Frobenius Manifolds: Natural submanifolds and induced bi-Hamiltonian structures

Details Frobenius Manifolds: Natural submanifolds and induced bi-Hamiltonian structures Frobenius Manifolds: Natural submanifolds and induced bi-Hamiltonian structures

2002-01-07

arxiv.org/abs/math/0201039v1

Mathematics

Differential Geometry

27 Pages, LaTeX, 1 figure

Scientific paper

Submanifolds of Frobenius manifolds are studied. In particular, so-called natural submanifolds are defined and, for semi-simple Frobenius manifolds, classified. These carry the structure of a Frobenius algebra on each tangent space, but will, in general, be curved. The induced curvature is studied, a main result being that these natural submanifolds carry a induced pencil of compatible metrics. It is then shown how one may constrain the bi-Hamiltonian hierarchies associated to a Frobenius manifold to live on these natural submanifolds whilst retaining their, now non-local, bi-Hamiltonian structure.

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