Frobenius manifolds are complex manifolds with a rich structure on the holomorphic tangent bundle, a multiplication and a metric which harmonize in the most natural way. They were defined by Dubrovin in 1991, motivated by the work of Witten, Dijkgraaf, E. Verlinde, and H. Verlinde on topological field theory. Originally coming from physics, Frobenius manifolds now turn up in very different areas of mathematics, giving unexpected relations between them, in quantum cohomology, singularity theory, integrable systems, symplectic geometry, and others. The isomorphy of certain Frobenius manifolds in quantum cohomology and in singularity theory is one version of mirror symmetry.

This book is devoted to the relations between Frobenius manifolds and singularity theory. It consists of two parts.

In part 1 F-manifolds are studied, manifolds with a multiplication on the tangent bundle with a natural integrability condition. They were introduced in [HM][Man2, I§5]. Frobenius manifolds are F-manifolds. Studying F-manifolds, one is led directly to discriminants, a classical subject of singularity theory, and to Lagrange maps and their singularities. Our development of the general structure of F-manifolds is at the same time an introduction to discriminants and Lagrange maps. As an application, we use some work of Givental to prove a conjecture of Dubrovin about Frobenius manifolds and Coxeter groups.

In part 2 we take up the construction of Frobenius manifolds in singularity theory.