Regular F-manifolds: initial conditions and Frobenius metrics

2014/11/17 by David, Liana, Hertling, Claus · 3 citations
#Differential Geometry (math.DG) #FOS: Mathematics

paper · doi:10.48550/arxiv.1411.4553

Abstract

A regular F-manifold is an F-manifold (with Euler field) (M, ∘, e, E), such that the endomorphism \mathcal U(X) := E ∘ X of TM is regular at any p∈ M. We prove that the germ ((M,p), ∘, e, E) is uniquely determined (up to isomorphism) by the conjugacy class of \mathcal U_p : T_pM → T_pM. We obtain that any regular F-manifold admits a preferred system of local coordinates and we find conditions, in these coordinates, for a metric to be Frobenius. We study the Lie algebra of infinitesimal symmetries of regular F-manifolds. We show that any regular F-manifold is locally isomorphic to the parameter space of a Malgrange universal connection. We prove an initial condition theorem for Frobenius metrics on regular F-manifolds.

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• Linear F-manifolds, a duality and the generalized tangent bundle
• Haantjes torsion and integrability: a proof of Bolsinov-Konyaev-Matveev's conjecture
• Hamiltonian formalism for non-diagonalisable systems of hydrodynamic type

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