Integrable hierarchies, Frölicher–Nijenhuis bicomplexes and Lauricella bi-flat F-manifolds

Abstract

Given the Frölicher-Nijenhuis bicomplex associated with a -tensor field L with vanishing Nijenhuis torsion, we define a multi-parameter family of bi-flat structures . This result is obtained by combining the construction of integrable hierarchies of hydrodynamic type starting from Frölicher-Nijenhuis bicomplexes with the construction of flat F-manifold structures from integrable systems of hydrodynamic type. By construction L is the operator of multiplication by the Euler vector field E and the number of parameters coincides with the number of Jordan blocks appearing in its Jordan normal form. We call these structures Lauricella bi-flat structures since in the n-dimensional semisimple case flat coordinates of are Lauricella functions. The -tensor fields defining the corresponding integrable hierarchies have a similar block diagonal structure.