Quantum integrability of the Beltrami-Laplace operator for geodesically equivalent metrics.

Vladimir S. Matveev and Peter Topalov

Abstract In this paper we show that if there exist a Riemannian metric ḡ on M^{n}, geodesically equivalent to the metric g, then it is possible to construct self-adjoint operators I_{0}, I_{1}, ... ,I_{n−1} : L_{2} → L_{2} such that the operators I_{k} commute with the operator Δ and pairwise commute. If the metrics g, ḡ are strictly non-proportional at least at one point then the operators I_{0}, I_{1}, ... ,I_{n−1} are linearly independent. If in addition the manifold is closed then the operators are self-adjoint and we have complete quantum integrability of the Beltrami-Laplace operator. We also formulate (without proofs) when the Liouville integrability of a geodesic flow implies the existence a geodesically equivalent metric.

Let g, ḡ be two Riemannian metrics on the same manifold M^{n}. They are geodesically equivalent, if they have the same geodesics, considered as unparameterized curves.

Consider the Beltrami-Laplace operator Δ := g^{ij}∇_{i}∇_{j}, where ∇ denotes the covariant derivative, corresponding to the metric g. The Beltrami-Laplace operator acts on the space L_{2} of measurable functions ψ : M^{n} → R such that the Lebesgue integral

∫_{M^{n}} ψ^{2}\sqrt{det(g)}dx^{1}dx^{2}...dx^{n}

is finite.

Denote by G the linear operator g^{−1}ḡ = (g^{iα}ḡ_{αj}). Consider the characteristic polynomial det(G − μE) = c_{0}μ^{n} + c_{1}μ^{n−1} + ... + c^{n}. The coefficients c_{1}, .., c_{n} are smooth functions on the manifold M^{n}, and c_{0} ≡ (−1)^{n}. Consider the linear operators

S_{0}, S_{1}, ..., S_{n−1} : T M^{n} → T M^{n}

given by the general formula

S_{k} =^{def} (\frac{det(g)}{det(ḡ)})^{\frac{k+2}{n+1}} ∑_{i=0}^{k} c_{i}G^{k−i+1}.

Consider the operators

I_{0}, I_{1}, ..., I_{n−1} : L_{2} → L_{2}  (1)

given by the general formula

I_{k} := g^{ij}∇_{i}(S_{k})_{j}^{p}∇_{p}.  (2)

Remark 1 . The operator I_{n−1} is exactly the operator −Δ.

Theorem 1 . If the metrics g and ḡ on M^{n} are geodesically equivalent then the operators I_{k} pairwise commute. In particular they commute with the Beltrami-Laplace operator Δ.

Theorem 1 is a ”quantum” analog of the following theorem.

Consider the functions I_{k} : T M^{n} → R, k = 0, ..., n − 1, given by formulae

I_{k}(x, ξ) = g(S_{k}ξ, ξ),  (3)

where g(ν, ξ) denotes the dot product of the tangent vectors ν, ξ ∈ T_{x_{0}}M^{n} in the metrics g.

Theorem 2 . If the metrics g and ḡ on M^{n} are geodesically equivalent then the functions I_{k} are integrals of the geodesic flow of the metric g and pairwise commute (if we canonically identify by the metric g the tangent and the cotangent bundles of M^{n} and consider the standard symplectic form on the cotangent bundle) .

Theorem 2 was proved in [19]. For closed two-dimensional manifolds Theorem 1 was essentially proved in [8].

Remark 2 ([19]) . Let metrics g, ḡ be geodesically equivalent. If at a point x_{0} ∈ M^{n} the number of different roots of the polynomial det(G − μE) is equal to n_{1} then at almost all points of T_{x_{0}}M^{n} the dimension of the linear space generated by dI_{0}, dI_{1}, ..., dI_{n−1} is greater or equal than n_{1}.

In particular, if the characteristic polynomial det(G − μE) has no multiple roots at a point x_{0} ∈ M^{n} then the functions I_{0}, ..., I_{n−1} are functionally independent almost everywhere in T U(x_{0}), where U(x_{0}) is a sufficiently small neighborhood of x_{0}.

If at every point of a neighborhood U(x_{0}) of a point x_{0} ∈ M^{n} the number of different eigenvalues of the polynomial det(G − μE) is less or equal than n_{1} then at all points of T U(x_{0}) the dimension of the linear space generated by dI_{0}, dI_{1}, ..., dI_{n−1} is less or equal than n_{1}.

The metrics g, ḡ are strictly non-proportional at a point x_{0} ∈ M^{n} if the characteristic polynomial det(G − μE) has no multiple roots at the point x_{0} ∈ M^{n}.

Corollary 1 . Suppose M^{n} is connected. Let metrics g, ḡ on M^{n} be geodesically equivalent and strictly non-proportional at least at one point of M^{n}. Then the metrics are strictly non-proportional almost everywhere.

More generally, suppose M^{n} is connected and the metrics g, ḡ on M^{n} are geodesically equivalent. If at every point of a neighborhood U ⊂ M^{n} the number of different eigenvalues of the polynomial det(G − μE) is less or equal than n_{1} then at every point of M^{n} the number of different eigenvalues of the polynomial det(G − μE) is less or equal than n_{1}.

Corollary 2 . Let metrics g, ḡ on a connected manifold M^{n} be geodesically equivalent. The are strictly non proportional at least at one point of M^{n} if and only if the operators I_{0}, ..., I_{n−1} are linearly independent.

Corollaries 1,2 easily follows from the following observation. If we have an integrable Hamiltonian systems then the dimension of the linear space generated by differentials of the integrals is constant along each orbit.

Proof of Corollary 1. Take a geodesic γ : R → M^{n} and consider the points x_{0} = γ(0), x_{1} = γ(1) ∈ M^{n}. Suppose that the metric ḡ is geodesically equivalent to the metric ḡ and strictly non-proportional with g at the point x_{0}. Let us prove that in each neighborhood of the point x_{1} there exist a point x suck that the metrics are strictly non-proportional at x.

The geodesic orbit of the geodesic γ is the curve (γ, γ̇) : R → T M^{n} (assuming γ̇ = γ̇(t) ∈ T_{γ(t)}M^{n} is the velocity vector of γ.) Take a small neighborhood O^{2n} ⊂ T M^{n} of the point (x_{0}, γ̇(0)) ∈ T M^{n}. Consider the union of all points of all geodesic orbits with start from the points of the neighborhood. Denote it by U. Since the metrics are strictly non-proportional at x_{0}, the differentials dI_{0}, dI_{1}, ..., dI_{n−1} of the integrals I_{1}, I_{2}, ..., I_{n−1} are linear independent at almost all points of U. Evidently the set U contains a neighborhood of the point (x_{1}, γ̇(1)). Then the integrals I_{0}, I_{1}, ..., I_{n−1} are functionally independent at almost all points of a neighborhood of the point (x_{1}, γ̇(1)). Using Remark 2 we have that in each neighborhood of x_{1} there exist a point where the metrics are strictly non-proportional.

Now suppose M^{n} is closed (= compact + without boundary) and connected. Let metrics g, ḡ on M^{n} be geodesically equivalent. Suppose the metrics are strictly non-proportional at least at one point of the manifold. Then we have complete Liouville integrability for the geodesic flow of the metric g and complete quantum integrability for the Beltrami-Laplace operator of the metric g. Liouville integrability means in particular that almost all connected components of level submanifolds {I_{0} = C_{0}, I_{1} = C_{1}, ..., I_{n−1} = C_{n−1}}, where C_{1}, C_{2}, ..., C_{n−1} are constants, are n-dimensional tori, and orbits on such tori are windings.

Quantum integrability means that there exists a countable basis

Φ = {ϕ_{1}, ϕ_{2}, ..., ϕ_{m}, ...}

of the space L_{2} such that each ϕ_{m} is an eigenfunction of each operator I_{k}.

Moreover, in our case the variables can be separated. More precisely, take any function ϕ from the basis Φ. Since ϕ is an eigenfunction of each operator I_{k} we have that ϕ is a solution of the system of n partial differential equations

I_{k}ϕ = λ_{k}ϕ, k = 0, 1, ..., n − 1.  (4)

The separation of variables means that in a neighborhood of almost any point there exist coordinates (x^{1}, x^{2}, ..., x^{n}) such that in this coordinates the system (4) is equivalent to the system

\frac{∂^{2}}{(∂x^{k})^{2}} ϕ = F_{k}(x^{k}, λ_{0}, λ_{1}, ..., λ_{n−1})ϕ, k = 0, 1, ..., n − 1,  (5)

where the function F_{k} depends on the variable x^{k} and on the parameters λ_{0}, λ_{1}, ..., λ_{n−1}. Then ϕ is the product

X_{1}(x^{1})X_{2}(x^{2})...X_{n}(x^{n}),

and each X_{k} is a solution of (5) so that we reduced the system of partial differential equations (4) to the system of ordinary differential equations

\frac{∂^{2}}{(∂x^{k})^{2}} X_{k}(x^{k}) = F_{k}(x^{k}, λ_{0}, λ_{1}, ..., λ_{n−1})X_{k}(x^{k}), k = 0, 1, ..., n − 1. (6)

Does the Liouville integrability of the geodesic flow implies the existence of a geodesically equivalent metric? Let g, ḡ be two metrics on M^{n}. Consider the functions I_{k} : T M^{n} → R, k = 0, ..., n − 1, given by (3). With the help of g identify the tangent and the cotangent bundles of M^{n} and consider the standard symplectic structure on T^{∗}M^{n}.

Theorem 3 . Let the functions I_{k} commute and let they be functionally independent almost everywhere. Then the metrics g, ḡ are geodesically equivalent.

Consider the operators I_{k} : L_{2} → L_{2}, k = 0, ..., n − 1, given by 1.

Corollary 3 . Suppose the manifold M^{n} is connected. Let the operators I_{k} commute and let they be linear independent. Then the metrics g, ḡ are geodesically equivalent.

Let the function I_{k} commute. If they are not functionally independent then the metrics can be not geodesically equivalent.

Consider the tangent bundle T M^{n}. Consider the operator G = g^{−1}ḡ. The operator G is evidently self-adjoint (with respect to both metrics). Then all eigenvalues of the operator G are real and we have a decomposition of any tangent space T_{x}M^{n} to the direct sum V_{x}^{1} ⊕ V_{x}^{2} ⊕ ... ⊕ V_{x}^{n_{1}} of eigenspaces.

Recall that the distribution

V_{x}^{1}, V_{x}^{2}, ..., V_{x}^{n_{1}} ⊂ T_{x}M^{n}, V_{x}^{1} ⊕ V_{x}^{2} ⊕ ... ⊕ V_{x}^{n_{1}} = T_{x}M^{n}

is integrable if at a neighborhood of almost every points of M^{n} there exist a coordinate system x^{1}, x^{2}, ..., x^{n} such that the vectors tangent to the coordinate lines lie in the elements of the distribution.

Theorem 4 . Let the functions I_{k} commute and let the distribution of the eigenspaces of the operator G is integrable. Then the metrics g, ḡ are geodesically equivalent.

Are there many interesting examples of geodesically equivalent metrics on closed manifolds? The following theorem (essentially due to N. S. Sinjukov [15]) gives us a construction that, given a pair of geodesically equivalent metrics, produces another pair of geodesically equivalent metrics. Starting from the metric of constant curvature on the sphere, we obtain the metric of the ellipsoid and the metric of the Poisson sphere.

Let g, ḡ be Riemannian metrics on the manifold M^{n}. Consider the linear

operator B : T M^{n} → T M^{n} given by B_{j}^{i} = (\frac{det(ḡ)}{det(g)})^{\frac{1}{n+1}} ḡ^{ip}g_{pj}. By definition put metric gB equal g_{ip}B_{j}^{p}. By definition put metric ḡB equal ḡ_{ip}B_{j}^{p}. In other words, the dot product gB(ξ, ν) of arbitrary vectors ξ, ν ∈ T_{x_{0}}M^{n} is equal to g(Bξ, ν) and the dot product ḡB(ξ, ν) is equal to ḡ(Bξ, ν). Evidently the linear operator B is self-adjoint with respect to the metrics g and ḡ and the metrics gB, ḡB are well-definite.

Theorem 5 . The metrics g and ḡ are geodesically equivalent if and only if the metrics gB and ḡB are geodesically equivalent.

Evidently the metrics g and ḡ are strictly non-proportional at a point x ∈ M^{n} if and only if the metrics gB and ḡB are strictly non-proportional at the point x.

Thus if we have a pair of geodesically equivalent metrics g, ḡ then we can construct the other pair of geodesically equivalent metrics gB, ḡB. We can apply the construction once more, the result is another pair of geodesically equivalent metrics. It is natural to denote it by gB^{2}, ḡB^{2} since this pair is given by

gB^{2}(ξ, ν) = g(B^{2}ξ, ν), ḡB^{2}(ξ, ν) = ḡ(B^{2}ξ, ν).

We can go in other direction and consider the metrics gB^{−1}, ḡB^{−1} given by

gB^{−1}(ξ, ν) = g(B^{−1}ξ, ν), ḡB^{−1}(ξ, ν) = ḡ(B^{−1}ξ, ν).

They are geodesically equivalent also.

To start the process, we need an initial pair of geodesically equivalent metrics g, ḡ. We take the following one (obtained by E. Beltrami [1], [2]). The metric ḡ is the restriction of the Euclidean metric

(dx^{1})^{2} + (dx^{2})^{2} + ... + (dx^{n+1})^{2}

to the standard sphere

S^{n} = {(x^{1}, x^{2}, ..., x^{n+1}) ∈ R^{n+1} : (x^{1})^{2} + (x^{2})^{2} + ... + (x^{n+1})^{2} = 1}.

The metric g is the pull-back l^{∗}ḡ, where the diffeomorphism l : S^{n} → S^{n} is given by

l(x) =^{def} \frac{Ax}{∥Ax∥},

where A is an arbitrary non-degenerate linear transformation of R^{n+1}, Ax means ”the transformation A applied to the vector x = (x^{1}, x^{2}, ..., x^{n+1})” and ∥x∥ is the standard norm \sqrt{(x^{1})^{2} + (x^{2})^{2} + ... + (x^{n+1})^{2}}.

Since the mapping l evidently takes the geodesics of the metric g to the geodesics of the metric g, the metrics g, ḡ are geodesically equivalent. For this choice of metrics g and ḡ the metric gB^{−1} is the metric of an ellipsoid and the metric ḡB^{2} is the metric of a Poisson sphere. Varying the linear operator L, we can obtain metrics of all possible ellipsoids and all possible Poisson spheres.

The metric of the ellipsoid

E =^{def} \{(x^{1}, ..., x^{n+1}) ∈ R^{n+1} : \frac{(x^{1})^{2}}{a_{1}} + \frac{(x^{2})^{2}}{a_{2}} + ... + \frac{(x^{n+1})^{2}}{a_{n+1}} = 1\}

is the restriction of the metric

(dx^{1})^{2} + (dx^{2})^{2} + ... + (dx^{n+1})^{2}

to the ellipsoid E. By the metric of the Poisson sphere we, follow [5], mean the restriction of the metric

\frac{1}{\frac{(x^{1})^{2}}{a_{1}^{2}} + \frac{(x^{2})^{2}}{a_{2}^{2}} + ... + \frac{(x^{n+1})^{2}}{a_{n+1}^{2}}} ((dx^{1})^{2} + ... + (dx^{n+1})^{2})

to the ellipsoid E.

The metric of Poisson sphere has the following mechanical sense. Consider the free motion of (n+1)-dimension rigid body in (n+1)-dimensional space around a fixed point. The configuration space of the correspondent Hamiltonian system is SO(n + 1) and the corresponding Hamiltonian is left-invariant. Consider the embedding SO(n) ⟼ SO(n + 1) as the stabilizer of a vector v ∈ R^{n+1}. Consider the action of the group SO(n) on SO(n+1) by left translations. The Hamiltonian of the motion is evidently invariant modulo this action, and the reduced system on T SO(n + 1)/SO(n) ≅ T S^{n} is the geodesic flow of the (appropriate) Poisson metric.

Theorem 6 ([19], independently obtained by S. Tabachnikov [16]). The restriction of the Euclidean metric

(dx^{1})^{2} + ... + (dx^{n+1})^{2}

to the standard ellipsoid

E^{n} = \{(x^{1}, x^{2}, ..., x^{n+1}) ∈ R^{n+1} : \frac{(x^{1})^{2}}{a_{1}} + ... + \frac{(x^{n+1})^{2}}{a_{n+1}} = 1\}

is geodesically equivalent to the restriction of the metric

\frac{1}{(\frac{x^{1}}{a_{1}} )^{2} + ... + (\frac{x^{n+1}}{a_{n+1}} )^{2}} (\frac{(dx^{1})^{2}}{a_{1}} + ... + \frac{(dx^{n+1})^{2}}{a_{n+1}} )  (7)

to the same ellipsoid.

Theorem 7 (Topalov, [20]). The restriction of the metric

\frac{1}{\frac{(x^{1})^{2}}{a_{1}^{2}} + ... + \frac{(x^{n+1})^{2}}{a_{n+1}^{2}}} ((dx^{1})^{2} + ... + (dx^{n+1})^{2})  (8)

to the ellipsoid

E^{n} = \{(x^{1}, x^{2}, ..., x^{n+1}) ∈ R^{n+1} : \frac{(x^{1})^{2}}{a_{1}} + ... + \frac{(x^{n+1})^{2}}{a_{n+1}} = 1\}

is geodesically equivalent to the restriction of the metric

a_{1}(dx^{1})^{2} + ... + a_{n+1}(dx^{n+1})^{2} − (x^{1}dx^{1} + x^{2}dx^{2} + ... + x^{n+1}dx^{n+1})^{2} (9)

to the same ellipsoid.

The authors are grateful to Professors W. Ballmann, V. Bangert, A.V. Bolsinov, V. Cortes, A. T. Fomenko, M. Gromov, E. Heintze, O. Perelomov, S. Tabachnikov for interesting discussions. The second author was partially supported by MESC’, grant number MM-810/98. The first author is grateful to Max-Planck Institute of Mathematics (Bonn) and Max-Planck-Gesellschaft for hospitality and for financial support.

Proof of Theorem 1.

Let G be the diagonal matrix diag(ρ_{1}, ρ_{2}, ..., ρ_{n}), where

ρ_{i} = \frac{1}{ϕ_{i}(ϕ_{1}ϕ_{2}ϕ_{3}...ϕ_{n})}.

Consider the characteristic polynomial det(G − μE) = c_{0}μ^{n} + c_{1}μ^{n−1} + ... + c^{n} of the matrix G. . Consider the matrixes S_{0}, S_{1}, ..., S_{n−1} given by the general formula

S_{k} =^{def} (\frac{det(g)}{det(ḡ)})^{\frac{k+2}{n+1}} ∑_{i=0}^{k} c_{i}G^{k−i+1}.

Denote by σ_{k} the elementary symmetric polynomial of degree k of n variables variables ϕ_{1}, ϕ_{2}, ..., ϕ_{n}. Denote by σ_{k}(ϕ̂_{i}) the elementary symmetric polynomial of degree k of n − 1 variables ϕ_{1}, ϕ_{2}, ..., ϕ_{i−1}, ϕ_{i+1}, ϕ_{n}.

Lemma 1 .

S_{k} = (−1)^{n−k}diag(σ_{n−k−1}(ϕ̂_{1}), σ_{n−k−1}(ϕ̂_{2}), ..., σ_{n−k−1}(ϕ̂_{n})).

Proof. Easy to see that the coefficients of the characteristic polynomial are given by

c_{k} = (−1)^{n−k} \frac{σ_{n−k}}{(ϕ_{1}ϕ_{2}...ϕ_{n})^{n+1}} .

In particular,

det(G) = \frac{1}{(ϕ_{1}ϕ_{2}...ϕ_{n})^{n+1}} .

Then for k = 0 we have

S_{0} = (\frac{1}{det(G)})^{\frac{2}{n+1}} c_{0}G
= (−1)^{n}(ϕ_{1}, ϕ_{2}, ..., ϕ_{n})^{2}diag(\frac{1}{ϕ_{1}(ϕ_{1}ϕ_{2}...ϕ_{n})}, \frac{1}{ϕ_{2}(ϕ_{1}ϕ_{2}...ϕ_{n})}, ..., \frac{1}{ϕ_{n}(ϕ_{1}ϕ_{2}...ϕ_{n})})
= (−1)^{n}diag(ϕ_{2}ϕ_{3}...ϕ_{n}, ϕ_{1}ϕ_{3}...ϕ_{n}, ..., ϕ_{1}ϕ_{2}...ϕ_{n−3}ϕ_{n−2}, ϕ_{1}ϕ_{2}...ϕ_{n−2}ϕ_{n−1})).

Suppose that the lemma is true for k = l − 1. Then for S_{l} we have

S_{l} = (\frac{1}{det(G)})^{\frac{1}{n+1}} G(S_{l−1} + (\frac{1}{det(G)})^{\frac{k+1}{n+1}} c_{l}E)
= diag(\frac{1}{ϕ_{1}} , \frac{1}{ϕ_{2}} , ..., \frac{1}{ϕ_{n}})diag((−1)^{n−l}σ_{n−l}(ϕ̂_{1}) + (−1)^{n−l+1}σ_{n−l},

(−1)^{n−l}σ_{n−l}(ϕ̂_{2}) + (−1)^{n−l+1}σ_{n−l}, ..., (−1)^{n−l}σ_{n−l}(ϕ̂_{n}) + (−1)^{n−l+1}σ_{n−l})

= (−1)^{n−l−1}diag(σ_{n−l−1}(ϕ̂_{1}), σ_{n−l−1}(ϕ̂_{2}), ..., σ_{n−l−1}(ϕ̂_{n})).

Q. E. D.

For simplicity we assume that the manifold is connected and that the metrics are strictly non-proportional at least at one point of the manifold. Then they are strictly non-proportional at almost all points of the manifold and it is sufficient to check the statement of the theorem in such points.

If geodesically equivalent metrics are strictly non-proportionally at a point x_{0} then in a neighborhood of x_{0} there exist coordinates x^{1}, x^{2}, ..., x^{n} such that the metrics are given by the formulae (see [7]

Π_{1}(dx^{1})^{2} + Π_{2}(dx^{2})^{2} + ⋯ + Π_{n}(dx^{n})^{2}  (10)
ρ^{1}Π_{1}(dx^{1})^{2} + ρ^{2}Π_{2}(dx^{2})^{2} + ⋯ + ρ^{n}Π_{n}(dx^{n})^{2},  (11)

respectively, where

Π_{i} =^{def} (ϕ_{i} − ϕ_{1}) ⋯ (ϕ_{i} − ϕ_{i−1})(ϕ_{i+1} − ϕ_{i}) ⋯ (ϕ_{m} − ϕ_{i}), nonumber (12)
ρ^{i} = \frac{1}{ϕ_{1}...ϕ_{m}} \frac{1}{ϕ_{i}}

and ϕ_{1}, ϕ_{2}, ..., ϕ_{n}, 0 < ϕ_{1} < ϕ_{2} < ... < ϕ_{n}, are smooth functions such that the function ϕ_{i} depends on x^{i} only.

First we show that in this coordinates the operators I_{k} have the form

(−1)^{n}I_{n−1} = Π_{1} \frac{∂^{2}}{(∂x^{1})^{2}} + ⋯ + Π_{n} \frac{∂^{2}}{(∂x^{n})^{2}}
(−1)^{n−2}I_{n−2} = σ_{1}(ϕ_{2}, ..., ϕ_{m})Π_{1} \frac{∂^{2}}{(∂x^{1})^{2}} + ⋯ + σ_{1}(ϕ_{1}, ..., ϕ_{m−1})Π_{n} \frac{∂^{2}}{(∂x^{n})^{2}} ,
(−1)^{n−3}I_{n−3} = σ_{2}(ϕ_{2}, ..., ϕ_{m})Π_{1} \frac{∂^{2}}{(∂x^{1})^{2}} + ⋯ + σ_{2}(ϕ_{1}, ..., ϕ_{m−1})Π_{n} \frac{∂^{2}}{(∂x^{n})^{2}} ,
⋮
−I_{0} = (ϕ_{2}...ϕ_{m})Π_{1} \frac{∂^{2}}{(∂x^{1})^{2}} + ⋯ + (ϕ_{1}...ϕ_{m−1})Π_{n} \frac{∂^{2}}{(∂x^{n})^{2}} ,

where σ_{k} denotes the elementary symmetric polynomial of degree k. Then we prove that the operators of this form commute.

Recall that for arbitrary function F : M^{n} → R and for arbitrary covector T = T_{i} covariant derivatives ∇_{j}F and ∇_{j}T_{i} are given by

∇_{j}F = \frac{∂F}{∂x^{j}}  (13)
∇_{j}T_{i} = \frac{∂T_{i}}{∂x^{j}} − ∑_{h=1}^{n} Γ_{ij}^{h}T_{h},  (14)

where the Christoffel symbols Γ_{jk}^{i} are given by the formula

Γ_{jk}^{i} = \frac{1}{2} ∑_{h=1}^{n} [g^{ih} (\frac{∂}{∂x^{j}} g_{hk} + \frac{∂}{∂x^{k}} g_{jh} − \frac{∂}{∂x^{h}} g_{jk})] .

In the Levi-Civita coordinates the matrix of the metric g is the diagonal matrix diag(Π_{1}, Π_{2}, ..., Π_{n}), so we do not need Γ_{jk}^{i} for j ≠ k to calculate the operator I_{k} := g^{ij}∇_{i}(S_{k})_{j}^{p}∇_{p}.

If i ≠ j we have

Γ_{jj}^{i} = \frac{1}{2} [g^{ii} (\frac{∂}{∂x^{j}} g_{ij} + \frac{∂}{∂x^{j}} g_{ji} − \frac{∂}{∂x^{i}} g_{jj})]
=  −\frac{1}{2}g^{ii} \frac{∂}{∂x^{i}} g_{jj}
=  \frac{1}{2} \frac{ϕ_{i}′}{Π_{i}} [\frac{1}{ϕ_{i} − ϕ_{j}} ] .  (15)

For i = j we have

Γ_{ii}^{i} = \frac{1}{2} [g^{ii} (\frac{∂}{∂x^{i}} g_{ii} + \frac{∂}{∂x^{i}} g_{ii} − \frac{∂}{∂x^{i}} g_{ii})]
=  \frac{1}{2}g^{ii} \frac{∂}{∂x^{i}} g_{ii}
= \frac{1}{2} \frac{ϕ_{i}′}{Π_{i}} [\frac{1}{ϕ_{i} − ϕ_{1}} + ... + \frac{1}{ϕ_{i} − ϕ_{i−1}} + \frac{1}{ϕ_{i} − ϕ_{i+1}} + ... + \frac{1}{ϕ_{i} − ϕ_{n}} ] .(16)

In particular, for each i we have

∑_{j=1}^{n} Γ_{jj}^{i} = 0.  (17)

Let us calculate I_{k} := ∑_{i=1}^{n} ∑_{j=1}^{n} g^{ij}∇_{i}(S_{k})_{j}^{p}∇_{p}. In view Lemma 1 the operator S_{k} is diagonal with the i-th element on the diagonal given by

(S_{k})_{i}^{i} = σ_{n−k−1}(ϕ_{1}, ϕ_{2}, ..., ϕ_{i−1}, ϕ_{i+1}, ..., ϕ_{n}).

Then

T_{j} = ∑_{p=1}^{n} [(S_{k})_{j}^{p}∇_{p}F = σ_{n−k−1}(ϕ_{1}, ϕ_{2}, ..., ϕ_{j−1}, ϕ_{j+1}, ..., ϕ_{n})\frac{∂F}{∂x^{j}} ]

and

I_{k}(F) = ∑_{i,j=1}^{n} [g^{ij}∇_{i} ∑_{p=1}^{n}[(S_{k})_{j}^{p}∇_{p}F]]

= ∑_{i,j=1}^{n}[g^{ij}∇_{i}T_{j}]

= g^{ii}σ_{n−k−1}(ϕ_{1}, ϕ_{2}, ..., ϕ_{i−1}, ϕ_{i+1}, ..., ϕ_{n})\frac{∂^{2}F}{(∂x^{i})^{2}}
− ∑_{i=1}^{n} ∑_{h=1}^{n} [Γ_{ii}^{h} \frac{∂F}{(∂x^{h})}]

= ∑_{i=1}^{n} [\frac{σ_{n−k−1}(ϕ_{1}, ϕ_{2}, ..., ϕ_{i−1}, ϕ_{i+1}, ..., ϕ_{n})}{Π_{i}} \frac{∂^{2}F}{(∂x^{i})^{2}} ]

− ∑_{i=1}^{n} ∑_{h=1}^{n} [Γ_{ii}^{h} \frac{∂F}{(∂x^{h})}] .

Since in view of (17) ∑_{i=1}^{n} ∑_{h=1}^{n} [Γ_{ii}^{h} \frac{∂F}{∂x^{h}} ] is zero we have

I_{k} = ∑_{i=1}^{n} [\frac{σ_{n−k−1}(ϕ_{1}, ϕ_{2}, ..., ϕ_{i−1}, ϕ_{i+1}, ..., ϕ_{n})}{Π_{i}} \frac{∂^{2}}{(∂x^{i})^{2}} ] .

Q.E.D.

Now let us prove that the operators I_{k} commute. Consider the operator

[I_{l+1−n}, I_{m+1−n}] = I_{l+1−n}I_{m+1−n} − I_{m+1−n}I_{l+1−n}.

Since each I_{k} is a differential operator of order 2, the operator [I_{l+1−n}, I_{m+1−n}] is a differential operator of order no greater than 4.

It is easy to see that the components of order 4, 3, 1, 0 of the operator [I_{l+1−n}, I_{m+1−n}] are zero. More precisely, the component of order 4 vanishes since for any i, j the operators \frac{∂^{2}}{(∂x^{i})^{2}} and \frac{∂^{2}}{(∂x^{j})^{2}} commute. The component of order 3 vanishes since the functions I_{l} and I_{m} are in involution. The components of order 1 and 0 vanish since the operators I_{l} and I_{m} have no component of order 1 and 0.

Thus we have to verify that the component of order 2 vanishes also. Evidently the coefficient by \frac{∂^{2}}{∂x^{j}∂x^{i}} is zero for i ≠ j. The coefficient by \frac{∂^{2}}{(∂x^{j})^{2}} is given by

∑_{i=1}^{n} [\frac{σ_{l}(ϕ_{1}, ϕ_{2}, ..., ϕ_{i−1}, ϕ_{i+1}, ..., ϕ_{n})}{Π_{i}} \frac{∂^{2}}{(∂x^{i})^{2}} \frac{σ_{m}(ϕ_{1}, ϕ_{2}, ..., ϕ_{j−1}, ϕ_{j+1}, ..., ϕ_{n})}{Π_{j}}

−\frac{σ_{m}(ϕ_{1}, ϕ_{2}, ..., ϕ_{i−1}, ϕ_{i+1}, ..., ϕ_{n})}{Π_{i}} \frac{∂^{2}}{(∂x^{i})^{2}} \frac{σ_{l}(ϕ_{1}, ϕ_{2}, ..., ϕ_{j−1}, ϕ_{j+1}, ..., ϕ_{n})}{Π_{j}}] .

So it is sufficient to prove that (for every i, j)

\frac{σ_{l}(ϕ_{1}, ϕ_{2}, ..., ϕ_{i−1}, ϕ_{i+1}, ..., ϕ_{n})}{Π_{i}} \frac{∂^{2}}{(∂x^{i})^{2}} \frac{σ_{m}(ϕ_{1}, ϕ_{2}, ..., ϕ_{j−1}, ϕ_{j+1}, ..., ϕ_{n})}{Π_{j}}

−\frac{σ_{m}(ϕ_{1}, ϕ_{2}, ..., ϕ_{i−1}, ϕ_{i+1}, ..., ϕ_{n})}{Π_{i}} \frac{∂^{2}}{(∂x^{i})^{2}} \frac{σ_{l}(ϕ_{1}, ϕ_{2}, ..., ϕ_{j−1}, ϕ_{j+1}, ..., ϕ_{n})}{Π_{j}}.

is zero. Since σ_{l}(ϕ_{1}, ϕ_{2}, ..., ϕ_{i−1}, ϕ_{i+1}, ..., ϕ_{n}) does not depend on x^{i}, it is sufficient to prove that

\frac{1}{Π_{j}} [σ_{l}(ϕ_{1}, ϕ_{2}..., ϕ_{i−1}, ϕ_{i+1}, ..., ϕ_{n})σ_{m}(ϕ_{1}, ϕ_{2}..., ϕ_{j−1}, ϕ_{j+1}, ..., ϕ_{n})

−σ_{m}(ϕ_{1}, ϕ_{2}..., ϕ_{i−1}, ϕ_{i+1}, ..., ϕ_{n})σ_{l}(ϕ_{1}, ϕ_{2}..., ϕ_{j−1}, ϕ_{j+1}, ..., ϕ_{n})] (18)

does not depend on x^{i}.

The last statement is trivial for i = j. Assume i < j. Denote by σ̄_{k} the elementary symmetric polynomial

σ_{k}(ϕ_{1}, ϕ_{2}, ..., ϕ_{i−1}, ϕ_{i+1}, ..., ϕ_{j−1}ϕ_{j+1}, ..., ϕ_{n}).

Substitute the trivial identities

σ_{l}(ϕ_{1}, ϕ_{2}, ..., ϕ_{i−1}, ϕ_{i+1}, ..., ϕ_{n}) = σ̄_{l−1}ϕ_{i} + σ̄_{l}
σ_{m}(ϕ_{1}, ϕ_{2}, ..., ϕ_{i−1}, ϕ_{i+1}, ..., ϕ_{n}) = σ̄_{m−1}ϕ_{i} + σ̄_{m}
σ_{l}(ϕ_{1}, ϕ_{2}, ..., ϕ_{j−1}, ϕ_{j+1}, ..., ϕ_{n}) = σ̄_{l−1}ϕ_{i} + σ̄_{l}
σ_{m}(ϕ_{1}, ϕ_{2}, ..., ϕ_{j−1}, ϕ_{j+1}, ..., ϕ_{n}) = σ̄_{m−1}ϕ_{i} + σ̄_{m}

in (18). After simple calculation we obtain

\frac{(σ̄_{l}σ̄_{m} − σ̄_{m}σ̄_{l})(ϕ_{i} − ϕ_{j})}{Π_{j}}

which does not depend on x^{i}. The statement is proved for i < j, the prove for i > j is essentially the same. Q.E.D.

If the manifold is closed then the operators I_{k} are self-adjoint. In this section we assume that the manifold M^{n} is closed. Let g and ḡ be

geodesically equivalent metrics on M^{n}. Consider the operators I_{k} given by (2). Recall that L_{2} is a Hilbert space with the following dot product:

< ϕ, ψ >= ∫_{M^{n}} ϕψdV,

where dV is the standard volume form dV = \sqrt{det(g)}dx^{1}dx^{2}...dx^{n} on M^{n}.

Lemma 2 . The operators I_{k} are self-adjoint.

In other words for any ϕ, ψ ∈ L_{2}

< ϕ, I_{k}(ψ) >=< I_{k}(ϕ), ψ >

for any ϕ, ψ ∈ L_{2}.

Proof. Recall that for each covector field T_{i} on the manifold and for each function ϕ we have

∫_{M^{n}} g^{ij}∇_{i}T_{j}dV = 0 (Divergence theorem).  (19)
ϕ∇_{i}T_{j} = ∇_{i}(ϕT_{j}) − T_{j}∇_{i}ϕ (Leibnitz rule).  (20)

Then

< ϕ, I_{k}(ψ) > = ∫_{M^{n}} ϕI_{k}(ψ)dV
= ∫_{M^{n}} ϕg^{ij}∇_{i}(S_{k})_{j}^{p}∇_{p}ψdV
= ∫_{M^{n}} g^{ij}∇_{i}((S_{k})_{j}^{p}ϕ∇_{p}ψ)dV − ∫_{M^{n}} g^{ij}(∇_{i}ϕ)((S_{k})_{j}^{p}∇_{p}ψ)dV
= − ∫_{M^{n}} g^{ip}(S_{k})_{p}^{j}(∇_{i}ϕ)(∇_{j}ψ).  (21)

Since the operator S_{k} is self-adjoint with respect to the metrics g, the twoform g^{ip}(S_{k})_{p}^{j} is symmetric and therefore the operator I_{k} is self-adjoint. Q. E. D.

Proof of Theorems 6,7.

Consider the space R^{n+1} with the standard coordinates x = (x^{1}, x^{2}, ..., x^{n+1}) and the sphere

S^{n} ⊂ R^{n+1}, S^{n} = {x ∈ R^{n+1} : (x^{1})^{2} + (x^{2})^{2} + ... + (x^{n+1})^{2} = 1}.

Let A be the diagonal matrix diag(\frac{1}{\sqrt{a_{1}}} , \frac{1}{\sqrt{a_{2}}} , ..., \frac{1}{\sqrt{a_{n+1}}} ), where 0 < a_{1} ≤ a_{2} ≤ ... ≤ a_{n+1}.

In order to prove Theorems 6,7 we consider the linear transformation x ⟼ Ax of the space R^{n+1} and construct the corresponding projective transformation l of the sphere S^{n} and the metric ḡ = l^{∗}g on the sphere (where g is the restriction of the Euclidean metric to the sphere). Then we construct the operator B and show that the metrics gB, ḡB are essentially the metrics from Theorem 6 and the metrics gB^{2}, ḡB^{2} are essentially the metrics from Theorem 7 so that Theorems 6,7 follow from Theorem 5.

Take an arbitrary point x = (x^{1}, x^{2}, ..., x^{n+1}) ∈ S^{n}. Denote by T_{x}R^{n+1} and T_{x}S^{n} the tangent spaces at the point x to R^{n+1} and S^{n}, respectively. Identify the tangent space T_{x}R^{n+1} with R^{n+1} by moving the origin of the coordinate system to the point x. Consider the matrix

Ā = \frac{1}{\sqrt{\frac{(x^{1})^{2}}{a_{1}} + \frac{(x^{2})^{2}}{a_{2}} + ... + \frac{(x^{n+1})^{2}}{a_{n+1}}}} A

and the matrix P with the general element P_{ij} = x^{i}x^{j}. The matrixes Ā, P depend on the choice of the point x. Easy to see that for each vector v ∈ T_{x}R^{n+1} the vector P v is the orthogonal projection of the vector v to the vector normal to the sphere. Then

P P = P,  (22)

since the square of the projection is the projection and

P Ā^{2}P = P,  (23)

since in view of (22) the matrix P Ā^{2}P commute with P, takes T_{x}S^{n} to zero and therefore is proportional to the projection P. Easy to se that the coefficient of proportionality is 1, since for the normal vector η = (x^{1}, x^{2}, ..., x^{n+1}) to the sphere we have P Ā^{2}P η = η.

Consider the mapping

L : R^{n+1} → R^{n+1}, L(x) = \begin{matrix}√ & \overline{(x^{1})^{2} + (x^{2})^{2} + ... + (x^{n+1})^{2}} \\ √ & \overline{\frac{(x^{1})^{2}}{a_{1}} + \frac{(x^{2})^{2}}{a_{2}} + ... + \frac{(x^{n+1})^{2}}{a_{n+1}}} \end{matrix} Ax.

The mapping L is a diffeomorphism on R^{n+1} \ 0, takes the sphere S^{n} to itself and coincide with the mapping l : x ⟼ \frac{Ax}{|Ax|} on the sphere.

The Jacobian J of the mapping L at the point x ∈ S^{n} is equal to

Ā − ĀP Ā^{2} + ĀP.

Then the pull-back of the Euclidean metric in R^{n+1} is given by

JTJ = (Ā − Ā^{2}P Ā + P Ā)(Ā − ĀP Ā^{2} + ĀP)
= Ā^{2} − Ā^{2}P Ā^{2} + Ā^{2}P
− Ā^{2}P Ā^{2} + Ā^{2}P Ā^{2}P Ā^{2} − Ā^{2}P Ā^{2}P
+ P Ā^{2} − P Ā^{2}P Ā^{2} + P Ā^{2}P
= Ā^{2} − Ā^{2}P Ā^{2} + P.

Let us show that the matrix

G^{−1} = Ā^{−2} − Ā^{−2}P − P Ā^{−2} + P Ā^{−2}P + P

is inverse to the matrix G = Ā^{2} − Ā^{2}P Ā^{2} + P.

Denote by 1 the identity (n + 1) × (n + 1) matrix diag(1, 1, ..., 1). We have

G^{−1}G = (Ā^{−2} − Ā^{−2}P − P Ā^{−2} + P Ā^{−2}P + P)(Ā^{2} − Ā^{2}P Ā^{2} + P)
= 1 − Ā^{−2}P Ā^{2} − P + P Ā^{−2}P Ā^{2} + P Ā^{2}
− P Ā^{2} + Ā^{−2}P Ā^{2}P Ā^{2} + P P Ā^{2} − P Ā^{−2}P Ā^{2}P Ā^{2}
+ Ā^{−2}P − Ā−2P P − P Ā^{−2}P + P Ā^{−2}P P + P P
= 1.

Since the Euclidean metric g_{euclid} is given by the identity matrix 1, then the matrix of the operator (l̄^{∗}g_{euclid})^{−1}g_{euclid} is the matrix G^{−1}. Let us show that G^{−1} takes tangent vectors to sphere to tangent vectors to sphere and therefore the restriction of the operator (l̄^{∗}g_{euclid})^{−1}g_{euclid} to T_{x}S^{n} coincide with the operator ḡ^{−1}g. Evidently T_{x}S^{n} is the eigenspace of the matrix P (with the eigenvalue 0). Then it is sufficient to show that the matrix P commute with G^{−1}. We have

G^{−1}P = (Ā^{−2} − Ā^{−2}P − P Ā^{−2} + P Ā^{−2}P + P)P
= Ā^{−2}P − Ā^{−2}P P − P Ā^{−2}P + P Ā^{−2}P P + P P
= Ā^{−2}P − Ā^{−2}P − P Ā^{−2}P + P Ā^{−2}P + P
= P̄A^{−2} − P Ā^{−2} − P Ā^{−2}P + P Ā^{−2}P + P
= P (Ā^{−2} − Ā^{−2}P − P Ā^{−2} + P Ā^{−2}P + P).

Thus the operator B at the tangent space T_{x}S^{n} is proportional to the restriction of the operator given by the matrix G^{−1} to the tangent space T_{x}S^{n}. Let us

find the coefficient of proportionality. By definition it is equal to (\frac{det(ḡ)}{det(g)})^{\frac{1}{n+1}} . It is easy to see that

\frac{det(ḡ)}{det(g)} = det(G).

More precisely, since the matrix P commute with the matrix G^{−1}, the tangent space T_{x}S^{n} is invariant under multiplication by the matrix G^{−1}. Since on T_{x}S^{n} the operator corresponding to the matrix G^{−1} coincide with the operator g^{−1}ḡ then all eigenvalues and eigenvectors of the operator g^{−1}ḡ are eigenvalues and eigenvectors of the matrix G and n eigenvectors of the matrix G lie in T_{x}S^{n}. The only eigenvector which does not lie in T_{x}S^{n} is the normal vector η to the sphere. The corresponding eigenvalue is evidently equal to 1. Using the fact that the determinant is equal to the product of all eigenvalues we obtain

\frac{det(ḡ)}{det(g)} = \frac{1}{det(ḡ^{−1}g)} = \frac{1}{det(G^{−1})} = det(G).

Let us find the determinant of the matrix G. We have

det(G) = det^{2}(J)
= det^{2}(Ā − ĀP Ā^{2} + ĀP)
= \frac{1}{det^{2}(A)}det^{2}(Ā^{2} − Ā^{2}P Ā^{2} + Ā^{2}P)

Let us show that

det(Ā^{2} − Ā^{2}P Ā^{2} + Ā^{2}P) = det(Ā^{2} − Ā^{2}P Ā^{2} + P).

Easy to check that

(Ā^{2} − Ā^{2}P Ā^{2} + Ā^{2}P)(1 − P) = (Ā^{2} − Ā^{2}P Ā^{2} + P)(1 − P)

so that the operators (Ā^{2} − Ā^{2}P Ā^{2} + Ā^{2}P) and (Ā^{2} − Ā^{2}P Ā^{2} + P) coincide on T S^{n}. Then there exist an eigenvector v ∉ T S^{n} of the operator (Ā^{2} − Ā^{2}P Ā^{2} + Ā^{2}P). The corresponding eigenvalue is equal to 1 since P (Ā^{2}−Ā^{2}P Ā^{2}+Ā^{2}P) = P. Since the determinant is the product of all eigenvalues we have

det(Ā^{2} − Ā^{2}P Ā^{2} + Ā^{2}P) = det(Ā^{2} − Ā^{2}P Ā^{2} + P).

Thus

\frac{1}{det^{2}(A)}det^{2}(G) = det(G)

and

det(G) = ( \frac{1}{\frac{(x^{1})^{2}}{a_{1}} + \frac{(x^{2})^{2}}{a_{2}} + ... + \frac{(x^{n+1})^{2}}{a_{n+1}}} )^{n+1} .

Finally the operator B is the restriction of the operator given by the matrix

( \frac{1}{\frac{(x^{1})^{2}}{a_{1}} + \frac{(x^{2})^{2}}{a_{2}} + ... + \frac{(x^{n+1})^{2}}{a_{n+1}}} ) G^{−1}

to T S^{n}. Then the metrics gB, ḡB are the restrictions to T S^{n} of the metrics given by the matrixes

( \frac{1}{\frac{(x^{1})^{2}}{a_{1}} + \frac{(x^{2})^{2}}{a_{2}} + ... + \frac{(x^{n+1})^{2}}{a_{n+1}}} ) G^{−1}, ( \frac{1}{\frac{(x^{1})^{2}}{a_{1}} + \frac{(x^{2})^{2}}{a_{2}} + ... + \frac{(x^{n+1})^{2}}{a_{n+1}}} ) 1,

respectively.

The restriction to the sphere of the metric with matrix

( \frac{1}{\frac{(x^{1})^{2}}{a_{1}} + \frac{(x^{2})^{2}}{a_{2}} + ... + \frac{(x^{n+1})^{2}}{a_{n+1}}} ) G^{−1}

coincides with the restriction to the sphere of the metric with the matrix

( \frac{1}{\frac{(x^{1})^{2}}{a_{1}} + \frac{(x^{2})^{2}}{a_{2}} + ... + \frac{(x^{n+1})^{2}}{a_{n+1}}} ) Ā^{−2} = A^{−2}

To show this it is sufficient to check that

(1 − P)G^{−1}(1 − P) = (1 − P)Ā(1 − P)

which is trivial.

Let us show that the restriction of the metric with the matrix A to the sphere is the metric of the ellipsoid. Consider the ellipsoid

\{\frac{(x^{1})^{2}}{a_{1}} + \frac{(x^{2})^{2}}{a_{2}} + ... + \frac{(x^{n+1})^{2}}{a_{n+1}} = 1\} .

Consider the mapping

m : R^{n+1} → R^{n+1}, m(x^{1}, x^{2}, ..., x^{n}) = (\frac{x^{1}}{a_{1}} , \frac{x^{2}}{a_{2}} , ..., \frac{x^{n+1}}{a_{n+1}} ).

This mapping takes the ellipsoid to the sphere.

The Jacobian of this mapping is the A. Then the pull-back metric of the metric with the matrix A^{−2} is the metric A^{T} A^{−2}A = 1.

Analogically the pull-back of the metric ḡB to the sphere gives us the metric (7). More precisely, the matrix of the pull-back of the metric with the matrix

( \frac{1}{\frac{(x^{1})^{2}}{a_{1}} + \frac{(x^{2})^{2}}{a_{2}} + ... + \frac{(x^{n+1})^{2}}{a_{n+1}}} ) 1

is given by the matrix

\frac{1}{\frac{(x^{1})^{2}}{a_{1}} + \frac{(x^{2})^{2}}{a_{2}} + ... + \frac{(x^{n+1})^{2}}{a_{n+1}}} A^{2}.

Theorem 6 is proved.

The metric gB^{2}, ḡB^{2} are the restriction to the sphere of the metrics with matrixes

( \frac{1}{\frac{(x^{1})^{2}}{a_{1}} + \frac{(x^{2})^{2}}{a_{2}} + ... + \frac{(x^{n+1})^{2}}{a_{n+1}}} )^{2} (G^{−1})^{2}, ( \frac{1}{\frac{(x^{1})^{2}}{a_{1}} + \frac{(x^{2})^{2}}{a_{2}} + ... + \frac{(x^{n+1})^{2}}{a_{n+1}}} )^{2} (G^{−1}),

respectively. Easy to check that the matrix (1 − P)(G^{−1})^{2}(1 − P) is the matrix Ā^{−4} − Ā^{−2}P Ā^{−2}. Then the pull-back of the corresponding metrics gives us the metrics with the matrix

A^{−2} − P, ( \frac{1}{\frac{(x^{1})^{2}}{a_{1}} + \frac{(x^{2})^{2}}{a_{2}} + ... + \frac{(x^{n+1})^{2}}{a_{n+1}}} ) 1,

which give us the metrics (8), (9). Q. E. D.

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