Compatible and Almost Compatible Pseudo-Riemannian Metrics

@article{2001CompatibleAA,
  title={Compatible and Almost Compatible Pseudo-Riemannian Metrics},
  author={Олег Иванович Мохов},
  journal={Functional Analysis and Its Applications},
  year={2001},
  volume={35},
  pages={100-110}
}
In the present paper, the notions of compatible and almost compatible Riemannian and pseudo-Riemannian metrics are introduced. These notions are motivated by the theory of compatible Poisson structures of hydrodynamic type (local and nonlocal) and generalize the notion of flat pencils of metrics, which plays an important role in the theory of integrable systems of hydrodynamic type and Dubrovin's theory of Frobenius manifolds. Compatible metrics generate compatible Poisson structures of… 
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References

SHOWING 1-10 OF 33 REFERENCES
Flat pencils of metrics and Frobenius manifolds
This paper is based on the author's talk at 1997 Taniguchi Symposium ``Integrable Systems and Algebraic Geometry''. We consider an approach to the theory of Frobenius manifolds based on the geometry
Hamiltonian systems of hydrodynamic type and their realization on hypersurfaces of a pseudo-Euclidean space
A canonical correspondence between Hamiltonian systems of differential equations of hydrodynamic type and hypersurfaces of a pseudo-Euclidean space is constructed. In this correspondence
A Simple model of the integrable Hamiltonian equation
A method of analysis of the infinite‐dimensional Hamiltonian equations which avoids the introduction of the Backlund transformation or the use of the Lax equation is suggested. This analysis is based
On the structure of symplectic operators and hereditary symmetries
In the last fifteen years, there has been a remarlmble development in the exact analysis of certain nonlinear evolution equations, like tbe Korteweg-de Vries equation. I t is weH known that among the
Multi‐Hamiltonian structure of equations of hydrodynamic type
The discussion of the Hamiltonian structure of two‐component equations of hydrodynamic type is completed by presenting the Hamiltonian operators for Euler’s equation governing the motion of plane
Classification results and the Darboux theorem for low-order Hamiltonian operators
Hamiltonian operators and their behavior under differential substitutions are studied. Scalar Hamiltonian operators are classified up to fifth order, and it is shown that all such operators may be
On a new class of completely integrable nonlinear wave equations. II. Multi‐Hamiltonian structure
The multi‐Hamiltonian structure of a class of nonlinear wave equations governing the propagation of finite amplitude waves is discussed. Infinitely many conservation laws had earlier been obtained
...
...