Riemannian F-manifolds, bi-flat F-manifolds, and flat pencils of metrics

TL;DR

This paper explores the relationships between Riemannian F-manifolds, bi-flat F-manifolds, and flat pencils of metrics, introducing canonical constructions and transformations that deepen understanding of their geometric structures.

Contribution

It introduces canonical methods to construct flat F-manifolds from Riemannian F-manifolds and develops Legendre transformations for these structures, advancing the theory of F-manifolds.

Findings

Canonical flat F-manifolds associated to Riemannian F-manifolds

Construction of homogeneous Riemannian F-manifolds from flat pencils

Development of Legendre transformations for Riemannian F-manifolds

Abstract

In this paper we study relations between various natural structures on F-manifolds. In particular, given an arbitrary Riemannian F-manifold we present a construction of a canonical flat F-manifold associated to it. We also describe a construction of a canonical homogeneous Riemannian F-manifold associated to an arbitrary exact homogeneous flat pencil of metrics satisfying a certain non-degeneracy assumption. In the last part of the paper we construct Legendre transformations for Riemannian F-manifolds.