An F-manifold is a manifold with a multiplication on the tangent bundle which satisfies a certain integrability condition. It is defined in section 2.3. Sections 2.4 and 2.5 give two reasons why this is a good notion. In section 2.4 it is shown that germs of F-manifolds decompose in a nice way. In section 2.5 the relation to connections and metrics is discussed. It turns out that the integrability condition is part of the potentiality condition for Frobenius manifolds. Therefore Frobenius manifolds are F-manifolds.

Section 2.1 is a self-contained elementary account of the structure of finite dimensional algebras in general (e.g. the tangent spaces of an F-manifold) and Frobenius algebras in particular. Section 2.2 discusses vector bundles with multiplication. There the caustic and the analytic spectrum are defined, two notions which are important for F-manifolds.

Finite-dimensional algebras

In this section (Q, o, e) is a ℂ-algebra of finite dimension (≥ 1) with commutative and associative multiplication and with unit e. The next lemma gives precise information on the decomposition of Q into irreducible algebras. The statements are well known and elementary. They can be deduced directly in the given order or from more general results in commutative algebra (Q is an Artin algebra). Algebra homomorphisms are always supposed to map the unit to the unit.