Quantum integrability of the Beltrami-Laplace operator for geodesically equivalent metrics
by
Vladimir S. Matveev
Chelyabinsk State University

Let g₁ be a Riemannian metric on a smooth manifold Mⁿ. Consider the Beltrami-Laplace operator \Delta(f) : = div(grad(f)), where grad(f) denotes the gradient of a function f from the space L₂ and div denotes the divergention. The Beltrami-Laplace operator acts on the space L₂. Let g₂ be another metric on the same manifold Mⁿ. Metrics g₁, g₂ are geodesically equivalent, if they have the same geodesics, considered as unparameterized curves.

Denote by G the linear (on each tangent space) operator : TMⁿ --> TMⁿ given by

(G)ij : = (g1-1 g2)ij = (g1)ih (g2)hj.

Consider the characteristic polynomial det(G- tE) = c₀tⁿ + c₁t^n-1 + ... + cⁿ. The coefficients c₁, ... , c_n are smooth functions on the manifold Mⁿ and c₀ = (-1)ⁿ. Consider the linear (on each tangent space) operators

S0, S1, ... , Sn-1 : TMn --> TMn

given by the general formula

Sk : = (det(g1)/det(g2))(k+2)/(n+1)(c0Gk+1+c1Gk + ... + ciGk-i+1 + ... + ckG).

Consider the operators

I0,  I1, ... , In-1 : L2 --> L2

given by the general formula

Ik(f) : = div(Sk(grad(f))).

Remark 1 . The operator I_n-1 is exactly the operator -\Delta.

Theorem 1 . If the metrics g₁ and g₂ on Mⁿ are geodesically equivalent then the operators I_k pairwise commute. In particular they commute with the Beltrami-Laplace operator \Delta. If the manifold is closed then the operators are self-adjoint.

The metrics g₁, g₂ are strictly non-proportional at a point x₀ from Mⁿ if the characteristic polynomial det(G- tE) has no multiple roots at the point x₀.

Corollary 1 . Suppose Mⁿ is connected. Let metrics g₁, g₂ on Mⁿ be geodesically equivalent and strictly non-proportional at least at one point of Mⁿ. Then the metrics are strictly non-proportional almost everywhere. In particular the operators are linear independent.

Thus if Mⁿ is closed and connected and if the metrics g₁, g₂ on Mⁿ are geodesically equivalent then the Beltrami-Laplace operator of the metric g₁ is completely quantum integrable. In particular there exists a countable basis

F = {f1, f2, ... , fm, ... }

of the space L₂ such that each f_m is an eigenfunction of each operator I_k.

Moreover, in our case the variables can be separated. More precisely, take any function f from the basis F. Since f is an eigenfunction of each operator I_k we have that f is a solution of the system of n partial differential equations

Ik(f) = ak f,   k = 0, 1, ... , n-1.

(1)

The separation of variables means that in a neighborhood of almost any point there exist coordinates (x¹, x², ... , xⁿ) such that in this coordinates the system (1) is equivalent to the system

fxk xk = Rk(xk, a0, a1, ... , an-1)f,   k = 0, 1, ... , n-1,

(2)

where the function R_k depends on the variable x^k and on the parameters a₀, a₁, ... , a_n-1. Then f is the product

X1(x1)X2(x2) ... Xn(xn),

and each X_k is a solution of the equation number k from (2) so that we reduced the system of partial differential equations (1) to the system of ordinary differential equations

X"k(xk) = Rk(xk, a0, a1, ... , an-1)Xk(xk),   k = 0, 1, ..., n-1.

Date received: April 30, 1999