Isoparametric hypersurfaces in spheres, integrable nondiagonalizable systems of hydrodynamic type, and N-wave systems

Abstract

Isoparametric hypersurface M n ⊂ S n + 1 can be defined as an intersection of the unit sphere r 2 = ( u 1) 2 + … + ( u n + 2 ) 2 = 1 with the level set F( u) = const of a homogeneous polynomial F of degree g, satisfying Cartan-Munzner equations (▽ F) 2 = g 2 r 2 g − 2 , Δ F = cr g − 2 c = const. We introduce a hamiltonian system of hydrodynamic type u i t = 1 g δ ij d dx ∂F ∂u j , with the hamiltonian operator δ ij d dx and the hamiltonian density F (u) g . Under the additional assumption of the homogeneity of the hypersurface M n , the restriction of this system to M n proves to be nondiagonalizable, but integrable and can be transformed to an appropriate integrable reduction of the N-wave system. Possible generalizations to isoparametric submanifolds (finite or infinite dimensional) are also briefly indicated.