Integrable systems, Nijenhuis geometry and Lauricella bi-flat structures

TL;DR

This paper introduces Lauricella bi-flat structures derived from integrable hydrodynamic systems and Nijenhuis geometry, linking flat structures with Lauricella functions in semisimple cases.

Contribution

It constructs multi-parameter bi-flat structures from Fr"olicher-Nijenhuis bicomplexes, connecting integrable systems, Nijenhuis geometry, and Lauricella functions.

Findings

Defined Lauricella bi-flat structures in integrable systems.

Connected flat coordinates to Lauricella functions in semisimple cases.

Established a new geometric framework linking integrable systems and special functions.

Abstract

Combining the construction of integrable systems of hydrodynamic type starting from the Fr\"olicher-Nijenhuis bicomplex $(d,d_L)$ associated with a (1,1)-tensor field $L$ with vanishing Nijenhuis torsion with the construction of flat structures starting from integrable systems of hydrodynamic type we define multi-parameter families of bi-flat structures $(\nabla,e,\circ,\nabla^*,*,E)$ associated with Fr\"olicher-Nijenhuis bicomplexes. We call these structures Lauricella bi-flat structures since in the n-dimensional semisimple case (n-1) flat coordinates of r are Lauricella functions.