In this section the relation between F-manifolds and symplectic geometry is discussed. The most crucial fact is shown in section 3.1: the analytic spectrum of a massive (i.e. with generically semisimple multiplication) F-manifold M is a Lagrange variety L ⊂ T*M; and a Lagrange variety L ⊂ T*M in the cotangent bundle of a manifold M supplies the manifold M with the structure of an F-manifold if and only if the map a : TM → π∗OL from (3.1) is an isomorphism.

The condition that this map a : TM → π∗OL is an isomorphism is close to Givental's notion of a miniversal Lagrange map [Gi2, ch. 13]. In section 3.4 the correspondence between massive F-manifolds and Lagrange maps is rewritten using this notion.

If E is an Euler field in a massive F-manifold M then the holomorphic function F := a–1(E) : L → ℂ satisfies dF|Lreg = α|Lreg (here α is the canonical 1-form on T*M). But as L may have singularities, the global existence of E and of such a holomorphic function is not clear. This is discussed in section 3.2.

Much weaker than the existence of E is the existence of a continuous function F : L → ℂ which is holomorphic on Lreg with dF|Lreg = α|Lreg. This is called a generating function for the massive F-manifold. It gives rise to the three notions bifurcation diagram, Lyashko–Looijenga map, and discriminant.