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

Given the Frölicher-Nijenhuis bicomplex ( d , d L ) associated with a ( 1 , 1 ) -tensor field L with vanishing Nijenhuis torsion, we define a multi-parameter family of bi-flat structures ( ∇ , e , ∘ , ∇ ∗ , ∗ , E ) . 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 ( n − 1 ) flat coordinates of ∇ are Lauricella functions. The ( 1 , 1 ) -tensor fields defining the corresponding integrable hierarchies have a similar block diagonal structure.