  We construct a functor from the derived category of homotopy Gerstenhaber
algebras with finite-dimensional cohomology to the purely geometric category of
so-called $F_{\infty}$-manifolds. The latter contains Frobenius manifolds as a
subcategory (so that a pointed Frobenius manifold is itself a homotopy
Gerstenhaber algebra). If a homotopy Gerstenhaber algebra happens to be formal
as a $L_{\infty}$-algebra, then its $F_{\infty}$-manifold comes equipped with
the Gauss-Manin connection. Mirror Symmetry implications are discussed.
