  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.
