Conformally flat pencils of metrics, Frobenius structures and a modified Saito construction

The structure of a Frobenius manifold encodes the geometry associated with a flat pencil of metrics. However, as shown in the authors’ earlier work [L. David, I.A.B. Strachan, Compatible metrics on manifolds and non-local bi-Hamiltoninan structures, Int. Math. Res. Notices 66 (2004) 3533–3557], much of the structure comes from the compatibility property of the pencil rather than from the flatness of the pencil itself. In this paper conformally flat pencils of metrics are studied and examples, based on a modification of the Saito construction, are developed.