  Manifolds with a commutative and associative multiplication on the tangent
bundle are called F-manifolds if a unit field exists and the multiplication
satisfies a natural integrability condition. They are studied here. They are
closely related to discriminants and Lagrange maps. Frobenius manifolds are
F-manifolds. As an application a conjecture of Dubrovin on Frobenius manifolds
and Coxeter groups is proved.
