SPECTRAL RIGIDITY OF A COMPACT

NEGATIVELY CURVED MANIFOLD

Christopher B. Croke, Vladimir A. Sharafutdinov

1. Introduction

In [1] V. Guillemin and D. Kazhdan introduced the following definition of spectral rigidity of a Riemannian manifold.

Let (M, g) be a compact boundaryless Riemannian manifold. A family g^{τ} of Riemannian metrics on M smoothly depending on the parameter τ ∈ [−ϵ, ϵ] is called the deformation of the metric g if g^{0} = g. A deformation is called trivial if there exists a one-parameter family of diffeomorphisms φ^{τ} : M → M such that φ^{0} = Id, and g^{τ} = (φ^{τ})^{∗}g^{0}. Given a deformation g^{τ} (−ϵ ≤ τ ≤ ϵ), let Δ^{τ} : C^{∞}(M) → C^{∞}(M) be the Laplace-Beltrami operator corresponding to the metric g^{τ}. The deformation is called isospectral if, for all −ϵ ≤ τ ≤ ϵ spectra of the operators Δ^{τ} and Δ^{0} coincide (counting multiplicities). A Riemannian manifold (M, g) is called spectrally rigid if it does not admit non-trivial isospectral deformations.

The main result of the present article is the following

Theorem 1.1. A compact negatively curved Riemannian manifold is spectrally rigid.

For two-dimensional manifolds, this result was obtained by V. Guillemin and D. Kazhdan [1]. The same authors proved this fact for n-dimensional manifolds [2] under a pointwise curvature pinching assumption. That result was later extended by Min-Oo [3] to the case where the curvature operator is negative definite. On the other hand, since [1] was published, a number of examples of isospectral deformations of compact manifolds have been given. The first such example was due to Gordon and Wilson [4]. Hence to rule out isospectral deformations there must be some extra assumption, such as curvature above. The examples of Vignéras in [5] show that even nonisometric surfaces of constant negative curvature can be isospectral. Thus the best rigidity one can hope for is: for a compact manifold of negative curvature the space of isospectral manifolds is finite. Theorem 1.1 can be considered an important step in that direction.

A compact Riemannian manifold is said to have a simple length spectrum if there do not exist two different closed geodesics such that the ratio of their lengths is a rational number. This is a generic condition.

Typeset by AMS-TEX

This work was partly supported by NSF grants #DMS-9505175 and #DMS-9626232 and, CRDF Grant #782.

Theorem 1.2. Let (M, g) be a compact negatively curved Riemannian manifold with simple length spectrum, and Δ : C^{∞}(M) → C^{∞}(M) be the corresponding Laplace-Beltrami operator. If real functions q_{1}, q_{2} ∈ C^{∞}(M) are such that the operators Δ + q_{1} and Δ + q_{2} have coincident spectra, then q_{1} ≡ q_{2}.

In [1,2] (as we will point out) Guillemin and Kazhdan show that Theorems 1.1 and 1.2 follow from Theorem 1.3 below, which is a generalization of Theorem 4 of [2] to the case of negatively curved manifolds.

In the statement below S^{m}τ′_{M} represents the symmetric m tensors, and d is the operator that symmetrizes the covariant derivative (see the next section for definitions). For f ∈ S^{m}τ′_{M} and ξ a vector we let ⟨f, ξ^{m}⟩ represent the action of f on the m-th tensor product of ξ with itself.

Theorem 1.3. Let (M, g) be a compact negatively curved Riemannian manifold. If a symmetric tensor field f ∈ C^{∞}(S^{m}τ′_{M}) is such that the integral

∫_{0}^{1}⟨f(γ(t)), γ̇^{m}(t)⟩ dt

is equal to zero for every closed geodesic γ : [0, 1] → M, then there exists a field v ∈ C^{∞}(S^{m−1}τ′_{M}) such that dv = f.

This result has three interesting special cases: namely for m = 0, 1, 2.

Corollary 1.4. Let M be a compact negatively curved manifold and f : M → R a C^{∞} function. If f integrates to zero along every closed geodesic then f must itself be zero. In particular a function is determined by its integral along closed geodesics.

Theorem 1.2 follows from this corollary (as in [1,2]) since in our case the spectrum determines the integral of the potential along each closed geodesic.

Note that the corollary is not true in positive curvature (even if closed geodesics are dense in unit tangent bundle) as the round sphere shows.

Corollary 1.5. Let M be a compact negatively curved manifold, ω a smooth one form (not a-priori closed) on M, and [α] a real 1-cohomology class. If for every closed geodesic γ the value of [α] on the homology class of γ is the same as the integral of ω around γ then ω is a closed form that represents [α].

To see this let α be a closed 1-form representing [α]. Then by Theorem 1.3 α−ω is exact, so the corollary follows.

Corollary 1.6. Let M be a compact negatively curved manifold and ω a smooth symmetric 2 form that integrates to 0 along every closed geodesic then ω = \frac{d}{dt}ϕ_{t}^{∗}g where ϕ_{t} is a smooth 1-parameter family of diffeomorphisms.

Theorem 1.3 gives us a one form α such that dα = ω (note d is not exterior derivative). Let V be the vector field dual to α, and ϕ_{t} be the 1-parameter group of diffeomorphisms corresponding to V/2. A standard computation yields the corollary. Theorem 1.1 follows from a similar argument as in [1,2].

The rest of the paper is devoted to the proof of Theorem 1.3. The proof is based on the ideas developed in [7] (also see [8]) where the corresponding theorem for convex domains with boundary and negative sectional curvature is proved. We pass from the convex with boundary case to the compact with no boundary case with the help of Livčic’s theorem [9,p. 566] in a way similar to [1,2].

2. Symmetric tensor fields on a compact negatively curved manifold

Given a manifold M, by τ_{M} = (T M, p, M) and τ′_{M} = (T′M, p′, M) we denote the tangent and cotangent vector bundles respectively. Points of the tangent space T M are denoted by pairs (x, ξ), where x ∈ M, ξ ∈ T_{x}M. Let

τ_{s}^{r}M = C ⊗ \underbrace{τ_{M} ⊗ ⋯ ⊗ τ_{m}}_{r} ⊗ \underbrace{τ′_{M} ⊗ ⋯ ⊗ τ′_{m}}_{s}  (2.1)

be the bundle of tensors of degree (r, s); here C is the trivial one-dimensional complex bundle, tensor products are taken over R. The canonical isomorphism τ_{s}^{r}M ⊗ τ_{s′}^{r′}M ≅ τ_{s+s′}^{r+r′}M allows us to introduce the tensor product, in fibers of the bundle τ_{∗}^{∗}M = ⊕_{r,s=0}^{∞}τ_{s}^{r}M, which turns the last bundle into the bundle of bigraded complex algebras. In such the way the section space C^{∞}(τ_{∗}^{∗}M) becomes the bigraded C^{∞}(M)-algebra that is called the algebra of tensor fields. Every permutation π of the set {1, …, r} (permutation π of the set {1, …, s}) defines an automorphism ρ^{π} (automorphism ρ_{π}) by the corresponding permutation of the first r (last s) factors in the right-hand side of (2.1). The automorphism ρ^{π} (ρ_{π}) is called the transposition of upper (lower) indices. Let S^{m}τ′_{M} be the subbundle of τ_{m}^{0}M consisting of tensors invariant with respect to all transpositions of lower indices. The canonical projection (symmetrization) σ : τ_{m}^{0}M → S^{m}τ′_{M} is defined by the equality σ = \frac{1}{m!} ∑_{π} ρ_{π}. The symmetric product is defined by the formula uv = σ(u ⊗ v). This product turns S^{∗}τ′_{M} = ⊕_{m=0}^{∞}S^{m}τ′_{M} into a bundle of commutative graded algebras. Therefore C^{∞}(S^{∗}τ′_{M}) is a commutative graded C^{∞}(M)-algebra.

We will also use the coordinate representation for tensor fields. Given a local coordinate system (x^{1}, …, x^{n}) with a domain U ⊂ M, a section u ∈ C^{∞}(τ_{s}^{r}M; U) of the bundle τ_{s}^{r}M over U can be uniquely represented in the form

u = u_{j_{1}...j_{s}}^{i_{i}...i_{r}} \frac{∂}{∂x^{i_{1}}} ⊗ ⋯ ⊗ \frac{∂}{∂x^{i_{r}}} ⊗ dx^{j_{1}} ⊗ ⋯ ⊗ dx^{j_{s}}.  (2.2)

The coefficients u_{j_{1}...j_{s}}^{i_{i}...i_{r}} ∈ C^{∞}(U) of the representation are called the coordinates (or components) of the field u with respect to the given coordinate system. Assuming the choice of coordinate system to be clear from context (or arbitrary), we will abbreviate formula (2.2) as follows: u = (u_{j_{1}...j_{s}}^{i_{i}...i_{r}}).

Now let (M, g) be a Riemannian manifold. By

ΩM = {(x, ξ) ∈ T M | |ξ|^{2} = ⟨ξ, ξ⟩ = g_{ij}(x)ξ^{i}ξ^{j} = 1}

we denote the manifold of unit tangent vectors. The metric g establishes a canonical isomorphism of the bundles τ_{s}^{r}M ≅ τ_{0}^{r+s}M ≅ τ_{r+s}^{0}M which is expressed in coordinate form by the well-known operations of raising and lowering indices

u^{i_{i}...i_{m}} = g^{i_{1}j_{1}} … g^{i_{m}j_{m}}u_{j_{1}...j_{m}}, u_{i_{1}...i_{m}} = g_{i_{1}j_{1}} … g_{i_{m}j_{m}}u^{j_{1}...j_{m}}.

Therefore in the case of a Riemannian manifold one does not need to distinguish coand contravariant tensors, and can talk about co- and contravariant components of the same tensor. The scalar product is introduced in fibers of the bundle τ_{m}^{0}M by the formula ⟨u, v⟩ = u^{i_{1}...i_{m}} v̄_{i_{1}...i_{m}}; it is evidently independent of the choice of local coordinates. It turns S^{m}τ′_{M} into a Hermitian vector bundle and allows us to define the space L_{2}(S^{m}τ′_{M}) with the scalar product

(u, v)_{L_{2}(S^{m}τ′_{M})} = ∫_{M}⟨u(x), v(x)⟩ dV^{n}(x),  (2.3)

where dV^{n} is the Riemannian volume form on M. The lower index in (2.3) will sometimes be omitted. For u ∈ C^{∞}(S^{m}τ′_{M}), the equality

⟨u(x), ξ^{m}⟩ = u_{i_{1}...i_{m}}(x)ξ^{i_{1}} … ξ^{i_{m}}

is valid which we will make use of abbreviation various formulas.

Given a Riemannian manifold (M, g), let

∇ : C^{∞}(τ_{s}^{r}M) → C^{∞}(τ_{s+1}^{r}M), ∇ : (u_{j_{1}...j_{s}}^{i_{i}...i_{r}}) ↦ (∇_{k}u_{j_{1}...j_{s}}^{i_{i}...i_{r}})

be the Levi-Civita covariant derivative. Inner differentiation

d : C^{∞}(S^{m}τ′_{M}) → C^{∞}(S^{m+1}τ′_{M})

is defined by the equality d = σ∇. The divergence

δ : C^{∞}(S^{m+1}τ′_{M}) → C^{∞}(S^{m}τ′_{M})

is defined in coordinate form by the formula (δu)_{i_{1}...i_{m}} = g^{jk}∇_{j}u_{ki_{1}...i_{m}}. The operators d and −δ are adjoints with respect to the scalar product (2.3).

Given a complete Riemannian manifold (M, g), we will let G^{t} : T M → T M denote the geodesic flow, and H denote the vector field, on the manifold T M, generating the flow G^{t}. Like every vector field, H can be considered as a first order differential operator

H : C^{∞}(T M) → C^{∞}(T M).

Since the manifold ΩM is invariant under the geodesic flow, H can also be considered as a differential operator

H : C^{∞}(ΩM) → C^{∞}(ΩM).

Given a local coordinate system (x^{1}, …, x^{n}) in M with a domain U ⊂ M, the functions (x^{1}, …, x^{n}, ξ^{1}, …, ξ^{n}) constitute a local coordinate system in T M with domain p^{−1}(U), where p is the projection of the tangent bundle. (Strictly speaking, we should write x^{i} ∘ p above; nevertheless, we will use the abbreviated notation x^{i} instead of x^{i} ∘ p, hoping that it will not lead to misunderstanding.) These are the only coordinate systems on T M that are used in the present article. The vector field H in coordinate form is:

H = ξ^{i} \frac{∂}{∂x^{i}} − Γ_{jk}^{i}(x)ξ^{j}ξ^{k} \frac{∂}{∂ξ^{i}} ,

where Γ_{jk}^{i} are the Christoffel symbols. A direct calculation in coordinates shows that for a symmetric tensor field v ∈ C^{∞}(S^{m}τ′_{M}),

H (⟨v(x), ξ^{m}⟩) = ⟨(dv)(x), ξ^{m+1}⟩.  (2.4)

This shows how the differential operator H is related to the inner differentiation d.

Lemma 2.1. Let a complete Riemannian manifold (M, g) be such that there exists an orbit of the geodesic flow which is dense in ΩM. If a symmetric tensor field v ∈ C^{∞}(S^{m}τ′_{M}) satisfies the equation

dv = 0,  (2.5)

then (i) if m is odd, v is identically zero; (ii) if m = 2l is even, v is of the form v = cg^{l}, where c is a constant.

It is well-known [6] that in the case of a compact negatively curved Riemannian manifold almost all orbits of the geodesic flow on ΩM are dense in ΩM. Therefore the conclusion of Lemma 2.1 holds for such manifolds.

Proof of Lemma 2.1. Define the function φ ∈ C^{∞}(T M) by the equality φ(x, ξ) = ⟨v(x), ξ^{m}⟩. It follows from (2.4) and (2.5) that φ is constant on every orbit of the geodesic flow. Therefore, the restriction of φ to ΩM is constant. From this, taking the homogeneity of φ(x, ξ) in its second argument into account, we obtain

⟨v(x), ξ^{m}⟩ = c|ξ|^{m}.

This equality clearly implies the claim of the lemma.

Given a compact manifold M and an integer k ≥ 0, by H^{k}(S^{m}τ′_{M}) we denote the topological Hilbert space of sections of S^{m}τ′_{M} whose components have locally quadratically integrable derivatives up to order k with respect to any local coordinates. Let ∥ · ∥_{k} denote one of equivalent norms of the space.

Theorem 2.2. Let a compact Riemannian manifold (M, g) be such that there exists an orbit of the geodesic flow which is dense in ΩM, and let k ≥ 1 be an integer.

1. For even m, every symmetric tensor field f ∈ H^{k}(S^{m}τ′_{M}) can be uniquely represented in the form

f = dv + f̃,  (2.6)

where v ∈ H^{k+1}(S^{m−1}τ′_{M}), and the field f̃ ∈ H^{k}(S^{m}τ′_{M}) is solenoidal, i.e., satisfies the equation

δf̃ = 0.  (2.7)

These fields satisfy the estimates

∥v∥_{k+1} ≤ C∥δf∥_{k−1},  ∥f̃∥_{k} ≤ C∥f∥_{k}  (2.8)

with a constant C independent of f.

2. For odd m = 2l + 1, the previous claim is also valid under the additional assumption that v satisfies the relation

(v, g^{l})_{L_{2}(S^{2l}τ′_{M})} = 0.  (2.9)

In particular, in both 1 and 2 above if f is smooth then f̃ and v are also smooth.

The terms of decomposition (2.6) are called the potential and solenoidal parts of the symmetric tensor field f respectively.

Proof. Assume existence of symmetric tensor fields v and f̃ satisfying (2.6) and (2.7), and apply the operator δ to the first of the equalities to obtain

δdv = δf.  (2.10)

Conversely, if equation (2.10) has a solution satisfying the first of estimates (2.8), then, putting f̃ = f − dv, we would arrive at the claim of the theorem.

As can be easily shown [8, p. 89], the operator

δd : H^{k+1}(S^{m−1}τ′_{M}) → H^{k−1}(S^{m−1}τ′_{M})  (2.11)

is elliptic. Therefore, its kernel Ker(δd) is a finite-dimensional vector space consisting of smooth fields; the image Im(δd) is a closed subspace in H^{k−1}(S^{m−1}τ′_{M}); the orthogonal complement (Im(δd))^{⊥} is a finite-dimensional vector space consisting of smooth fields; and operator (2.11) induces an isomorphism of the topological Hilbert spaces

H^{k+1}(S^{m−1}τ′_{M})/Ker(δd) → Im(δd).  (2.12)

Let us show that

Ker(δd) = (Im(δd))^{⊥} = {v ∈ C^{∞}(S_{m−1}τ′_{M}) | dv = 0}.  (2.13)

Indeed, if v ∈ Ker(δd), then

(dv, dv) = −(v, δdv) = 0.

If v ∈ (Im(δd))^{⊥}, then for every u ∈ C^{∞}(S^{m−1}τ′_{M})

(δdv, u) = (v, δdu) = 0.

Therefore, δdv = 0, i.e., v ∈ Ker(δd).

Observe that the right-hand side of equation (2.10) belongs to Im(δd) since (δf, v) = −(f, dv) = 0 if dv = 0. Therefore, equation (2.10) has a solution for every f ∈ H^{k}(S^{m}τ′_{M}).

In the case of even m, equalities (2.13) with the help of Lemma 2.1 imply that Ker(δd) = 0. Thus, equation (2.10) has a unique solution for every f ∈ H^{k}(S^{m}τ′_{M}). Since (2.11) is an isomorphism, the first of estimates (2.8) holds.

In the case of odd m = 2l+1, equalities (2.13) with the help of Lemma 2.1 imply that Ker(δd) consists of the fields cg^{l}. Therefore, equation (2.10) has a unique solution satisfying condition (2.9). The first of the estimates (2.8) also holds for the solution. Thus the theorem is proved.

Theorem 2.3. Let (M, g) be a compact non-positively curved Riemannian manifold such that there exists an orbit of the geodesic flow which is dense in ΩM. If a function u ∈ C^{∞}(ΩM) and a symmetric tensor field f ∈ C^{∞}(S^{m}τ′_{M}) satisfy the equation

Hu(x, ξ) = ⟨f(x), ξ^{m}⟩  (2.14)

on ΩM, then the field f is potential, i.e., there exists a symmetric tensor field v ∈ C^{∞}(S^{m−1}τ′_{M}) such that dv = f.

We first show that Theorem 2.3 follows from the following special case.

Lemma 2.4. Let (M, g) be as in Theorem 2.3. If a symmetric tensor field f ∈ C^{∞}(S^{m}τ′_{M}) is solenoidal, i.e., satisfies the equation

δf = 0,  (2.15)

and if there exists a function u ∈ C^{∞}(ΩM) satisfying (2.14), then f ≡ 0.

Proof. Indeed, let the assumptions of Theorem 2.3 be fulfilled, and let (2.6) be the decomposition of the field f into potential and solenoidal parts. Putting

ũ(x, ξ) = u(x, ξ) − ⟨v(x), ξ^{m−1}⟩

from (2.4) and (2.14) we derive

Hũ(x, ξ) = ⟨f̃(x), ξ^{m}⟩.

Assuming Lemma 2.4 to be valid, the last equality implies f̃ = 0. Now formula (2.6) yields dv = f.

Before proving Lemma 2.4, we recall some more notions of tensor analysis.

Given a manifold M, the bundle β_{s}^{r}M = p^{∗}(τ_{s}^{r}M) over T M, where p : T M → M is the projection of the tangent bundle, is called the bundle of semibasic tensors of degree (r, s). Sections of the bundle are called semibasic tensor fields. For such a field u ∈ C^{∞}(β_{s}^{r}M), the coordinate representation u = (u_{j_{i}...j_{s}}^{i_{1}...i_{r}}(x, ξ)) holds in the domain of a local coordinate system on T M. Under a change of local coordinates, the components of a semibasic tensor field are transformed according to the same formula as for an ordinary tensor field. In particular, C^{∞}(β_{0}^{0}M) = C^{∞}(T M).

The vertical covariant derivative

∇^{v} : C^{∞}(β_{s}^{r}M) → C^{∞}(β_{s+1}^{r}M)

is defined in coordinate form by the formula

∇^{v}_{k}u_{j_{i}...j_{s}}^{i_{1}...i_{r}} = \frac{∂}{∂ξ^{k}} u_{j_{i}...j_{s}}^{i_{1}...i_{r}}.

Thus for a tangent vectors X, ξ ∈ T_{x}M, (∇^{v}_{X}u)(ξ) is just \frac{d}{dt}u(ξ + tX). For a Riemannian manifold (M, g), the horizontal covariant derivative

∇^{h} : C^{∞}(β_{s}^{r}M) → C^{∞}(β_{s+1}^{r}M)

is defined in coordinate form by the equality

∇^{h}_{k}u_{j_{i}...j_{s}}^{i_{1}...i_{r}} = \frac{∂}{∂x^{k}} u_{j_{i}...j_{s}}^{i_{1}...i_{r}} − Γ_{kq}^{p}ξ^{q} \frac{∂}{∂ξ^{p}} u_{j_{i}...j_{s}}^{i_{1}...i_{r}}+

+ ∑_{m=1}^{r} Γ_{kp}^{i_{m}}u_{j_{i}...j_{s}}^{i_{1}...i_{m−1}pi_{m+1}...i_{r}} − ∑_{m=1}^{s} Γ_{kj_{m}}^{p}u_{j_{i}...j_{m−1}pj_{m+1}...j_{s}}^{i_{1}...i_{r}}.

For tangent vectors X, ξ ∈ T_{x}M let γ be a curve in M with γ′(0) = X, and γ̃(t) be the horizontal lift of γ to T M such that γ̃(0) = ξ. Then u(γ̃(t)) is a tensor field

along γ(t) starting with u(ξ), and (∇^{h}_{X}u)(ξ) is the covariant derivative of this field.

As can be easily shown, ∇^{v} and ∇^{h} are well-defined differential operators, i.e., are independent of the choice of local coordinates. For a semibasic vector fields v we

define the horizontal divergence, div^{h} v, by

div^{h} v = ∇^{h}_{i}v^{i}.

Similarly define the vertical divergence div^{v} v.

A major role in our proof of Lemma 2.4 is played by Pestov’s identity [8, p. 122] asserting that, for every real function u ∈ C^{∞}(T M), the equality

2⟨∇^{h}u, ∇^{v}(Hu)⟩ = |∇^{h}u|^{2} + div^{h} v + div^{v} w − ⟨R(ξ, ∇^{v}u)ξ, ∇^{v}u⟩  (2.16)

holds on T M, where R is the curvature tensor, and the semibasic vector fields v and w are given by the formulas

v = ⟨∇^{h}u, ∇^{v}u⟩ξ − ⟨ξ, ∇^{h}u⟩∇^{v}u,  (2.17)
w = ⟨ξ, ∇^{h}u⟩∇^{h}u.  (2.18)

We point out for future reference that ⟨w, ξ⟩ = |Hu|^{2}.

Observe that the last term in the right-hand side of (2.16) is the sectional cur-

vature multiplied by |ξ ∧ ∇^{v}u|^{2}. Therefore, since the curvature is nonpositive, (2.16) implies the inequality

|∇^{h}u|^{2} ≤ 2⟨∇^{h}u, ∇^{v}(Hu)⟩ − div^{h} v − div^{v} w.  (2.19)

Proof of Lemma 2.4. We extend the function u(x, ξ) onto T M \ 0 in such a way that the function becomes positively homogeneous of degree m − 1 in its second argument. Then u ∈ C^{∞}(T M \ 0), and equation (2.14) holds on T M \ 0. Using (2.14), we transform the first term in the right-hand side of inequality (2.19) as

follows (hereafter the notation ∇^{v}^{i} = g^{ij}∇^{v}_{j} and ∇^{h}^{i} = g^{ij}∇^{h}_{j} are used):

2⟨∇^{h}u, ∇^{v}(Hu)⟩ = 2∇^{h}^{j}u · \frac{∂}{∂ξ^{j}} (f_{i_{1}...i_{m}}ξ^{i_{1}} … ξ^{i_{m}}) = 2m∇^{h}^{j}u · f_{ji_{2}...i_{m}}ξ^{i_{2}} … ξ^{i_{m}} =

= ∇^{h}^{j}(2muf_{ji_{2}...i_{m}}ξ^{i_{2}} … ξ^{i_{m}}) − 2mu(δf)_{i_{2}...i_{m}}ξ^{i_{2}} … ξ^{i_{m}}.

Using condition (2.15), we obtain

2⟨∇^{h}u, ∇^{v}(Hu)⟩ = ∇^{h}_{i}ṽ^{i},  (2.20)

where

ṽ^{i} = 2mug^{ij}f_{ji_{2}...i_{m}}ξ^{i_{2}} … ξ^{i_{m}}.

That is ṽ is the vector field dual to the one form 2muf(·, ξ, …, ξ). Replacing the first term in the right-hand side of (2.19) by its value (2.20), we obtain

|∇^{h}u|^{2} ≤ div^{h}(ṽ − v) − div^{v} w.  (2.21)

We multiply inequality (2.21) by the symplectic volume form dΣ = dΣ^{2n−1} and integrate the result over ΩM. Transforming the integrals on the right-hand side of the so-obtained inequality by the Gauss-Ostrogradskiĭ formulas for vertical and horizontal divergences [8, p. 110], we arrive at the relation

∫_{ΩM} |∇^{h}u|^{2} dΣ ≤ −(n + 2m − 2) ∫_{ΩM}⟨w, ξ⟩ dΣ.  (2.22)

The constant n + 2m − 2 above comes from the fact that the field w(x, ξ) is homogeneous of degree 2m − 1 in its second argument, as can be seen from (2.18). Further, since ⟨w, ξ⟩ = |Hu|^{2}, inequality (2.22) takes the form

∫_{ΩM} |∇^{h}u|^{2} dΣ + (n + 2m − 2) ∫_{ΩM} |Hu|^{2} dΣ ≤ 0.

Consequently, Hu ≡ 0. Now (2.14) implies that f ≡ 0. The lemma is thus proved.

Proof of Theorem 1.3:. The condition of the theorem means that the integral of the function

F (x, ξ) = ⟨f(x), ξ^{m}⟩

over every closed orbit of the geodesic flow is equal to zero. Recall that the geodesic flow of a compact negatively curved manifold has Anosov’s type [6]. By the smooth version of Livčic’s theorem [9, p. 566], there exists a function u ∈ C^{∞}(ΩM) satisfying equation (2.14). Applying Theorem 2.3, we arrive at the claim of Theorem 2.5.

As was mentioned above, Theorems 1.1 and 1.2 can be proved using Theorem 1.3 in the same way as in [1,2].

References

1. V. Guillemin, D. Kazhdan, Some inverse spectral results for negatively curved 2-manifolds, Topology 19 (1980), 301–312.

2. V. Guillemin, D. Kazhdan, Some inverse spectral results for negatively curved n-manifolds, Proceedings of Symposia in Pure Math. 36 (1980), 153–180.

3. M. Min-Oo, Spectral rigidity for manifolds with negative curvature operator, Contemp. Math. 51 (1986), 99-103.

4. M. Vignéras, Variétés riemanniennes isospectrales et non isométriques, Ann. of Math. 110 (1980), 21-32.

5. C. Gordon and E. Wilson, Isospectral deformations of compact solvmanifolds, J. Diff. Geom. 19 (1984), 241-256.

6. D. Anosov, Geodesic Flows on Closed Riemannian Manifolds with Negative Curvature, Proc. Steclov Inst. of Math., vol. 90, 1967.

7. L. Pestov and A. Sharafutdinov, Integral geometry of tensor fields on a manifold of negative curvature, Novosibirsk (transl. from Sibirskii Math. Zhurnal) 29, No.3 (1988), 114-130.

8. V. A. Sharafutdinov, Integral Geometry of Tensor Fields, VSP, Utrecht, the Netherlands, 1994.

9. R. de la Llave, J. M. Marco, R. Moriyon, Canonical perturbation theory of Anosov Systems and regularity results for the Livsic cohomology equation, Annals of Math. 123 (1986), 537– 611.

Department of Mathematics University of Pennsylvania Philadelphia, PA 19104 E-mail address: ccroke@math.upenn.edu

Institute of Mathematics, Universitetskiy pr., 4, Novosibirsk, 630090, Russia E-mail address: sharaf@math.nsc.su