Geodesic equivalence and integrability.

Peter J. Topalov, Vladimir S. Matveev

Abstract

We suggest a construction that, given a trajectorial diffeomorphism between two Hamiltonian systems, produces integrals of them. As the main example we treat geodesic equivalence of metrics. We show that the existence of a non-trivially geodesically equivalent metric leads to Liouville integrability, and present explicit formulae for integrals.

1 Introduction

Integrals of a system are closely related to symmetries. A classical example is Noether’s theorem: if a vector field X on a manifold M preserves a Lagrangian L : T M → R , then the function I_{X} =^{def} \frac{∂L}{∂ẋ}(x, ẋ)X(x) is a first integral of the corresponding Lagrangian system.

There are many generalizations of Noether’s theorem, we recall the following two. In the paper [2] it was shown that the existence of a vector field on T^{∗}M which commutes with a Hamiltonian vector field allows one to construct a (multi-valued) integral of the Hamiltonian system. In the paper [11] the result of [2] was generalized to tensor fields. It was shown that if a Hamiltonian flow preserves a tensor field on T^{∗}M, then there exists an (also multi-valued) integral of the Hamiltonian system.

In our paper we, following ideas of [11], present a construction which, given a diffeomorphism between two Hamiltonian systems that takes the trajectories and the isoenergy surfaces of the first Hamiltonian system to the trajectories and the isoenergy surfaces of the second one, produces n integrals of the first system, where n is the number of the degrees of freedom of the system.

The construction is applied to geodesically equivalent metrics. Let g = (g_{ij}) and ḡ = (ḡ_{ij}) be smooth metrics on the same manifold M^{n}.

Definition 1. The metrics g and ḡ are geodesically equivalent, if they have the same geodesics (considered as unparameterized curves).

This is rather classical material. In 1869 Dini [3] formulated the problem of local classification of geodesically equivalent metrics, and solved it for dimension two. In 1896 Levi-Civita [4] got a local description of geodesically equivalent metrics on manifolds of arbitrary dimension. In the paper [6] a family of (nontrivial) examples of geodesically equivalent metrics on closed manifolds was constructed.

For geodesically equivalent metrics, a trajectorial diffeomorphism Φ is given by Φ(x, ξ) = (x, \frac{∥ξ∥_{g}}{∥ξ∥ḡ} ξ). Here (x, ξ) ∈ T M^{n}, x is a point of M^{n} and ξ ∈ T_{x}M^{n}.

Theorem 1. Let metrics g and ḡ on M^{n} be geodesically equivalent. 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}. Then the functions I_{k} =

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

of the geodesic flow of the metric g and pairwise commute.

Remark 1. The integral I_{0} = (\frac{det(g)}{det(ḡ)})^{\frac{2}{n+1}} ḡ(ξ, ξ) was obtained by Painlevé, see [4]. The integral I_{n−1} is the energy integral (multiplied by minus two).

The integrals I_{1}, I_{2}, ..., I_{n−2} seem to be new, although in each Levi-Civita chart the integrals are linear combinations of Levi-Civita integrals (see Section 3 for definitions). We touch on the connection between the integrals I_{0}, ..., I_{n−1} and Levi-Civita integrals in Section 5.

Metrics g, ḡ on M^{n} are strictly non-proportional at a point x ∈ M^{n}, if the characteristic polynomial \frac{1}{det(g)}det(ḡ − tg))_{|x} has no multiple root.

Corollary 1. Let M^{n} be a closed real-analytic manifold supplied with two realanalytic metrics g, ḡ such that the metrics g, ḡ are geodesically equivalent and strictly non-proportional at least at one point. Then the fundamental group π_{1}(M^{n}) of the manifold M^{n} contains a commutative subgroup of finite index, and the dimension of the homology group H_{1}(M^{n}; Q) is no greater than n.

For dimension two the converse of Theorem 1 is also true, and the condition of Corollary 1 can be weakened.

Corollary 2. Metrics g and ḡ on a surface M^{2} are geodesically equivalent, if

and only if the function (\frac{det(g)}{det(ḡ)})^{\frac{2}{3}} ḡ(ξ, ξ) is an integral of the geodesic flow of the metric g.

Corollary 3. Let metrics g, ḡ on a closed surface of negative Euler characteristic be geodesically equivalent. Then g = Cḡ, where C is a constant.

Corollary 4. Let metrics g, ḡ on the torus T^{2} be geodesically equivalent. If they are proportional at a point x ∈ T^{2}, then g = Cḡ, where C is a positive constant.

Corollary 5. Let metrics g, ḡ on the sphere S^{2} be geodesically equivalent. Then there are three possibilities.

1. The metrics are proportional at exactly two points

2. The metrics are proportional at exactly four points.

3. The metrics are completely proportional, i.e. g = Cḡ, where C is a positive constant.

In the first case the metrics admit a Killing vector field.

Recall that a vector field on M^{n} is Killing (with respect to a metric), if the flow of the field preserves the metric.

Corollary 6. Let metrics g, ḡ on a surface M^{2} be geodesically equivalent. If the metrics are proportional at each point of an open non-empty domain U ⊂ M^{2}, then g = Cḡ, where C is a positive constant.

Corollary 7. If metrics g, ḡ on a surface M^{n} are geodesically equivalent, and if the metric g admits a non-trivial Killing vector field, then the metric ḡ also admits a non-trivial Killing vector field.

One of the most famouse integrable geodesic flow on closed surface is the geodesic flow of the metric on ellipsoid (see [7]). Consider the ellipsoid ∑_{i=1}^{n} \frac{(x^{i})^{2}}{a_{i}} = 1, where a_{i} > 0, i = 1, ..., n.

Theorem 2. The restriction of the metric ∑_{i=1}^{n}(dx^{i})^{2} to the ellipsoid ∑_{i=1}^{n} \frac{(x^{i})^{2}}{a_{i}} = 1 is geodesically equivalent to the restriction of the metric

\frac{1}{∑{n \atop i=1} (\frac{x^{i}}{a_{i}} )2} (∑_{i=1}^{n} \frac{(dx^{i})^{2}}{a_{i}} )

to the ellipsoid.

The paper is organized as follows. In Section 2 we present the announced construction. Theorem 3 there gives an explicit formula for a one-parameter family of first integrals, if a trajectorial diffeomorphism between two Hamiltonian systems is given.

In Section 3, for use in Sections 4, 5, 7 we formulate Levi-Civita and Painlevé results about a local form of geodesically equivalent metrics.

In Section 4 we apply the construction to geodesically equivalent metrics, and prove that the functions I_{0}, ..., I_{n−1} from Theorem 1 are integrals of the geodesic flow of the metric g.

In Section 5 we prove that the integrals I_{0}, ..., I_{n−1} are in involution.

In Section 6 we prove Corollaries 1, 2, 3, 4, 5, 6, 7.

In Section 7 we prove Theorem 2.

The authors are grateful to A. V. Bolsinov, A. T. Fomenko, V. V. Kozlov and I. A. Taimanov for useful discussions. Essential part of the results were obtained during a 4-week visit of P. Topalov to Bremen University. Authors are grateful to the Institute of Theoretical Physics of the Bremen University for the hospitality and to the Deutsche Forschungsgemeinschaft for partial financial support.

2 Trajectorial diffeomorphisms and integrals

Let v and v̄ be Hamiltonian systems on symplectic manifolds (M, ω) and (M̄, ω̄) with Hamiltonians H and H̄ respectively. Consider the isoenergy surfaces

Q =^{def} {x ∈ M : H(x) = h} , Q̄ =^{def} {x ∈ M : H̄(x) = h̄} ,

where h and h̄ are regular values of the functions H, H̄ respectively. Let U(Q) ⊂ M and U(Q̄) ⊂ M̄ be neighborhoods of the isoenergy surfaces Q and Q̄.

Definition 2. A diffeomorphism Φ : U(Q) ⟶ U(Q̄), Φ(Q) = Q̄, is said to be trajectorial on Q, if the restriction Φ|_{Q} takes the trajectories of the system v to the trajectories of the system v̄.

Denote the restriction Φ|_{Q} by ϕ. Since ϕ takes the trajectories of v to the trajectories of v̄, it takes the vector field v to the vector field that is proportional to v̄. Denote by a_{1} : Q → R the coefficient of proportionality, i.e. ϕ_{∗}(v) = a_{1}v̄. Since Φ takes Q to Q̄, it takes the differential dH to a form that is proportional to dH̄. Denote by a_{2} : Q → R the coefficient of proportionality, i.e. ϕ_{∗}dH = a_{2}dH̄. By a we denote the product a_{1}a_{2}. We denote the Pfaffian of a skew-symmetric matrix X by Pf(X).

Theorem 3. Let a diffeomorphism Φ : U(Q) → U(Q̄), Φ(Q) = Q̄, be trajectorial on Q. Then for each value of the parameter t the polynomial

P^{n−1}(t) =^{def} \frac{Pf (Φ^{∗}ω̄ − tω)}{Pf (ω) (t − a)}  (1)

is an integral of the system v on Q. In particular, all the coefficients of the polynomial P^{n−1}(t) are integrals.

Proof. Denote by σ, σ̄ the restrictions of the forms ω, ω̄ to Q, Q̄ respectively. Consider the form ϕ^{∗}σ̄ on Q.

Lemma 1 (Topalov, [11]). The flow v preserves the form ϕ^{∗}σ̄.

Proof of Lemma 1. The Lie derivative L_{v} of the form ϕ^{∗}σ̄ along the vector field v satisfies

L_{v}ϕ^{∗}σ̄ = d [ı_{v}ϕ^{∗}σ̄] + ı_{v}d [ϕ^{∗}σ̄] .

On the right side both terms vanish. More precisely, for an arbitrary vector u ∈ T_{x}Q at an arbitrary point x ∈ Q we have

ı_{v}ϕ^{∗}σ̄(u) = σ̄(ϕ_{∗}(v), ϕ_{∗}(u)) =
= σ̄(a_{1}v̄, ϕ_{∗}(u)) =
= −a_{1}dH̄ (ϕ_{∗}(u)) = 0.

Since the form ω̄ is closed, the form σ̄ is also closed and d [ϕ^{∗}σ̄] = ϕ^{∗}(dσ̄) = 0, q. e. d.

It is obvious that the kernels of the forms σ and ϕ^{∗}σ̄ coincide (in the space T_{x}Q at each point x ∈ Q) with the linear span of the vector v. Therefore these forms induce two non-degenerate tensor fields on the quotient bundle T Q/⟨v⟩. We shall denote the corresponding forms on T Q/⟨v⟩ also by the letters σ, σ̄.

Lemma 2. The characteristic polynomial of the operator (σ)^{−1}(ϕ^{∗}σ̄) on T Q/⟨v⟩ is preserved by the flow v.

Proof of Lemma 2. Since the flow v preserves the Hamiltonian H and the form ω, the flow v preserves the form σ. Since the flow v preserves both forms, it preserves the characteristic polynomial of the operator (σ)^{−1}(ϕ^{∗}σ̄), q. e. d.

Since both forms are skew-symmetric, each root of the characteristic polynomial of the operator (σ)^{−1}(Φ^{∗}σ̄) has an even multiplicity. Then the characteristic polynomial is the square of a polynomial δ^{n−1}(t) of degree n − 1. Hence the polynomial δ^{n−1}(t) is also preserved by the flow v. It is obvious that

δ^{n−1}(t) = (−1)^{n−1} \begin{matrix}Pf & (ϕ^{∗}σ̄ − tσ) \\ Pf & (σ) \end{matrix}.  (2)

The last step of the proof is to verify that

(t − a)δ^{n−1} = \begin{matrix}Pf & (Φ^{∗}ω̄ − tω) \\ Pf & (ω) \end{matrix} =^{def} Δ^{n}.

Take an arbitrary point x ∈ Q. Consider the form Φ^{∗}ω̄ − aω on T_{x}M. The form ı_{v}(Φ^{∗}ω̄ − aω) equals zero. More precisely, for any vector u ∈ T_{x}M we have

ı_{v}(Φ^{∗}ω̄ − aω) = ω̄(Φ_{∗}(v), Φ_{∗}(u)) − aω(v, u) =
= ω̄(a_{1}v, Φ_{∗}(u)) − aω(v, u) =
= −a_{1}dH̄(Φ_{∗}(u)) + adH =
= −adH + adH = 0.

There exists a vector A ∈ T_{x}M such that ω(A, v) ≠ 0 and the restriction of the form ı_{A}(Φ^{∗}ω̄−aω) to the space T_{x}M equals zero. More precisely, since the forms Φ^{∗}ω̄, ω are skew-symmetric, then the kernel K_{Φ^{∗}ω̄−aω} of the form Φ^{∗}ω̄ − aω has an even dimension, and the kernel of the restriction of the form Φ^{∗}ω̄ − aω to T_{x}Q has an odd dimension. Thus the intersection K_{Φ^{∗}ω̄−aω} ∩(T_{x}M \T_{x}Q) is not empty. For each vector A from the intersection we obviously have ω(A, v) ≠ 0 and ı_{A}(Φ^{∗}ω̄ − aω) = 0. Without loss of generality we can assume ω(A, v) = 1.

Consider a basis (v, e_{1}, ..., e_{2n−2}) for the space T_{x}Q. The set (A, v, e_{1}, ..., e_{2n−2}) is a basis for the space T_{x}M. In this basis we have

det(Φ^{∗}ω̄ − tω) = det | \begin{matrix}0 & a − t & (∗) \\ −(a − t) & 0 & 0 ⋯ 0 \\ −(∗) & 0 & (Φ^{∗}ω̄ − tω)_{⟨e_{1},...,e_{2n−2}⟩} \end{matrix} |

= (a − t)^{2} det((Φ^{∗}ω̄ − tω)_{⟨e_{1},...,e_{2n−2}⟩})
= (a − t)^{2} det(ϕ^{∗}σ̄ − tσ),

where (Φ^{∗}ω̄ − tω)_{⟨e_{1},...,e_{2n−2}⟩} is the matrix of the form Φ^{∗}ω̄ − tω in the basis (e_{1}, ...e_{2n−2}). Finally, δ^{n−1} = P^{n−1}, q. e. d .

3 Levi-Civita theorem

Let g and ḡ be smooth metrics on a manifold M^{n}. Denote by ρ^{1}, ..., ρ^{m} (1 ≤ m ≤ n) the common eigenvalues of the metrics g and ḡ. Suppose the functions ρ^{1}, ..., ρ^{m} are differ at every point of an open domain D ⊂ M^{n}. In the paper [4], Levi-Civita proved that for every point P ∈ D there is an open neighborhood U(P) ⊂ D and a coordinate system x̄ = (x̄_{1}, ..., x̄_{m}) (in U(P)), where x̄_{i} = (x_{i}^{1}, ..., x_{i}^{k_{i}}), (1 ≤ i ≤ m), such that the quadratic forms of the metrics g and ḡ have the following form:

g( ˙x̄, ˙ x̄) = Π_{1}(x̄)A_{1}(x̄_{1}, ˙ x̄_{1}) + Π_{2}(x̄)A_{2}(x̄_{2}, ˙ x̄_{2}) + ⋯ +
+ Π_{m}(x̄)A_{m}(x̄_{m}, ˙ x̄_{m}),  (3)
ḡ( ˙x̄, ˙ x̄) = ρ^{1}Π_{1}(x̄)A_{1}(x̄_{1}, ˙ x̄_{1}) + ρ^{2}Π_{2}(x̄)A_{2}(x̄_{2}, ˙ x̄_{2}) + ⋯ +
+ ρ^{m}Π_{m}(x̄)A_{m}(x̄_{m}, ˙ x̄_{m}),  (4)

where A_{i}(x̄_{i}, ˙ x̄_{i}) are positive-definite quadratic forms in the velocities ˙ x̄_{i} with coefficients depending on x̄_{i},

Π_{i} =^{def} (ϕ_{i} − ϕ_{1}) ⋯ (ϕ_{i} − ϕ_{i−1})(ϕ_{i+1} − ϕ_{i}) ⋯ (ϕ_{m} − ϕ_{i})  (5)

and ϕ_{1}, ϕ_{2}, ..., ϕ_{m}, 0 < ϕ_{1} < ϕ_{2} < ... < ϕ_{m}, are smooth functions such that

ϕ_{i} = \{ \begin{matrix}ϕ_{i}(x̄_{i}), if k_{i} = 1 \\ constant, else. \end{matrix}

It is easy to see that the functions ρ^{i} as functions of ϕ_{i} and the function ϕ_{i} as functions of ρ^{i} are given by

ρ^{i} = \frac{1}{ϕ_{1}...ϕ_{m}} \frac{1}{ϕ_{i}}
ϕ_{i} = \frac{1}{ρ_{i}} (ρ_{1}ρ_{2}...ρ_{m})^{\frac{1}{m+1}}

Definition 3. Let metrics g and ḡ be given by formulae (3) and (4) in a coordinate chart U. Then we say that the metrics g and ḡ have Levi-Civita local form (of type m), and the coordinate chart U is Levi-Civita coordinate chart (with respect to the metrics).

Levi-Civita proved that the metrics g and ḡ given by formulae (3) and (4) are geodesically equivalent. If we replace ϕ_{i} by ϕ_{i} + c, i = 1, ..., m, where c is a (positive for simplicity) constant, in (3) and (4), we obtain the following oneparameter family of metrics, geodesically equivalent to g:

g_{c}( ˙x̄, ˙ x̄) = \frac{1}{(ϕ_{1} + c) ⋯ (ϕ_{m} + c)} \{\frac{1}{ϕ_{1} + c}Π_{1}A_{1} + ⋯ + \frac{1}{ϕ_{m} + c}Π_{m}A_{m}\} . (6)

The next theorem is essentially due to Painlevé, see [4].

Theorem 4. If the metrics g and ḡ are geodesically equivalent, then the function

I_{0} =^{def} (\frac{det(g)}{det(ḡ)})^{\frac{2}{n+1}} ḡ( ˙x̄, ˙ x̄),  (7)

is an integral of the geodesic flow of the metric g.

Substituting g_{c} instead of ḡ in (7), we obtain the following one-parameter family of integrals

I_{c} =^{def} (\frac{det(g)}{det(g_{c})})^{\frac{2}{n+1}} g_{c}( ˙x̄, ˙ x̄) =
= C[(ϕ_{1} + c) ⋯ (ϕ_{m} + c)] \{\frac{1}{ϕ_{1} + c}Π_{1}A_{1} + ⋯ + \frac{1}{ϕ_{m} + c}Π_{m}A_{m}\}
= C{L_{1}c^{m−1} + L_{2}c^{m−2} + ⋯ + L_{m}},

where

L_{1} = Π_{1}A_{1} + ⋯ + Π_{m}A_{m} — the twice energy integral,
L_{2} = σ_{1}(ϕ_{2}, ..., ϕ_{m})Π_{1}A_{1} + ⋯ + σ_{1}(ϕ_{1}, ..., ϕ_{m−1})Π_{m}A_{m},
L_{3} = σ_{2}(ϕ_{2}, ..., ϕ_{m})Π_{1}A_{1} + ⋯ + σ_{2}(ϕ_{1}, ..., ϕ_{m−1})Π_{m}A_{m},
⋮
L_{m} = (ϕ_{2}...ϕ_{m})Π_{1}A_{1} + ⋯ + (ϕ_{1}...ϕ_{m−1})Π_{m}A_{m},

σ_{k} denotes the elementary symmetric polynomial of degree k, and C =^{def} [(ϕ_{1} + c)^{k_{1}−1} ⋯ (ϕ_{m} + c)^{k_{m}−1}]^{\frac{2}{n+1}} is a constant. Therefore the functions L_{k}, k = 1, ..., m, are integrals of the geodesic flows of the metric g. We call these integrals Levi-Civita integrals.

From the results of [8] it follows that Levi-Civita integrals are in involution. More precisely, let D = (d_{j}^{i}) be an m × m matrix. Suppose that for any i, j the element d_{j}^{i} depends only on the variables x̄_{j}. Denote by Δ the determinant of the matrix D and by Δ_{j}^{i} the minor of the element d_{j}^{i}. In the paper [8] it was shown that, for arbitrary functions A_{i}(x̄_{i}, ˙ x̄_{i}), quadratic in velocities ˙ x̄_{i}, the Lagrangian system with Lagrangian

T_{1} = Δ (\frac{A_{1}(x̄_{1}, ˙ x̄_{1})}{Δ_{1}^{1}} + \frac{A_{2}(x̄_{2}, ˙ x̄_{2})}{Δ_{2}^{1}} + ... + \frac{A_{m}(x̄_{m}, ˙ x̄_{m})}{Δ_{m}^{1}})

admits (m − 1) integrals

T_{i} = Δ (A_{1}(x̄_{1}, ˙ x̄_{1})\frac{Δ_{1}^{i}}{(Δ_{1}^{1})^{2}} + A_{2}(x̄_{2}, ˙ x̄_{2})\frac{Δ_{2}^{i}}{(Δ_{2}^{1})^{2}} + ... + A_{m}(x̄_{m}, ˙ x̄_{m})\frac{Δ_{m}^{i}}{(Δ_{m}^{1})^{2}} ) ,

where i = 2, ..., m, and if we identify the tangent and cotangent bundles the Lagrangian T_{1} and consider the standart symplectic form on the cotangent bundle, then the integrals are in involution.

If we take d_{j}^{i} = (ϕ_{j})^{m−i}, then Δ and Δ_{j}^{i} are given by

Δ_{j}^{i} = (−1)^{m−1}σ^{i−1}(ϕ_{1}, ϕ_{2}, ..., ϕ_{j−1}, ϕ_{j+1}, ..., ϕ_{m}) ∏_{α>β≥1,α≠j,β≠j}(ϕ_{α} − ϕ_{β}),

Δ = (−1)^{m} ∏_{α>β≥1}(ϕ_{α} − ϕ_{β}).

Therefore,

\frac{ΔΔ_{j}^{i}}{(Δ_{j}^{1})^{2}} = σ^{i−1}(ϕ_{1}, ϕ_{2}, ..., ϕ_{j−1}, ϕ_{j+1}, ..., ϕ_{m})Π_{j},

so T_{i} = −L_{i} and thus the integrals L_{i} are in involution, q. e. d.

4 Geodesic equivalence and corresponding integrals

Let metrics g and ḡ on a manifold M (of dimension n) be geodesically equivalent. By definition, put

U_{g}^{r}M =^{def} {(x, ξ) ∈ T M : ||ξ||_{g} = r} ,

where x ∈ M, ξ ∈ T_{x}M and ||ξ||_{g} =^{def} \sqrt{g(ξ, ξ)} = \sqrt{g_{ij}ξ^{i}ξ^{j}} is the norm of the

vector ξ in the metric g.

By the geodesic flow of the metric g we mean the Lagrangian system of differential equations \frac{d}{dt} (\frac{∂L}{∂ẋ}) − \frac{∂L}{∂x} = 0 on T M with Lagrangian L =^{def} \frac{1}{2}g_{ij}ẋ^{i}ẋ^{j}. Because of the Legendre transformation, the geodesic flow could be considered as a Hamiltonian system on T M (as a symplectic form we take ω_{g} =^{def} d[g_{ij}ξ^{j}dx^{i}]) with the Hamiltonian H_{g} =^{def} \frac{1}{2}g_{ij}ξ^{i}ξ^{j}.

Since the metrics g, ḡ are geodesically equivalent, the mapping Φ : T M → T M, Φ(x, ξ) = (x, ξ \frac{||ξ||_{g}}{||ξ||ḡ} ), takes the trajectories of the geodesic flow of the metric g to the trajectories of the geodesic flow of the metric ḡ. This mapping is a diffeomorphism (for r ≠ 0), takes U_{g}^{r}M to U r ; ḡM and is trajectorial on U_{g}^{r}M. Obviously the surfaces U_{g}^{r}, U r ; ḡ are regular isoenergy surfaces \{H_{g} = \frac{r}{2}\}, \{Hḡ = \frac{r}{2}\}.

By Theorem 3, in order to obtain a family of first integrals we have to find the polynomial Δ^{n}(t) and divide it by (t−a). In our case H_{g} = Hḡ∘Φ. Therefore the function a from Theorem 3 equals to \frac{||ξ||ḡ}{||ξ||_{g}}. In coordinates we have

ω_{g} = d[g_{ij}ξ^{j}dx^{i}]

and

ωḡ = d[ḡ_{ij}ξ^{j}dx^{i}].

Therefore,

Φ^{∗}ωḡ = d [\frac{||ξ||_{g}}{||ξ||ḡ} ḡ_{ij}ξ^{j}dx^{i}] =
= \frac{∂}{∂x^{k}} [\frac{||ξ||_{g}}{||ξ||ḡ} ḡ_{ij}ξ^{j}] dx^{k} ∧ dx^{i} −
− \frac{∂}{∂ξ^{k}} [\frac{||ξ||_{g}}{||ξ||ḡ} ḡ_{ij}ξ^{j}] dx^{i} ∧ dξ^{k}.

It is easy to see that at a point ξ ∈ T_{x}M the quantities

A_{ik} =^{def} −\frac{∂}{∂ξ^{k}} [\frac{||ξ||_{g}}{||ξ||ḡ} ḡ_{ij}ξ^{j}]

form an element of T_{x}M ⊗ T_{x}M. Without loss of generality we can assume that in the space T_{x}M the metrics g and ḡ are given in principal axes. Then

A_{ij} =^{def} −ρ^{i}(x)\frac{∂}{∂ξ^{j}} (ξ^{i} \begin{matrix}√ & \overline{ξ^{1}^{2} + ... + ξ^{n}^{2}} \\ √ & \overline{ρ^{1}ξ^{1}^{2} + ... + ρ^{n}ξ^{n}^{2}} \end{matrix} ) =

= ρ^{i}δ_{j}^{i} \frac{||ξ||_{g}}{||ξ||ḡ} − ρ^{i}ξ^{i} ( \frac{\frac{||ξ||ḡ}{||ξ||_{g}} − ρ^{j} \frac{||ξ||_{g}}{||ξ||ḡ}}{||ξ||2 ; ḡ} ξ^{j}) =
= diag(μ_{1}, ..., μ_{n}) − A ⊗ B.

Here ρ^{i}, i = 1, ..., n are common eigenvalues (here we allow ρ^{i} to be equal to ρ^{j} for some i, j) of the metrics g and ḡ, μ_{i} =^{def} −ρ^{i} \frac{||ξ||_{g}}{||ξ||ḡ}, A_{i} =^{def} ρ^{i}ξ^{i} and

B_{i} =^{def} \frac{\frac{||ξ||ḡ}{||ξ||_{g}} − ρ^{i} \frac{||ξ||_{g}}{||ξ||ḡ}}{||ξ||2 ; ḡ} ξ^{i}.

We have

det(Φ^{∗}ωḡ − tω_{g}) = det | \begin{matrix}(∗) & (A_{ij} + tδ_{ij}) \\ −(A_{ij} + tδ_{ij}) & 0 \end{matrix} |
= det(A_{ij} + tδ_{ij})^{2}.

Therefore,

Δ^{n}(t) = det (diag(t + μ_{1}, ..., t + μ_{n}) − a ⊗ b) .  (8)

Lemma 3. The following relation holds:

Δ^{n}(t) = (t + μ_{1}) ⋯ (t + μ_{n}) − (a_{1}b_{1})(t + μ_{2}) ⋯ (t + μ_{n}) − …
− (t + μ_{1}) ⋯ (t + μ_{n−1})(a_{n}b_{n}).  (9)

The lemma follows from induction considerations.

To divide the polynomial by (t−a) we shall use the Horner scheme. Suppose that Δ^{n}(t) = t^{n} + a_{n−1}t^{n−1} + ⋯ + a_{0} and δ^{n−1}(t) = t^{n−1} + b_{n−2}t^{n−2} + ⋯ + b_{0}. Then we have

b_{n−1} = a_{n} = 1,  (10)
b_{n−2} = a_{n−1} + a,  (11)
⋯
b_{k} = a_{k+1} + ab_{k+1},  (12)
⋯
0 = a_{0} + ab_{0}.  (13)

It follows from lemma 3 that

a_{0} = (μ_{1}...μ_{n}) − (A_{1}B_{1})(μ_{2}...μ_{n}) − ⋯ − (μ_{1}...μ_{n−1})A_{n}B_{n} =
= (−1)^{n} (\frac{||ξ||_{g}}{||ξ||ḡ} )^{n} (ρ^{1} ⋯ ρ^{n}).

Combining with (13) we get

b_{0} = −\frac{a_{0}}{a} = (−1)^{n+1} (\frac{||ξ||_{g}}{||ξ||ḡ} )^{n+1} (ρ^{1} ⋯ ρ^{n}).

Since \frac{1}{2}g_{ij}ξ^{i}ξ^{j} is an integral of the geodesic flow of the metric g, the function

I_{0} =^{def} (ρ^{1} ⋯ ρ^{n})^{− \frac{2}{n+1}} ḡ(ξ, ξ)  (14)

is also an integral of the geodesic flow of the metric g. Using Lemma 3 we have

a_{n−1} = (μ_{1} + ... + μ_{n}) − (A_{1}B_{1} + ... + A_{n}B_{n}) =
= \frac{||ξ||_{g}}{||ξ||3 ; ḡ} \{(ρ^{1}^{2}ξ^{1}^{2} + ... + ρ^{n}^{2}ξ^{n}^{2})−
− (ρ^{1} + ... + ρ^{n})(ρ^{1}ξ^{1}^{2} + ... + ρ^{n}ξ^{n}^{2})\} − \frac{||ξ||ḡ}{||ξ||_{g}} .

Using (11) we get

b_{n−2} = a_{n−2} + a =
= \frac{||ξ||_{g}}{||ξ||3 ; ḡ} \{(ρ^{1}^{2}ξ^{1}^{2} + ... + ρ^{n}^{2}ξ^{n}^{2})−
− (ρ^{1} + ... + ρ^{n})(ρ^{1}ξ^{1}^{2} + ... + ρ^{n}ξ^{n}^{2})}

Therefore, the function

I_{1} =^{def} (ρ^{1} ⋯ ρ^{n})^{− \frac{3}{n+1}} \{(ρ^{1}^{2}ξ^{1}^{2} + ... + ρ^{n}^{2}ξ^{n}^{2})−
− (ρ^{1} + ... + ρ^{n})(ρ^{1}ξ^{1}^{2} + ... + ρ^{n}ξ^{n}^{2})}

is an integral. (It is easy to see that \frac{||ξ||_{g}^{2}}{||ξ||2 ; ḡ} = (ρ^{1} ⋯ ρ^{n})^{− \frac{2}{n+1}} \frac{||ξ||_{g}^{2}}{I_{0}} .)

Arguing as above, we see that the functions

I_{k} =^{def} (ρ^{1} ⋯ ρ^{n})^{− \frac{k+2}{n+1}} \{(ρ^{1}^{k+1}ξ^{1}^{2} + ... + ρ^{n}^{k+1}ξ^{n}^{2})−
− (ρ^{1} + ... + ρ^{n})(ρ^{1}^{k}ξ^{1}^{2} + ... + ρ^{n}^{k}ξ^{n}^{2}) + ⋯
+ (−1)^{k}σ_{k}(ρ^{1}, ..., ρ^{n})(ρ^{1}ξ^{1}^{2} + ... + ρ^{n}ξ^{n}^{2})} ,

are integrals of the geodesic flow of the metric g, where by σ_{k} we denote the elementary symmetric polynomial of degree k. It is obvious that (−1)^{k}σ_{k} =

c_{k} from Theorem 1, and therefore I_{k} = (\frac{det(g)}{det(ḡ)})^{\frac{k+2}{n+1}} ḡ(S_{k}ξ, ξ). Thus I_{k}, k = 0, ..., n − 1, are integrals of the geodesic flow of the metric g, q. e. d.

5 Liouville integrability

The last step of the proof of Theorem 1 is to verify that the integrals I_{0}, ..., I_{n−1} are in involution. We proceed along the following plan. First we show that it is sufficient to prove the involutivity in each Levi-Civita chart. Then we prove that in each Levi-Civita chart the integrals I_{0}, ..., I_{n−1} are linear combinations of Levi-Civita integrals, and therefore commute.

Let g, ḡ be metrics on M. A point x ∈ M is called stable, if in a neighborhood of x the number of different eigenvalues of the metrics g, ḡ does not depend of a point.

Denote by M the set of stable points of M. The set M is an open subset of M. Obviously

M = ⨆_{1≤q≤n} M^{q},  (15)

where M^{q} denotes the set of stable points whose number of distinct common eigenvalues equals q. Points x ∈ M \ M are called points of bifurcation.

Lemma 4. The set M is everywhere dense in M.

Proof of Lemma 4. Denote by N(x) the number of distinct common eigenvalues of the metrics g, ĝ at a point x. Recall that the common eigenvalues of the metrics g, ĝ at a point x ∈ M are roots of the characteristic polynomial P_{x}(t) = det (G − tE)_{|x}, where G = (g^{iα}ḡ_{αj}). In particular, all roots of P_{x}(t) are real.

Let us prove that, for a sufficiently small neighborhood of an arbitrary point x ∈ M, for any y from the neighborhood the number N(x) is no less than N(y). Take a small ϵ > 0 and an arbitrary root ρ of P_{x}(t). Let us prove that for a sufficiently small neighborhood U(x) ⊂ M, for any y ∈ U(x) there is a root ρ_{y}, ρ − ϵ < ρ_{y} < ρ + ϵ, of the polynomial P_{y}(t). If ϵ is small, then for a sufficiently small neighborhood U(x) of the point x, for any y ∈ U(x) the numbers ρ+ϵ and ρ − ϵ are not roots of P_{y}(t). Consider the circle S_{ϵ} =^{def} {z ∈ C : |z − ρ| = ϵ} on the complex plane C. Clearly the number of roots (with multiplicities) of the polynomial P_{y} inside the circle is equal to

\frac{1}{2πi} ∫_{S_{ϵ}} \frac{P′_{y}(z)}{P_{y}(z)}dz.

Since for any y ∈ U(x) there are no roots of P_{y} on the circle S_{ϵ}, then the function

\frac{1}{2πi} ∫_{S_{ϵ}} \frac{P′_{y}(z)}{P_{y}(z)}dz

continuously depends on y ∈ U(x), and therefore is a constant. Clearly it is positive. Thus for any y ∈ U(x) there is at least one root of P_{y} that lies between ρ + ϵ and ρ − ϵ. Then for any y from a sufficiently small neighborhood of x we have N(y) ≥ N(x).

Now let us prove the lemma. Evidently the set M is an open subset of M. Then it is sufficient to prove that for any open subset U ⊂ M there is a stable point x ∈ U. Suppose otherwise, i.e. let all the points of U be points of bifurcation. Take a point y ∈ M with maximal value of the function N on it. We have that in a neighborhood U(y) of the point y the function N is constant and equals N(y). Then the point y is a stable point, and we get a contradiction, q. e. d.

Now let the metrics g, ḡ be geodesically equivalent. Since the set of points of bifurcation is nowhere dense, it is sufficient to prove the involutivity in each Levi-Civita chart. Let the metrics g and ḡ be given by

g( ˙x̄, ˙ x̄) = Π_{1}(x̄)A_{1}(x̄_{1}, ˙ x̄_{1}) + Π_{2}(x̄)A_{2}(x̄_{2}, ˙ x̄_{2}) + ⋯ +
+ Π_{m}(x̄)A_{m}(x̄_{m}, ˙ x̄_{m}),  (16)
ḡ( ˙x̄, ˙ x̄) = ρ^{1}Π_{1}(x̄)A_{1}(x̄_{1}, ˙ x̄_{1}) + ρ^{2}Π_{2}(x̄)A_{2}(x̄_{2}, ˙ x̄_{2}) + ⋯
+ ρ^{m}Π_{m}(x̄)A_{m}(x̄_{m}, ˙ x̄_{m}).  (17)

We show that the integrals I_{k} are linear combinations of the Levi-Civita integrals. We have

Ḡ = diag(\underbrace{ρ_{1}, ..., ρ_{1}}_{k_{1}}, ..., \underbrace{ρ_{m}, ..., ρ_{m}}_{k_{m}}),  (18)

where ρ_{k} = \frac{1}{(ϕ_{1}...ϕ_{m})} \frac{1}{ϕ_{k}}. It is easy to check that

S_{k} = (−1)^{k}diag(\underbrace{σ_{k}^{1}, ..., σ_{k}^{1}}_{k_{1}}, ..., \underbrace{σ_{k}^{m}, ..., σ_{k}^{m}}_{k_{m}}),  (19)

where

σ_{k}^{l} =^{def} σ_{k}(\underbrace{ρ_{1}, ..., ρ_{1}}_{k_{1}}, ..., \underbrace{ρ_{l}, ..., ρ_{l}}_{k_{l}−1}, ..., \underbrace{ρ_{m}, ..., ρ_{m}}_{k_{m}}).  (20)

We have

σ_{k}^{1} = \frac{1}{(ϕ_{1}...ϕ_{m})^{k}} σ_{k}(\underbrace{\frac{1}{ϕ_{1}} , ..., \frac{1}{ϕ_{1}}}_{k_{1}−1}, ..., \underbrace{\frac{1}{ϕ_{m}} , ..., \frac{1}{ϕ_{m}}}_{k_{m}}) =  (21)

= \frac{1}{(ϕ_{1}...ϕ_{m})^{k}} ∑_{|α|=k} \binom{k_{1} − 1}{α_{1}}\binom{k_{2}}{α_{2}} ⋯ \binom{k_{m}}{α_{m}} \frac{1}{ϕ_{1}^{α_{1}}} \frac{1}{ϕ_{2}^{α_{2}}} ⋯ \frac{1}{ϕ_{m}^{α_{m}}} , (22)

(23)

where |α| =^{def} α_{1} + ⋯ + α_{m} and α_{i} ≥ 0. Substituting \binom{k_{l}−1}{α_{l}} + \binom{k_{l}−1}{α_{l}−1} for \binom{k_{l}}{α_{l}}

(we assume that \binom{k}{0} = 1, \binom{k}{−1} = 0, k ≥ 0) for 2 ≤ l ≤ m we obtain

σ_{k}^{1} = \frac{1}{(ϕ_{1}...ϕ_{m})^{k}} (B_{k} + B_{k−1}σ_{1}(\frac{1}{ϕ_{2}} , ..., \frac{1}{ϕ_{m}} ) + ⋯ +
+ B_{k−m+1}σ_{m−1}(\frac{1}{ϕ_{2}} , ..., \frac{1}{ϕ_{m}} )) ,

where

B_{k} =^{def} ∑_{|α|=k} \binom{k_{1} − 1}{α_{1}} ⋯ \binom{k_{m} − 1}{α_{m}} \frac{1}{ϕ_{1}^{α_{1}}} ⋯ \frac{1}{ϕ_{m}^{α_{m}}} .  (24)

Note that

(\frac{det(g)}{det(ḡ)})^{\frac{k+2}{n+1}} = C_{k}(ϕ_{1}...ϕ_{m})^{k+2},  (25)

where C_{k} = [ϕ_{1}^{k_{1}−1}...ϕ_{m}^{k_{m}−1}]^{\frac{k+2}{n+1}} . Therefore,

I_{k} =^{def} (\frac{det(g)}{det(ḡ)})^{\frac{k+2}{n+1}} ḡ(S_{k} ˙x̄, ˙ x̄) =
= (−1)^{k}C_{k}(ϕ_{1}...ϕ_{m})^{k+2} {ρ^{1}σ_{k}^{1}Π_{1}A_{1} + ⋯ + ρ^{m}σ_{k}^{m}Π_{m}A_{m}} =
= (−1)^{k}C_{k}(ϕ_{1}...ϕ_{m})^{k+2} \{\frac{1}{ϕ_{1}...ϕ_{m}} \frac{1}{ϕ_{1}} \{\frac{1}{(ϕ_{1}...ϕ_{m})^{k}} (B_{k}+
+ ⋯ + B_{k−m+1}σ_{m−1}(\frac{1}{ϕ_{2}} , ..., \frac{1}{ϕ_{m}} ))\} Π_{1}A_{1} + ⋯\} =
= (−1)^{k}C_{k} {B_{k}L_{m} + B_{k−1}L_{m−1} + ⋯ + B_{k−m+1}L_{1}} ,  (26)

where L_{i} are Levi-Civita integrals.

Finally, since the integrals I_{0}, ..., I_{n−1} are linear combinations of Levi-Civita integrals with constant coefficients, and since Levi-Civita integrals commute, then the integrals I_{0}, ..., I_{n−1} also commute, q. e. d.

Remark 2. Let m be the number of distinct common eigenvalues of geodesically equivalent metrics g, ḡ at a point x. Then in a neighborhood U of the point x the number of functionally independent almost everywhere Levi-Civita integrals is no less than m. Therefore the dimension of the space generated by the differentials (dI_{0}, dI_{1}, ..., dI_{n−1}) no less than m in almost all points of T U.

6 Topological obstructions

Corollary 1 follows immediately from the following theorem. Recall that a group G is almost commutative, if there exists a commutative subgroup P ⊂ G of finite index.

Theorem 5 (Taimanov, [10]). If a real-analytic closed manifold M^{n} with a real-analytic metric satisfies at least one of the conditions:

a) π_{1}(M^{n}) is not almost commutative

b) dimH_{1}(M^{n}; Q) > dimM^{n},

then the geodesic flow on M^{n} is not analytically integrable.

If metrics g, ḡ are real-analytic and geodesically equivalent, then the integrals I_{0}, ..., I_{n−1} are also real-analytic. If the metrics are strictly non-proportional at least at one point of M^{n}, then the integrals are functionally independent almost everywhere in a neighborhood of that point. Since the integrals are realanalytic, then they are functionally independent almost everywhere and we can apply Theorem 5, q. e. d.

Proof of Corollary 2. Let metrics g, ḡ on a surface M^{2} be geodesically equiv-

alent. Using Theorem 1 we have that the function I_{0} = (\frac{det(g)}{det(ḡ)})^{\frac{2}{n+1}} ḡ(ξ, ξ) is an integral of the geodesic flow of the metric g. In one direction Corollary 2 is proved. In other direction the statement of Corollary 2 can be verified by direct calculation, and it was done in [12].

Proof of Corollaries 3, 4, 5, 6. Let g be a metric on a surface M^{2}. The following lemma is essentially due to [1], see also [5]. For simplicity assume that the surface M^{2} is oriented, otherwise finitely cover the surface by an oriented one. Consider the complex structure on M^{2} corresponding to the metric g. Let z be a complex coordinate in a open domain U ⊂ M^{2}. Consider the complex momentum p. We shall denote by z̄ and p̄ the complex conjugation of z and p respectively. In complex variables the Hamiltonian H : T^{∗}M^{2} → R of the geodesic flow of the metric g reads \frac{pp̄}{λ(z)}, where λ is a real-valued function. Suppose that the real-valued function

F = A(z)p^{2} + B(z)pp̄ + Ā(z)p̄^{2}

is an integral of the geodesic flow of the metric g.

Lemma 5. The form \frac{1}{A(z)}dzdz is meromorphic.

Remark 3. If the Hamiltonian and the integral are proportional at each point of M^{2}, i.e. if F ≡ α(z)H, where α : M^{2} → R, then by definition put \frac{1}{A(z)}dzdz equal zero.

Proof of Lemma 5. Since F is an integral of the Hamiltonian system with the Hamiltonian H, the Poisson bracket {H, F } equals zero. We have

{H, F } = H_{p}F_{z} − H_{z}F_{p} + Hp̄Fz̄ − Hz̄Fp̄ = 0  (27)

On the right side of (27) each term is a polynomial of third degree in momenta. Then the bracket is also a polynomial of third degree in momenta. In order for a polynomial to equal zero, all coefficients must be zero, in particular the coefficient of p^{3}. Thus \frac{Az̄}{λ} equals zero, and A is holomorphic. Then \frac{1}{A(z)} is meromorphic, q. e. d.

Let g, ḡ be geodesically equivalent metrics on a closed surface M^{2} of Euler

characteristic χ(M^{2}). Then the function I_{0} = (\frac{det(g)}{det(ḡ)})^{\frac{2}{n+1}} ḡ(ξ, ξ) is an integral of the geodesic flow of the metric g, and is quadratic in momenta (if we identify with the help of the metric g the tangent and cotangent bundles of M^{2}). Consider the form \frac{1}{A(z)}dzdz corresponding to the integral I_{0}. Suppose that the form is not identical zero. For a meromorphic 2-form on a closed Riemann surface, the number of poles P minus the number of zeros Z is equal to twice the Euler characteristic. It is easy to see that the form \frac{1}{A(z)}dzdz has no zeros (otherwise the metric ḡ has singularities). Then P = 2χ(M^{2}), and the Euler characteristic χ(M^{2}) can not be negative, q. e. d. Now assume the metrics are proportional at each point of an open subset U ⊂ M^{2}. Since the form is meromorphic, it must be zero. Thus ḡ = α(z)g, where α is a function on M^{2}. Let us show that the function α is constant. Actually, I_{0} = 2 (\frac{1}{α})^{\frac{1}{3}} H (here we identify T^{∗}M and T M with the help of the metric g). We have

\{H, I_{0}\} = \{H, 2 (\frac{1}{α})^{\frac{1}{3}} H\} = \{H, H\}2 (\frac{1}{α})^{\frac{1}{3}} + 2H\{(\frac{1}{α})^{\frac{1}{3}} , H\}.

Since \{H, H\} equals zero, we have that \{(\frac{1}{α})^{\frac{1}{3}} , H\} equals zero and the function α is constant, q. e. d.

Remark 4. For non-orientable surfaces the sign of the Euler characteristic coincides with the sign of the Euler characteristic of the oriented covering. Therefore Corollary 3 is true also for non-orientable surfaces.

It is easy to see that the form \frac{1}{A(z)}dzdz has poles precisely at points, where the metrics are proportional. If the surface M^{2} is the torus, then χ(M^{2}) = 0 and either the metrics g, ḡ are proportional at every point, or there are no points of proportionality of the metrics, q. e. d.

The following lemma is essentially due to Kolokol’tzov [5]. It completes the proof of Corollary 5.

Lemma 6. On the sphere S^{2} there are the following three possibilities for the form \frac{1}{A(z)}dzdz.

1. The form \frac{1}{A(z)}dzdz is identical zero.

2. The form \frac{1}{A(z)}dzdz has exactly two zeros (both zeros are of multiplicity two).

3. The form \frac{1}{A(z)}dzdz has exactly four zeros.

In the second case the metric g admits a non-trivial Killing vector field.

Proof of Corollary 7. Because of Noether’s theorem, if a metric admits a (non-trivial) Killing vector field, then the geodesic flow of the metric admits a (non-trivial) integral, linear in velocities, and vice versa.

Suppose the function

F_{1} = ∑_{i=1}^{n} a_{i}(x)ξ^{i}

is constant on the trajectories of the geodesic flow of the metric ḡ. Then the function

Φ^{∗}F_{1} = \frac{||ξ||_{g}}{||ξ||ḡ} ∑_{i=1}^{n} a_{i}(x)ξ^{i}

is constant on the trajectories of the geodesic flow of the metric g. Since the

function I_{0} = (\frac{det(g)}{det(ḡ)} ^{\frac{2}{n+1}} ) ḡ(ξ, ξ) is an integral of the geodesic flow of the metric

g, and since the function ||ξ||_{g} = \sqrt{g(ξ, ξ)} is also an integral of the geodesic flow of the metric g, then the function

\begin{matrix}√ & \overline{G(ξ, ξ)} \\ √ & \overline{I_{0}} \end{matrix}Φ^{∗}F_{1} = (\frac{det(g)}{det(ḡ)})^{\frac{1}{n+1}} ∑_{i=1}^{n} a_{i}(x)ξ^{i},

linear in velocities, is also an integral of the geodesic flow of the metric g, q. e. d.

7 Geodesically equivalent metrics on the ellipsoid.

Proof of Theorem 2. We show that in the elliptic coordinate system the restriction of the metrics

ds^{2} =^{def} ∑_{i=1}^{n}(dx^{i})^{2} and dr^{2} =^{def} \frac{1}{∑{n \atop i=1} (\frac{x^{i}}{a_{i}} )2} (∑_{i=1}^{n} \frac{(dx^{i})^{2}}{a_{i}} )

to the ellipsoid ∑_{i=1}^{n} \frac{(x^{i})^{2}}{a_{i}} = 1 have Levi-Civita local form, and therefore are geodesically equivalent. More precisely, consider elliptic coordinates ν^{1}, ..., ν^{n}. Without loss of generality we can assume that a^{1} < a^{2} < ... < a^{n}. Then the relation between the elliptic coordinates ν̄ and the Cartesian coordinates x̄ is given by

x^{i} = \sqrt{\frac{∏_{j=1}^{n}(a^{i} − ν^{j})}{∏_{j=1,j≠i}^{n}(a^{i} − a^{j})}}.  (28)

Recall that the elliptic coordinates are non-degenerate almost everywhere, and the set

{ν^{1} = 0, a_{1} < ν^{2} < a_{2}, a_{2} < ν^{3} < a_{3}, ..., a_{n−1} < ν^{n} < a^{n}}

is the part of the ellipsoid, lieing in the quadrant {x^{1} > 0, x^{2} > 0, ..., x^{n} > 0}. Since for any i the symmetry x^{i} → −x^{i} takes the ellipsoid to the ellipsoid and preserves the metrics ds^{2} and dr^{2}, it is sufficent to check the statement of the theorem only in the quadrant {x^{1} > 0, x^{2} > 0, ..., x^{n} > 0}.

In the elliptic coordinates the restriction of the metric ds^{2} to the ellipsoid has the following form

∑_{i=1}^{n} Π_{i}A_{i}(dν^{i})^{2},  (29)

where Π_{i} =^{def} ∏_{j=1,j≠i}^{n}(ν^{i} − ν^{j}), and A_{i} =^{def} \frac{ν^{i}}{∏_{j=1}^{n}(a^{i}−ν^{j})}. The restriction of the

metric dr^{2} to the ellipsoid is

(a^{1}a^{2}...a^{n}) ∑_{i=1}^{n} ρ^{i}Π_{i}A_{i}(dν^{i})^{2},  (30)

where ρ^{i} =^{def} \frac{1}{ν^{i}(ν^{1}ν^{2}...ν^{n})}. We see that the metrics ds^{2}, dr^{2} have Levi-Civita local form, and therefore are geodesically equivalent, q. e. d.

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