  Given a flat metric one may generate a local Hamiltonian structure via the
fundamental result of Dubrovin and Novikov. More generally, a flat pencil of
metrics will generate a local bi-Hamiltonian structure, and with additional
quasi-homogeneity conditions one obtains the structure of a Frobenius manifold.
With appropriate curvature conditions one may define a curved pencil of
compatible metrics and these give rise to an associated non-local
bi-Hamiltonian structure. Specific examples include the F-manifolds of Hertling
and Manin equipped with an invariant metric. In this paper the geometry
supporting such compatible metrics is studied and interpreted in terms of a
multiplication on the cotangent bundle. With additional quasi-homogeneity
assumptions one arrives at a so-called weak $\F$-manifold - a curved version of
a Frobenius manifold (which is not, in general, an F-manifold). A submanifold
theory is also developed.
