Regular F-manifolds: initial conditions and Frobenius metrics

TL;DR

This paper characterizes regular F-manifolds through their endomorphism conjugacy classes, establishes preferred coordinates, and provides conditions for Frobenius metrics, linking them to Malgrange universal connections.

Contribution

It introduces a classification of regular F-manifolds via conjugacy classes and derives conditions for Frobenius metrics, expanding understanding of their local structure.

Findings

Regular F-manifolds are uniquely determined by conjugacy classes of endomorphisms.

Existence of preferred local coordinates for regular F-manifolds.

Conditions for metrics to be Frobenius in these coordinates.

Abstract

A regular F-manifold is an F-manifold (with Euler field) (M, \circ, e, E), such that the endomorphism {\mathcal U}(X) := E \circ X of TM is regular at any p\in M. We prove that the germ ((M,p), \circ, e, E) is uniquely determined (up to isomorphism) by the conjugacy class of {\mathcal U}_{p} : T_{p}M \rightarrow T_{p}M. 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.