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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.

Riemannian F-Manifolds, Bi-Flat F-Manifolds, and Flat Pencils of Metrics