# 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…

## 29 Citations

On the compatible weakly nonlocal Poisson brackets of hydrodynamic type

- Mathematics, Physics
- 2001

We consider the pairs of general weakly nonlocal Poisson brackets of hydrodynamic type (Ferapontov brackets) and the corresponding integrable hierarchies. We show that, under the requirement of the…

Compatible Metrics of Constant Riemannian Curvature: Local Geometry, Nonlinear Equations, and Integrability

- Mathematics
- 2002

The description problem is solved for compatible metrics of constant Riemannian curvature. Nonlinear equations describing all nonsingular pencils of compatible metrics of constant Riemannian…

Integrability of the Equations for Nonsingular Pairs of Compatible Flat Metrics

- Mathematics
- 2000

We solve the problem of describing all nonsingular pairs of compatible flat metrics (or, in other words, nonsingular flat pencils of metrics) in the general N-component case. This problem is…

Quasi-Frobenius Algebras and Their Integrable N-Parameter Deformations Generated by Compatible N×N Metrics of Constant Riemannian Curvature

- Mathematics
- 2002

We prove that the equations describing compatible N×N metrics of constant Riemannian curvature define a special class of integrable N-parameter deformations of quasi-Frobenius (in general,…

Riemann invariants of semisimple non-locally bi-Hamiltonian systems of hydrodynamic type and compatible metrics

- Mathematics
- 2010

In this paper the diagonalizability of an arbitrary non-singular (semisimple) non-locally bi-Hamiltonian system of hydrodynamic type is proved and for each such system a full set of Riemann…

The Liouville Canonical Form for Compatible Nonlocal Poisson Brackets of Hydrodynamic Type and Integrable Hierarchies

- Mathematics
- 2002

We reduce an arbitrary pair of compatible nonlocal Poisson brackets of hydrodynamic type generated by metrics of constant Riemannian curvature (compatible Mokhov–Ferapontov brackets) to a canonical…

On the geometry of compatible Poisson and Riemannian structures

- Mathematics
- 2017

We consider compatibility conditions between Poisson and Riemannian structures on smooth manifolds by means of a contravariant partially complex structure, or $f$-structure, introducing the notion of…

Compatible Poisson brackets of hydrodynamic type

- Mathematics
- 2001

Some general properties of compatible Poisson brackets of hydrodynamic type are discussed, in particular: (a) an invariant differential-geometric criterion of the compatibility based on the Nijenhuis…

Compatible Dubrovin–Novikov Hamiltonian Operators, Lie Derivative, and Integrable Systems of Hydrodynamic Type

- Mathematics
- 2002

We prove that two Dubrovin–Novikov Hamiltonian operators are compatible if and only if one of these operators is the Lie derivative of the other operator along a certain vector field. We consider the…

On Algebraic-Geometric Methods for Constructing Submanifolds with Flat Normal Bundle and Holonomic Net of Curvature Lines

- Mathematics
- 2020

In this paper we propose a generalization of Krichever’s algebraic-geometric construction of orthogonal coordinate systems in a flat space. In the theory of integrable systems of hydrodynamic type a…

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