Mathematics – Differential Geometry
Scientific paper
2004-06-15
Functional Analysis and its Applications, Vol. 40, No. 1, 2006, 11-23.
Mathematics
Differential Geometry
23 pages
Scientific paper
We solve the problem of describing all nonlocal Hamiltonian operators of hydrodynamic type with flat metrics. This problem is also equivalent to the description of all flat submanifolds with flat normal bundle in a pseudo-Euclidean space. It is proved that every such Hamiltonian operator (or the submanifold corresponding to the operator) gives a pencil of compatible Poisson brackets, generates bi-Hamiltonian integrable hierarchies of hydrodynamic type, and also defines a family of integrals in involution. We prove that a natural special class of such Hamiltonian operators (submanifolds) is described exactly by the equations of associativity of two-dimensional topological quantum field theory (the Witten--Dijkgraaf--Verlinde--Verlinde and Dubrovin equations). It is shown that locally any N-dimensional Frobenius manifold can be presented by a certain special flat N-dimensional submanifold with flat normal bundle in a 2N-dimensional pseudo-Euclidean space. This submanifold is defined uniquely up to motions.
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