Reciprocal $F$-manifolds

Abstract

We consider the action of a special class of reciprocal transformation on the principal hierarchy associated to a semisimple $F$-manifold with compatible flat structure $(M,\circ ,\nabla ,e)$. Under some additional assumptions, the hierarchy obtained applying these reciprocal transformations is also associated to an $F$-manifold with compatible flat structure that we call reciprocal $F$-manifold. We also consider the special case of bi-flat $F$-manifolds $(M,\circ ,\ast ,{\nabla }^{(1)},{\nabla }^{(2)},e,E)$ and we study reciprocal transformations preserving flatness of both the connections ${\nabla }^{(1)}$ and ${\nabla }^{(2)}$ and how they act on corresponding solutions of an augmented Darboux-Egorov system.

32 pages

References in corpus (5)

• From Darboux-Egorov system to bi-flat $F$-manifolds
• $F$-manifolds with eventual identities, bidifferential calculus and twisted Lenard-Magri chains
• F-manifolds with flat structure and Dubrovin's duality
• Reciprocal transformations and local Hamiltonian structures of hydrodynamic type systems
• Reciprocal transformations and flat metrics on Hurwitz spaces

Cited by in corpus (2)

• Darboux-Egorov system, bi-flat $F$-manifolds and Painlevé VI
• Miura-reciprocal transformations and localizable Poisson pencils