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. K 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 diagonallycyclidic 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. K 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. K 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. K Parallels between Lie sphere geometry and projective differential geometry of surfaces are drawn in the conclusion.
