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, much of the structure comes from the compatibility properties 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 studied. 
Liana David and Ian A. B. Strachan