Compatible metrics and integrable systems Oleg I. Mokhov Department - PowerPoint PPT Presentation

Compatible metrics and integrable systems Oleg I. Mokhov Department of Geometry and Topology, Faculty of Mechanics and Mathematics, Moscow State University Conference in honour of Franco Magri’s 65th birthday Bi-Hamiltonian Systems and All That Milan - Bergamo, Italy, September 27 - October 1, 2011 Oleg I. Mokhov Compatible metrics and integrable systems

◮ Classification (an integrable description) of compatible Dubrovin–Novikov brackets (flat pencils of metrics or quasi-Frobenius manifolds) ◮ Classification (an integrable description) of compatible nonlocal Poisson brackets of hydrodynamic type (Mokhov–Ferapontov and general Ferapontov type) ◮ Classification of multi-dimensional Dubrovin–Novikov brackets of hydrodynamic type ◮ Riemann invariants for nonlocally bi-Hamiltonian systems of hydrodynamic type ◮ Integrable classes of compatible metrics Oleg I. Mokhov Compatible metrics and integrable systems

Compatible and almost compatible metrics. Definitions Definition Two Riemannian or pseudo-Riemannian contravariant metrics g ij 1 ( u ) and g ij 2 ( u ) are called compatible if for any linear combination of these metrics g ij ( u ) = λ 1 g ij 1 ( u ) + λ 2 g ij 2 ( u ) , where λ 1 and λ 2 are arbitrary constants, the coefficients of the corresponding Levi–Civita connections and the components of the corresponding Riemann curvature tensors are related by the same linear formula: Γ ij k ( u ) = λ 1 Γ ij 1 , k ( u ) + λ 2 Γ ij 2 , k ( u ) , R ij kl ( u ) = λ 1 R ij 1 , kl ( u ) + λ 2 R ij 2 , kl ( u ) . Oleg I. Mokhov Compatible metrics and integrable systems

Compatible and almost compatible metrics. Definitions Definition Two Riemannian or pseudo-Riemannian contravariant metrics g ij 1 ( u ) and g ij 2 ( u ) are called almost compatible if for any linear combination of these metrics g ij ( u ) = λ 1 g ij 1 ( u ) + λ 2 g ij 2 ( u ) , where λ 1 and λ 2 are arbitrary constants, the coefficients of the corresponding Levi–Civita connections are related by the same linear formula: Γ ij k ( u ) = λ 1 Γ ij 1 , k ( u ) + λ 2 Γ ij 2 , k ( u ) . Any two almost compatible metrics g ij 1 ( u ) and g ij 2 ( u ) form a pencil of almost compatible metrics . Oleg I. Mokhov Compatible metrics and integrable systems

Compatible metrics of constant Riemannian curvature Consider two flat metrics g ij 1 ( u ) and g ij 2 ( u ) . In this case, the condition R ij kl ( u ) = λ 1 R ij 1 , kl ( u ) + λ 2 R ij 2 , kl ( u ) means exactly that any of the metrics of the pencil g ij ( u ) = λ 1 g ij 1 ( u ) + λ 2 g ij 2 ( u ) , where λ 1 and λ 2 are arbitrary constants, is also flat. Thus, any two compatible flat metrics g ij 1 ( u ) and g ij 2 ( u ) form a pencil of compatible flat metrics . Generally speaking, it is not true for almost compatible flat metrics: for example, the flat two-component metrics g ij 1 ( u ) = exp ( u 1 u 2 ) δ ij , 1 ≤ i , j ≤ 2, and g ij 2 = δ ij , 1 ≤ i , j ≤ 2, are almost compatible but they are not compatible and do not form a pencil of almost compatible flat metrics. Oleg I. Mokhov Compatible metrics and integrable systems

Compatible metrics of constant Riemannian curvature Consider two metrics g ij 1 ( u ) and g ij 2 ( u ) of constant Riemannian curvature K 1 and K 2 , respectively, that is, R ij l δ j k δ j R ij l δ j k δ j 1 , kl ( u ) = K 1 ( δ i k − δ i 2 , kl ( u ) = K 2 ( δ i k − δ i l ) , l ) . Here, K 1 and K 2 are arbitrary constants. In this case, the condition R ij kl ( u ) = λ 1 R ij 1 , kl ( u ) + λ 2 R ij 2 , kl ( u ) means exactly that any of the metrics of the pencil g ij ( u ) = λ 1 g ij 1 ( u ) + λ 2 g ij 2 ( u ) , where λ 1 and λ 2 are arbitrary constants, is a metric of constant Riemannian curvature λ 1 K 1 + λ 2 K 2 . Thus, any two compatible metrics of constant Riemannian curvature form a pencil of compatible metrics of constant Riemannian curvature . Oleg I. Mokhov Compatible metrics and integrable systems

Motivation. Compatible Poisson brackets of hydrodynamic type A Hamiltonian operator given by an arbitrary matrix homogeneous first-order ordinary differential operator, that is, a Hamiltonian operator of the form P ij [ u ( x )] = g ij ( u ( x )) d dx + b ij k ( u ( x )) u k x , is called a local Hamiltonian operator of hydrodynamic type or Dubrovin–Novikov Hamiltonian operator . The operator is called nondegenerate if det ( g ij ( u )) �≡ 0 . If det ( g ij ( u )) �≡ 0 , then operator is Hamiltonian if and only if 1) g ij ( u ) is an arbitrary contravariant flat pseudo-Riemannian metric (a metric of zero Riemannian curvature), 2) b ij k ( u ) = − g is ( u )Γ j sk ( u ) , where Γ j sk ( u ) is the Levi-Civita connection generated by the metric g ij ( u ) (the Dubrovin–Novikov theorem). Oleg I. Mokhov Compatible metrics and integrable systems

Motivation. Compatible Poisson brackets of hydrodynamic type For any nondegenerate local Hamiltonian operator of hydroodynamic type there always exist local coordinates v 1 , ..., v N (flat coordinates of the metric g ij ( u ) ) in which all the coefficients of the operator are constant: g ij ( v ) = η ij = const , � jk ( v ) = 0 , � b ij Γ i � k ( v ) = 0 , that is the corresponding Poisson bracket has the form � δ v i ( x ) η ij d δ I δ J { I , J } = δ v j ( x ) dx , dx where ( η ij ) is a nondegenerate symmetric constant matrix: η ij = η ji , η ij = const , det ( η ij ) � = 0 . Oleg I. Mokhov Compatible metrics and integrable systems

Two nondegenerate Dubrovin–Novikov Hamiltonian operators P ij 1 [ u ( x )] and P ij 2 [ u ( x )] given by flat metrics g ij 1 ( u ) and g ij 2 ( u ) are compatible if and only if 1) any linear combination of these metrics g ij ( u ) = λ 1 g ij 1 ( u ) + λ 2 g ij 2 ( u ) , where λ 1 and λ 2 are arbitrary constants, is a flat metric, 2) the coefficients of the corresponding Levi-Civita connections are related by the same linear formula: Γ ij k ( u ) = λ 1 Γ ij 1 , k ( u ) + λ 2 Γ ij 2 , k ( u ) . These conditions on flat metrics g ij 1 ( u ) and g ij 2 ( u ) define a pencil of compatible flat metrics (Dubrovin’s flat pencil of metrics or quasi-Frobenius manifold). So the problem of description of compatible nondegenerate Dubrovin–Novikov brackets is exactly the problem of description of pencils of compatible flat metrics. Oleg I. Mokhov Compatible metrics and integrable systems

Compatible and almost compatible metrics and the Nijenhuis tensor Consider two arbitrary contravariant Riemannian or pseudo-Riemannian metrics g ij 1 ( u ) and g ij 2 ( u ) . Introduce the affinor v i j ( u ) = g is 1 ( u ) g 2 , sj ( u ) , where g 2 , sj ( u ) is the covariant metric inverse to the metric g ij 2 ( u ) : g is 2 ( u ) g 2 , sj ( u ) = δ i j . Consider the Nijenhuis tensor of this affinor ∂ v k ∂ v s j ( u ) ∂ v k s ( u ) ∂ v s j j N k ij ( u ) = v s ∂ u s − v s ∂ u s + v k i ∂ u j − v k i i ( u ) s ( u ) ∂ u i . Theorem Any two metrics g ij 1 ( u ) and g ij 2 ( u ) are almost compatible if and only if the corresponding Nijenhuis tensor N k ij ( u ) vanishes. Oleg I. Mokhov Compatible metrics and integrable systems

Compatible and almost compatible metrics and the Nijenhuis tensor Definition Two Riemannian or pseudo-Riemannian metrics g ij 1 ( u ) and g ij 2 ( u ) are called a nonsingular (semisimple) pair of metrics if the eigenvalues of this pair of metrics, that is, the roots of the equation det ( g ij 1 ( u ) − λ g ij 2 ( u )) = 0 , are distinct. A pencil of metrics is called nonsingular if it is formed by a nonsingular pair of metrics. Theorem If a pair of metrics g ij 1 ( u ) and g ij 2 ( u ) is nonsingular, then the metrics g ij 1 ( u ) and g ij 2 ( u ) are compatible if and only if the Nijenhuis tensor of the affinor v i j ( u ) = g is 1 ( u ) g 2 , sj ( u ) vanishes. Thus, a nonsingular pair of metrics is compatible if and only if the metrics are almost compatible. Oleg I. Mokhov Compatible metrics and integrable systems

Compatible and almost compatible metrics and the Nijenhuis tensor Assume that a pair of metrics g ij 1 ( u ) and g ij 2 ( u ) is nonsingular and the corresponding Nijenhuis tensor vanishes. The eigenvalues of the pair of metrics g ij 1 ( u ) and g ij 2 ( u ) coincide with the eigenvalues of the affinor v i j ( u ) = g is 1 ( u ) g 2 , sj ( u ) . If all eigenvalues of an affinor are distinct, then by the Nijenhuis theorem the vanishing of the Nijenhuis tensor of this affinor implies that there exist local coordinates such that, in these coordinates, the affinor reduces to a diagonal form in the corresponding neighbourhood. So we can consider that the affinor v i j ( u ) is diagonal in the local coordinates u 1 , ..., u N , that is, v i j ( u ) = λ i ( u ) δ i j . Oleg I. Mokhov Compatible metrics and integrable systems

Compatible and almost compatible metrics and the Nijenhuis tensor The eigenvalues λ i ( u ) , i = 1 , ..., N , are distinct: λ i � = λ j if i � = j . Lemma If the affinor v i j ( u ) = g is 1 ( u ) g 2 , sj ( u ) is diagonal in local coordinates and all its eigenvalues are distinct, then, in these coordinates, the metrics g ij 1 ( u ) and g ij 2 ( u ) are also diagonal. We have g ij 1 ( u ) = λ i ( u ) g ij 2 ( u ) . It follows from symmetry of the metrics g ij 1 ( u ) and g ij 2 ( u ) that for any indices i and j ( λ i ( u ) − λ j ( u )) g ij 2 ( u ) = 0 , that is, g ij 2 ( u ) = g ij 1 ( u ) = 0 if i � = j . Oleg I. Mokhov Compatible metrics and integrable systems