-itzeica system. In the framework of the canonical correspondence between Hamiltonian system s of hydrodynamic type and hypersurfaces in Lie sphere geometry, it is poin ted out that invariants of Lie-geometric hypersurfaces coincide with the re ciprocal invariants of hydrodynamic type systems. Integrable evolutions of surfaces in Lie sphere geometry are introduced. Th is provides an interpretation of the simplest Lie-invariant functional as t he 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.