The definition and elementary properties of a Frobenius manifold M are put together in section 9.1. Sections 9.2, 9.4, and 9.5 are devoted to their second structure connections. These are connections over ℙ1 × M on the lifted tangent bundle of M with logarithmic poles along certain hypersurfaces. They come from some twists of the original flat structure by the multiplication and the Euler field. To know them is very instructive for the construction of Frobenius manifolds for singularities, because in that case one of them turns out to be isomorphic to an extension of the Gauß–Manin connection.

Sections 9.2, 9.4, and 9.5 build on the definition and discussion of the second structure connections in [Man2] for the case of semisimple Frobenius manifolds, on results in [Du3], and on [SK9, §5], where they together with many properties had been established much earlier implicitly in the case of singularities.

The second structure connections have some counterparts, the first structure connections, which are better known. The latter are partly Fourier duals. The main purpose of their treatment in section 9.3 (and in section 9.4) is to compare them with the second structure connections.

Definition of Frobenius manifolds

Frobenius manifolds were defined by Dubrovin [Du1][Du3]. We follow the notations in Manin's book [Man2, chapters I and II].