Geodesic
Nonlocal Hamiltonian Operators of Hydrodynamic Type with Flat Metrics, Integrable Hierarchies, and the Associativity Equations
Funkcionalʹnyj analiz i ego priloženiâ, Tome 40 (2006) no. 1, pp. 14-29 
O. I. Mokhov. Nonlocal Hamiltonian Operators of Hydrodynamic Type with Flat Metrics, Integrable Hierarchies, and the Associativity Equations. Funkcionalʹnyj analiz i ego priloženiâ, Tome 40 (2006) no. 1, pp. 14-29. http://geodesic.mathdoc.fr/item/FAA_2006_40_1_a1/
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We solve the problem of describing all nonlocal Hamiltonian operators of hydrodynamic type with flat metrics. This problem is equivalent to describing all flat submanifolds with flat normal bundle in a pseudo-Euclidean space. We prove that every such Hamiltonian operator (or the corresponding submanifold) specifies a pencil of compatible Poisson brackets, generates bihamiltonian integrable hierarchies of hydrodynamic type, and also defines a family of integrals in involution. We prove that there is a natural special class of such Hamiltonian operators (submanifolds) exactly described by the associativity equations of two-dimensional topological quantum field theory (the Witten–Dijkgraaf–Verlinde–Verlinde and Dubrovin equations). We show that each $N$-dimensional Frobenius manifold can locally be represented by a special flat $N$-dimensional submanifold with flat normal bundle in a $2N$-dimensional pseudo-Euclidean space. This submanifold is uniquely determined up to motions.

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