• Corpus ID: 17808717

$F$-manifolds and integrable systems of hydrodynamic type

@article{Lorenzoni2009FmanifoldsAI,
  title={\$F\$-manifolds and integrable systems of hydrodynamic type},
  author={Paolo Lorenzoni and Marco Pedroni and Andrea Raimondo},
  journal={arXiv: Differential Geometry},
  year={2009}
}
We investigate the role of Hertling-Manin condition on the structure constants of an associative commutative algebra in the theory of integrable systems of hydrodynamic type. In such a framework we introduce the notion of F-manifold with compatible connection generalizing a structure introduced by Manin. 

Natural Connections for Semi-Hamiltonian Systems: The Case of the $${\epsilon}$$-System

Given a semi-Hamiltonian system, we construct an F-manifold with a connection satisfying a suitable compatibility condition with the product. We exemplify this procedure in the case of the so-called

Bi-Hamiltonian structure of the oriented associativity equation

The oriented associativity equation plays a fundamental role in the theory of integrable systems. In this paper we prove that the equation, besides being Hamiltonian with respect to a first-order

Symmetries of F-manifolds with eventual identities and special families of connections

We construct a duality for F-manifolds with eventual identities and certain special families of connections and we study its interactions with several well-known constructions from the theory of

Poisson bracket on 1-forms and evolutionary partial differential equations

We introduce a bracket on 1-forms defined on , i.e. the infinite jet extension of the space of loops, and prove that it satisfies the standard properties of a Poisson bracket. Using this bracket, we

Killing tensors with nonvanishing Haantjes torsion and integrable systems

The second-order integrable Killing tensor with simple eigenvalues and vanishing Haantjes torsion is the key ingredient in construction of Liouville integrable systems of Stäckel type. We present two

(TE)-structures over the irreducible 2-dimensional globally nilpotent F-manifold germ

We find formal and holomorphic normal forms for a class of meromorphic connections (the so-called $(TE)$-structures) over the irreducible $2$-dimensional globally nilpotent $F$-manifold germ

Flat $F$-manifolds, Miura invariants and integrable systems of conservation laws

We extend some of the results proved for scalar equations in [3,4], to the case of systems of integrable conservation laws. In particular, for such systems we prove that the eigenvalues of a matrix

Flat F-Manifolds, F-CohFTs, and Integrable Hierarchies

We define the double ramification hierarchy associated to an F-cohomological field theory and use this construction to prove that the principal hierarchy of any semisimple (homogeneous) flat

Riemannian F-Manifolds, Bi-Flat F-Manifolds, and Flat Pencils of Metrics

In this paper, we study relations between various natural structures on F-manifolds. In particular, given an arbitrary Riemannian F-manifold, we present a construction of a canonical flat

Primitive Forms without Higher Residue Structure and Integrable Hierarchies (I).

We introduce primitive forms with or without higher residue structure and explore their connection with the flat structures with or without a metric and integrable hierarchies of KdV type. Just as

References

SHOWING 1-10 OF 27 REFERENCES

THE GEOMETRY OF HAMILTONIAN SYSTEMS OF HYDRODYNAMIC TYPE. THE GENERALIZED HODOGRAPH METHOD

It is proved that there exists an infinite involutive family of integrals of hydrodynamic type for diagonal Hamiltonian systems of quasilinear equations; the completeness of the family is also

Hamiltonian Structures of Reductions of the Benney System

We show how to construct the Hamiltonian structures of any reduction of the Benney chain (dispersionless KP). The construction follows the scheme suggested by Ferapontov, leading in general to

Tri-Hamiltonian Structures of Egorov Systems of Hydrodynamic Type

We prove a simple condition under which the metric corresponding to a diagonalizable semi-Hamiltonian hydrodynamic type system belongs to the class of Egorov (potential) metrics. For Egorov diagonal

Integrability of the Egorov systems of hydrodynamic type

We present integrability criterion for the Egorov systems of hydrodynamic type. We find the general solution by the generalized hodograph method and give examples. We discuss a description of

ON ALMOST DUALITY FOR FROBENIUS MANIFOLDS

We present a universal construction of almost duality for Frobenius man- ifolds. The analytic setup of this construction is described in details for the case of semisimple Frobenius manifolds. We

Weak Frobenius manifolds

We establish a new universal relation between the Lie bracket and $\circ$-multiplication of tangent fields on any Frobenius (super)manifold. We use this identity in order to introduce the notion of

MULTIPLICATION ON THE TANGENT BUNDLE

Manifolds with a commutative and associative multi- plication on the tangent bundle are called F-manifolds if a unit field exists and the multiplication satisfies a natural integrability con- dition.