Two dimensional compact simple Riemannian manifolds are boundary distance rigid

Leonid Pestov^{∗}

Institute of Computational Mathematics and Mathematical Geophysics Russian Academy of Sciences Novisibirsk 630090, Russia

Gunther Uhlmann^{†} Department of Mathematics University of Washington Seattle, WA 98195, USA

Abstract

We prove that knowing the lengths of geodesics joining points of the boundary of a two-dimensional, compact, simple Riemannian manifold with boundary, we can determine uniquely the Riemannian metric up to the natural obstruction.

1 Introduction and statement of the results

Let (M, g) be a compact Riemannian manifold with boundary ∂M. Let d_{g}(x, y) denote the geodesic distance between x and y. The inverse problem we address in this paper is whether we can determine the Riemannian metric g knowing d_{g}(x, y) for any x ∈ ∂M, y ∈ ∂M. This problem arose in rigidity questions in Riemannian geometry [M], [C], [Gr]. For the case in which M is a bounded domain of Euclidean space and the metric is conformal to the Euclidean one, this problem is known as the inverse kinematic problem which arose in Geophysics and has a long history (see for instance [R] and the references cited there).

The metric g cannot be determined from this information alone. We have d_{ψ^{∗}g} = d_{g} for any diffeomorphism ψ : M → M that leaves the boundary pointwise fixed, i.e., ψ|_{∂M} = Id, where Id denotes the identity map and ψ^{∗}g is the pull-back of the metric g. The natural question is whether this is the only obstruction to unique identifiability of the metric. It is easy to see that this is not the case. Namely one can construct a metric g and find a point x_{0} in M so that d_{g}(x_{0}, ∂M) > sup _{x,y∈∂M}d_{g}(x, y). For such a metric, d_{g} is independent of a change of g in a neighborhood of x_{0}. The hemisphere of the round sphere is another example.

∗Part of this work was done while the author was visiting MSRI and the University of Washington.

†Partly supported by NSF and a John Simon Guggenheim Fellowship.

Therefore it is necessary to impose some a-priori restrictions on the metric. One such restriction is to assume that the Riemannian manifold is simple, i.e., given two points there is a unique geodesic joining the points and ∂M is strictly convex. ∂M is strictly convex if the second fundamental form of the boundary is positive definite in every boundary point.

R. Michel conjectured in [M] that simple manifolds are boundary distance rigid that is d_{g} determines g uniquely up to an isometry which is the identity on the boundary. This is known for simple subspaces of Euclidean space (see [Gr]), simple subspaces of an open hemisphere in two dimensions (see [M]), simple subspaces of symmetric spaces of constant negative curvature [BCG], simple two dimensional spaces of negative curvature (see [C1] or [O]).

In this paper we prove that simple two dimensional compact Riemannian manifolds are boundary distance rigid. More precisely we show

Theorem 1.1. Let (M, g_{i}), i = 1, 2 be a two dimensional simple Riemannian compact Riemannian manifold. Assume

d_{g_{1}}(x, y) = d_{g_{2}}(x, y) ∀(x, y) ∈ ∂M × ∂M

then there exists a diffeomorphism ψ : M → M, ψ|_{∂M} = Id, so that

g_{2} = ψ^{∗}g_{1}.

As it has been shown in [Sh], Theorem 1.1 follows from

Theorem 1.2. Let (M, g_{i}), i = 1, 2 be a two dimensional simple Riemannian compact Riemannian manifold. Assume

d_{g_{1}}(x, y) = d_{g_{2}}(x, y) ∀(x, y) ∈ ∂M × ∂M

and

g_{1}|_{∂M} = g_{2}|_{∂M}.

then there exists a diffeomorphism ψ : M → M, ψ|_{∂M} = Id, ψ′|_{∂M} = Id, so that

g_{2} = ψ^{∗}g_{1}.

We will prove Theorem 1.2. The function d_{g} measures the travel times of geodesics joining points of the boundary. In the case that both g_{1} and g_{2} are conformal to the Euclidean metric e (i.e., (g_{k})_{ij} = α_{k}δ_{ij}, k = 1, 2 with δ_{ij} the Krönecker symbol), as mentioned earlier, the problem we are considering here is known in seismology as the inverse kinematic problem. In this case, it has been proven by Mukhometov in two dimensions [Mu] that if (M, g_{i}), i = 1, 2 is simple and d_{g_{1}} = d_{g_{2}}, then g_{1} = g_{2}. More generally the same method of proof shows that if (M, g_{i}), i = 1, 2 are simple compact Riemannian manifolds with boundary and they are in the same conformal class, i.e. g_{1} = αg_{2} for a positive function α and d_{g_{1}} = d_{g_{2}} then g_{1} = g_{2} [Mu1]. In this case the diffeomorphism ψ must be the identity. For related results and generalizations see [B], [BG], [C], [GN], [MR].

We mention a closely related inverse problems. Suppose we have a Riemannian metric in Euclidean space which is the Euclidean metric outside a compact set. The inverse scattering problem for metrics is to determine the Riemannian metric by measuring the scattering operator (see [G]). A similar obstruction occurs in this case with ψ equal to the identity outside a compact set. It was proven in [G] that from the wave front set of the scattering operator one can determine, under some non-trapping assumptions on the metric, the scattering relation on the boundary of a large ball. We proceed to define in more detail the scattering relation and its relation with the boundary distance function.

Let ν denote the unit-inner normal to ∂M. We denote by Ω (M) → M the unit-sphere bundle over M:

Ω(M) = ⋃_{x∈M} Ω_{x}, Ω_{x} = {ξ ∈ T_{x}(M) : |ξ|_{g} = 1}.

Ω(M) is a (2 dim M −1)-dimensional compact manifold with boundary, which can be written as the union ∂Ω (M) = ∂_{+}Ω (M) ∪ ∂_{−}Ω (M)

∂_{±}Ω (M) = {(x, ξ) ∈ ∂Ω (M) , ± (ν (x) , ξ) ≥ 0 }.

The manifold of inner vectors ∂_{+}Ω (M) and outer vectors ∂_{−}Ω (M) intersect at the set of tangent vectors

∂_{0}Ω (M) = {(x, ξ) ∈ ∂Ω (M) , (ν (x) , ξ) = 0 }.

Let (M, g) be an n-dimensional compact manifold with boundary. We say that (M, g) is non-trapping if each maximal geodesic is finite. Let (M, g) be non-trapping and the boundary ∂M is strictly convex. Denote by τ(x, ξ) the length of the geodesic γ(x, ξ, t), t ≥ 0, starting at the point x in the direction ξ ∈ Ω_{x}. These function is smooth on Ω(M)\∂_{0}Ω(M). The function τ^{0} = τ|_{∂Ω(M)} is equal zero on ∂_{−}Ω(M) and is smooth on ∂_{+}Ω(M). Its odd part with respect to ξ

τ_{−}^{0}(x, ξ) = \frac{1}{2} (τ^{0}(x, ξ) − τ^{0} (x, −ξ))

is the smooth function.

Definition 1.1. Let (M, g) be non-trapping with strictly convex boundary. The scattering relation α : ∂Ω (M) → ∂Ω (M) is defined by

α(x, ξ) = (γ(x, ξ, 2τ_{−}^{0}(x, ξ)), γ̇(x, ξ, 2τ_{−}^{0}(x, ξ))).

The scattering relation is a diffeomorphism ∂Ω (M) → ∂Ω (M) . Notice that α|_{∂_{+}Ω(M)} : ∂_{+}Ω (M) → ∂_{−}Ω (M) , α|_{∂_{−}Ω(M)} : ∂_{−}Ω (M) → ∂_{+}Ω (M) are diffeomorphisms as well. Obviously, α is an involution, α^{2} = id and ∂_{0}Ω (M) is the hypersurface of its fixed points, α(x, ξ) = (x, ξ), (x, ξ) ∈ ∂_{0}Ω (M) .

A natural inverse problem is whether the scattering relation determines the metric g up to an isometry which is the identity on the boundary. In the case that (M, g) is a simple manifold, and we know the metric at the boundary, knowing the scattering relation is equivalent to knowing the boundary distance function ([M]). We show in this paper that if we know the scattering relation we can determine the Dirichlet-to-Neumann (DN) map associated to the Laplace-Beltrami operator of the metric. We proceed to define the DN map.

Let (M, g) be a compact Riemannian manifold with boundary. The Laplace-Beltrami operator associated to the metric g is given in local coordinates by

Δ_{g}u = \frac{1}{\sqrt{det g}} ∑_{i,j=1}^{n} \frac{∂}{∂x_{i}} (\sqrt{det g}g^{ij} \frac{∂u}{∂x_{j}} )

where (g^{ij}) is the inverse of the metric g. Let us consider the Dirichlet problem

Δ_{g}u = 0 on M, u|_{∂M} = f.

We define the DN map in this case by

Λ_{g}(f) = (ν, ∇u|_{∂M})

The inverse problem is to recover g from Λ_{g}.

In the two dimensional case the Laplace-Beltrami operator is conformally invariant. More precisely

Δ_{βg} = \frac{1}{β} Δ_{g}

for any function β, β ≠ 0. Therefore we have that for n = 2

Λ_{β(ψ^{∗}g)} = Λ_{g}

for any non-zero β satisfying β|_{∂M} = 1.

Therefore the best that one can do in two dimensions is to show that we can determine the conformal class of the metric g up to an isometry which is the identity on the boundary That this is the case is a result proven in [LeU] for simple metrics and for general connected two dimensional Riemannian manifolds with boundary in [LaU].

In this paper we prove:

Theorem 1.3. Let (M, g_{i}), i = 1, 2 be compact, simple two dimensional Riemannian manifolds with boundary. Assume that α_{g_{1}} = α_{g_{2}}. Then Λ_{g_{1}} = Λ_{g_{2}}.

The proof of Theorem 1.2 is reduced then to the proof of Theorem 1.3. In fact from Theorem 1.3 and the result of [LaU] we get that we can determine the conformal class of the metric up to an isometry which is the identity on the boundary. Now by Mukhometov’s result we have that the conformal factor must be one proving that the metrics are isometric via a diffeomorphism which is the identity at the boundary. In other words d_{g_{1}} = d_{g_{2}} implies that α_{g_{1}} = α_{g_{2}}. By Theorem 1.3 Λ_{g_{1}} = Λ_{g_{2}}. By the result of [LeU], [LaU], there exists a diffeomorphism ψ : M ⟶ M, ψ|_{∂M} = Identity and a function β ≠ 0, β|_{∂M} = identity such that g_{1} = βψ^{∗}g_{2}. By Mukhometov’s theorem β = 1 showing that g_{1} = ψ^{∗}g_{2} proving Theorem 1.2. and Theorem 1.1.

The proof of Theorem 1.3 consists in showing that from the scattering relation we can determine the traces at the boundary of conjugate harmonic functions, which is equivalent information to knowing the DN map associated to the Laplace-Beltrami operator. The steps to accomplish this are outlined below. It relies in a connection between the Hilbert transform and geodesic flow.

We embed (M, g) into a compact Riemannian manifold (S, g) with no boundary. Let φ_{t} be the geodesic flow on Ω(S) and H = \frac{d}{dt}φ_{t}|_{t=0} be the geodesic vector field. Introduce the map ψ : Ω(M) → ∂_{−}Ω(M) defined by

ψ(x, ξ) = φ_{τ(x,ξ)}(x, ξ), (x, ξ) ∈ Ω(M).

The solution of the boundary value problem for the transport equation

Hu = 0, u|_{∂_{+}Ω(M)} = w

can be written in the form

u = w_{ψ} = w ∘ α ∘ ψ.

Let u^{f} be the solution of the boundary value problem

Hu = −f, u|_{∂_{−}Ω(M)} = 0,

which we can write as

u^{f}(x, ξ) = ∫_{0}^{τ(x,ξ)}f(φ_{t}(x, ξ))dt, (x, ξ) ∈ Ω(M).

In particular

Hτ = −1.

The trace

If = u^{f}|_{∂_{+}Ω(M)}

is called the geodesic X-ray transform of the function f. By the fundamental theorem of calculus we have

IHf = (f ∘ α − f)|_{∂_{+}Ω(M)}.  (1.1)

In what follows we will consider the operator I acting only on functions that do not depend on ξ, unless otherwise indicated. Let L_{μ}^{2}(∂_{+}Ω(M)) is the real Hilbert space, with scalar product given by

(u, v)_{L_{μ}^{2}(∂_{+}Ω(M))} = ∫_{∂_{+}Ω(M)} μuvdΣ, μ = (ξ, ν).

Here the measure dΣ = d(∂M) ∧ dΩ_{x} where d(∂M) is the induced volume form on the boundary by the standard measure on M and

dΩ_{x} = ∑_{k=1}^{n}(−1)^{k+1}ξ^{k}dξ^{1} ∧ ... ∧ d̂ξ^{k} ∧ ...dξ^{n}.

As usual the scalar product in L^{2}(M) is defined by

(u, v) = ∫_{M} uv\sqrt{detg}dx.

The operator I is a bounded operator from L^{2}(M) into L_{μ}^{2}(∂_{+}Ω(M)). The adjoint I^{∗} : L_{μ}^{2}(∂_{+}Ω(M)) → L^{2}(M) is given by

I^{∗}w(x) = ∫_{Ω_{x}} w_{ψ}(x, ξ)dΩ_{x},

We will study the solvability of equation I^{∗}w = h with smooth right hand side. Let w ∈ C^{∞}(∂_{+}Ω(M)). Then the function w_{ψ} will not be smooth on Ω(M) in general. We have that w_{ψ} ∈ C^{∞}(Ω(M) \ ∂_{0}Ω(M)). We give below necessary and sufficient conditions for smoothness of w_{ψ} on Ω(M).

We introduce the operators of even and odd continuation with respect to α:

A_{±}w(x, ξ) = w(x, ξ), (x, ξ) ∈ ∂_{+}Ω (M) ,
A_{±}w(x, ξ) = ± (α^{∗}w) (x, ξ), (x, ξ) ∈ ∂_{−}Ω (M) .

The scattering relation preserves the measure |(ξ, ν)|dΣ and therefore the operators A_{±} : L_{μ}^{2}(∂_{+}Ω(M)) → L_{|μ|}^{2} (∂Ω (M)) are bounded, where L_{|μ|}^{2} (∂Ω (M)) is real Hilbert space with scalar product

(u, v)_{L_{|μ|}^{2}(∂Ω(M))} = ∫_{∂Ω(M)} |μ| uvdΣ, μ = (ξ, ν).

The adjoint of A_{±} is a bounded operator A_{±}^{∗} : L_{|μ|}^{2} (∂Ω (M)) → L_{μ}^{2}(∂_{+}Ω(M)) given by

A_{±}^{∗}u = (u ± u ∘ α)|_{∂_{+}Ω(M)}.

Using A_{−}^{∗} formula (1.1) can be written in the form

IHf = −A_{−}^{∗}f^{0}, f^{0} = f|_{∂Ω(M)}.  (1.2)

The space C_{α}^{∞} (∂_{+}Ω (M)) is defined by

C_{α}^{∞} (∂_{+}Ω (M)) = {w ∈ C^{∞} (∂_{+}Ω (M)) : w_{ψ} ∈ C^{∞} (Ω (M))}.

We have the following characterization of the space of smooth solutions of the transport equation

Lemma 1.1.

C_{α}^{∞}(∂_{+}Ω(M)) = {w ∈ C^{∞}(∂_{+}Ω(M)) : A_{+}w ∈ C^{∞}(∂Ω(M))}.

Now we can state the main theorem for solvability for I^{∗}.

Theorem 1.4. Let (M, g) be a simple, compact two dimensional Riemannian manifold with boundary. Then the operator I^{∗} : C_{α}^{∞}(∂_{+}Ω(M)) → C^{∞}(M) is onto.

Now we define the Hilbert transform:

Hu(x, ξ) = \frac{1}{2π} ∫_{Ω_{x}} \frac{1 + (ξ, η)}{(ξ_{⊥}, η)}u(x, η)dΩ_{x}(η), ξ ∈ Ω_{x},  (1.3)

where the integral is understood as a principle value integral. Here ⊥ means a 90^{o} degree rotation. In coordinates (ξ_{⊥})_{i} = ε_{ij}ξ^{j}, where

ε = \sqrt{det g} (\begin{matrix}0 & 1 \\ −1 & 0 \end{matrix}) .

The Hilbert transform H transforms even (respectively odd) functions with respect to ξ to even (respectively odd) ones. If H_{+} (respectively H_{−}) is the even (respectively odd) part of the operator H:

H_{+}u(x, ξ) = \frac{1}{2π} ∫_{Ω_{x}} \frac{(ξ, η)}{(ξ_{⊥}, η)}u(x, η)dΩ_{x}(η),

Hu_{−}(x, ξ) = \frac{1}{2π} ∫_{Ω_{x}} \frac{1}{(ξ_{⊥}, η)}u(x, η)dΩ_{x}(η)

and u_{+}, u_{−} are the even and odd parts of the function u, then H_{+}u = Hu_{+}, H_{−}u = Hu_{−}.

We introduce the notation H_{⊥} = (ξ_{⊥}, ∇) = −(ξ, ∇_{⊥}), where ∇_{⊥} = ε∇ and ∇ is the covariant derivative with respect to the metric g. The following commutator formula for the geodesic vector field and the Hilbert transform is very important in our approach.

Theorem 1.5. Let (M, g) be a two dimensional Riemannian manifold. For any smooth function u on Ω(M) we have the identity

[H, H]u = H_{⊥}u_{0} + (H_{⊥}u)_{0}  (1.4)

where

u_{0}(x) = \frac{1}{2π} ∫_{Ω_{x}} u(x, ξ)dΩ_{x}

is the average value.

Now we can prove the Theorem 1.3.

Separating the odd and even parts with respect to ξ in (1.4) we obtain the identities:

H_{+}Hu − HH_{−}u = (H_{⊥}u)_{0}, H_{−}Hu − HH_{+}u = H_{⊥}u_{0}.

Let (M, g) be a non-trapping strictly convex manifold. Take u = w_{ψ}, w ∈ C_{α}^{∞}(∂_{+}(Ω)). Then

2πHH_{+}w_{ψ} = −H_{⊥}I^{∗}w

and using the formula (1.2) we conclude

2πA_{−}^{∗}H_{+}A_{+}w = IH_{⊥}I^{∗}w,  (1.5)

since w_{ψ}|_{∂Ω(M)} = A_{+}w.

Let (h, h_{∗}) be a pair of conjugate harmonic functions on M,

∇h = ∇_{⊥}h_{∗}, ∇h_{∗} = −∇_{⊥}h.

Notice, that δ∇ = △ is the Laplace-Beltrami operator and δ∇_{⊥} = 0. Let I^{∗}w = h. Since IH_{⊥}h = IHh_{∗} = −A_{−}^{∗}h_{∗}^{0}, where h_{∗}^{0} = h_{∗}|_{∂M}, we obtain from (1.5)

2πA_{−}^{∗}H_{+}A_{+}w = −A_{−}^{∗}h_{∗}^{0}.  (1.6)

The following theorem gives the key to obtain the DN map from the scattering relation.

Theorem 1.6. Let M be a 2-dimensional simple manifold. Let w ∈ C_{α}^{∞}(∂_{+}Ω(M)) and h_{∗} is harmonic continuation of function h_{∗}^{0}. Then the equation (1.6) holds iff the functions h = I^{∗}w and h_{∗} are conjugate harmonic functions.

Proof. The necessity has already been established. Using (1.2) and (1.5) the equality (1.6) can be written in the form

IH_{⊥}h = IHq,

where q is an arbitrary smooth continuation onto M of the function h_{∗}^{0} and h = I^{∗}w. Thus, the ray transform of the vector field ∇q + ∇_{⊥}h equal 0. Consequently, this field is potential ([An]), that is, ∇q + ∇_{⊥}h = ∇p and p|_{∂M} = 0. Then the functions h and h_{∗} = q − p are conjugate harmonic functions and h_{∗}|_{∂M} = h_{∗}^{0}. We have finished the proof of the main theorem.

In summary we have the following procedure to obtain the DN map from the scattering relation. For an arbitrary given smooth function h_{∗}^{0} on ∂M we find a solution w ∈ C_{α}^{∞}(∂_{+}Ω(M)) of the equation (1.6). Then the functions h^{0} = 2π(A_{+}w)_{0} (notice, that 2π(A_{+}w)_{0} = I^{∗}w|_{∂M}) and h_{∗}^{0} are the traces of conjugate harmonic functions. It gives the map

h_{∗}^{0} → (ν_{⊥}, ∇^{∥}h^{0}) = (ν, ∇h_{∗})|_{∂M}.

which is the DN map. This proves Theorem 1.3. For the definition of ∇^{∥} see section 2.

A brief outline of the paper is as follows. In section 2 we collect some facts and definition we need later. In section 3 we study the solvability of I^{∗}w = h on Sobolev spaces and prove Theorem 1.4. In section 4 we make a detailed study of the scattering relation and prove Lemma 1.1. In section 5 we prove Theorem 1.5.

2 Preliminaries and Notation

Here we will give some definitions and formulas, that we need in what follows. For further references see [E], [J], [K], [Sh]. Let π : T (M) → M be the tangent bundle over an ndimensional Riemannian manifold (M, g). We will denote points of the manifold T (M) by pairs (x, ξ). The connection map K : T (T (M)) → T (M) is defined by its local representation

K(x, ξ, y, η) = (x, η + Γ(x)(y, ξ)), (Γ(x)(y, ξ))^{i} = Γ_{jk}^{i}(x)y^{j}ξ^{k}, i = 1, ..., n,

where Γ_{jk}^{i} are the Christoffel symbols of the metric g

Γ_{jk}^{i} = \frac{1}{2}g^{il}(\frac{∂g_{jl}}{∂x^{k}} + \frac{∂g_{kl}}{∂x^{j}} − \frac{∂g_{jk}}{∂x^{l}}).

The linear map K(x, ξ) = K|_{(x,ξ)} : T_{(x,ξ)}(T (M)) → T_{x}M defines the horizontal subspace H_{(x,ξ)} = KerK(x, ξ). It can be identified with the tangent space T_{x}(M) using the operator

J_{(x,ξ)}^{h} = (π′(x, ξ)|_{H_{(x,ξ)}})^{−1} : T_{x}(M) → H_{(x,ξ)}.

The vertical space V_{(x,ξ)} = Kerπ′(x, ξ) can also identified with T_{x}(M) by using the operator

J_{(x,ξ)}^{v} = (K(x, ξ)|_{V_{(x,ξ)}})^{−1} : T_{x}(M) → V_{(x,ξ)}.

The tangent space T_{(x,ξ)}(T (M)) is the direct sum of the horizontal and vertical subspaces, T_{(x,ξ)}(T (M)) = H_{(x,ξ)} ⊕ V_{(x,ξ)}. An arbitrary vector X ∈ T_{(x,ξ)}(T (M)) can be uniquely decomposed in the form

X = J_{(x,ξ)}^{h}X_{h} + J_{(x,ξ)}^{v}X_{v},

where

X_{h} = π′(x, ξ)X, X_{v} = K(x, ξ)X.

We will call X_{h}, X_{v} the horizontal and vertical components of the vector X and use the notation X = (X_{h}, X_{v}). If in local coordinates X = (X^{1}, ..., X^{2n}) then X_{h}, X_{v} is given by

X_{h}^{i} = X^{i}, X_{v}^{i} = X^{i+n} + Γ_{jk}^{i}(x)X^{j}ξ^{k}, i = 1, ..., n.

The horizontal and vertical components Y^{h}, Y^{v} ∈ T_{x}^{∗}(M) of a covector Y ∈ T_{(x,ξ)}^{∗}(T (M)), Y = (Y^{h}, Y^{v}) are given by

⟨X, Y ⟩ = (X_{h}, Y^{h}) + (X_{v}, Y^{v}).

In local coordinates we have

Y_{j}^{h} = Y_{j} − Γ_{jk}^{i}(x)Y_{i+n}ξ^{k}, Y_{j}^{v} = Y_{j+n}, j = 1, ..., n.

Let N be a smooth manifold and f : T (M) → N a smooth map. Then the derivative f′(x, ξ) : T_{(x,ξ)}(T (M)) → T_{f(x,ξ)}(N) defines the horizontal ∇_{h}f(x, ξ) and vertical ∇_{v}f(x, ξ) derivatives:

∇_{h}f(x, ξ) = f′(x, ξ) ∘ J_{(x,ξ)}^{h} : T_{x}(M) → T_{f(x,ξ)}(N),
∇_{v}f(x, ξ) = f′(x, ξ) ∘ J_{(x,ξ)}^{v} : T_{x}(M) → T_{f(x,ξ)}(N).

We have then that

f′(x, ξ)X = (∇_{h}f(x, ξ), X_{h}) + (∇_{v}f(x, ξ), X_{v})  (2.1)

In local coordinates

∇_{hj}f^{(α)}(x, ξ) = (\frac{∂}{∂x^{j}} − Γ_{jk}^{i}(x)ξ^{k} \frac{∂}{∂ξ^{i}})f^{(α)}(x, ξ),

∇_{vj}f^{(α)}(x, ξ) = \frac{∂}{∂ξ^{j}} f^{(α)}(x, ξ), α = 1, ..., dimN.

Let e : Σ → T (M) be a smooth embedding and p : T (M) → Σ denotes a normal projection, i.e. pe = id. For a point (x, ξ) ∈ e(Σ), Ker p′(x, ξ) = N(Σ) ⋂ T_{(x,ξ)}(T (M)), where N(Σ) is the normal bundle of Σ.

Definition 2.1. The horizontal and vertical derivatives ∇_{h}^{∥}, ∇_{v}^{∥} on Σ are the tangent components of ∇_{h} and ∇_{v} with respect to Σ:

∇_{h}^{∥}f = ∇_{h}(f ∘ p)|_{Σ}, ∇_{v}^{∥}f = ∇_{v}(f ∘ p)|_{Σ}.

This definition is obviously independent of the choice of the normal projection. From the identity f = (f ∘ p) ∘ e and (2.1) we have for any vector X ∈ T_{(x,ξ)}(T (Σ)), by identifying X with its embedding e′(x, ξ)X, that

f′(x, ξ)X = (f ∘ p)′(x, ξ)X = (∇_{h}(f ∘ p)(x, ξ), X_{h}) + (∇_{v}(f ∘ p)(x, ξ), X_{v}).

Therefore

f′(x, ξ)X = (∇_{h}^{∥}f(x, ξ), X_{h}) + (∇_{v}^{∥}f(x, ξ), X_{v}).

Let us give an equivalent definition. Denote by i(x, ξ) the isomorphism

i (x, ξ) : T_{(x,ξ)} (T (M)) → T_{x}(M) × T_{x} (M) , defined by i (x, ξ) X = (X_{h}, X_{v})

Let i_{Σ} (x, ξ) be its restriction on T_{(x,ξ)} (Σ) , i_{Σ} (x, ξ) = i (x, ξ) |_{T_{(x,ξ)}(Σ)}. Denote by T_{x}^{(i)} (M) = pr_{i} Ran i_{Σ} (x, ξ) ⊂ T_{x}(M) × T_{x} (M) , i = 1, 2, the projections to the first and second component of the range of i_{Σ}. Then

∇_{h}^{∥}f (x, ξ) = ∇_{h}f̃(x, ξ) |_{T_{x}^{(1)}(M)}, ∇_{v}^{∥}f (x, ξ) = ∇_{v}f̃(x, ξ) |_{T_{x}^{(2)}(M)},

where f̃: T (M) → N is an arbitrary smooth continuation of f. Example 1

Let us consider the special case of Σ = Ω(M) the unit sphere bundle. Then p(x, ξ) = (x, ξ/|ξ|) is a normal projection. For simplicity of notation in the case of the manifold Ω(M) we will use notation ∂ instead of ∇_{v}^{∥}. Then

∂ = (\frac{∂}{∂ξ} − ξ(ξ, \frac{∂}{∂ξ}))|_{|ξ|=1}.

Clearly, (ξ, ∂) = 0. Notice, that the horizontal derivative is tangent to the submanifold Ω(M) since ∇_{h}|ξ| = 0. Because of this we keep the original notation for horizontal derivative on Ω(M) and moreover simplify this notation to just ∇. Thus, in the case of the manifold Ω(M) we will use notation ∇ and ∂ for the horizontal and vertical derivatives. We will also use ∇ for horizontal derivative on T (M). Notice relation

T_{(x,ξ)}(Ω(M)) = {X ∈ T_{(x,ξ)}(T (M)) : (ξ, X_{v}) = 0}.

Example 2

Let Γ be a smooth hypersurface in M with normal ν and Σ its lift to Ω(M), Σ = π_{1}^{−1}(Γ) = {(x, ξ) : x ∈ Γ, ξ ∈ Ω_{x}(M)}, where π_{1} = π|_{Ω(M)}. Then

T_{(x,ξ)}(π_{1}^{−1}(Γ)) = {X ∈ T_{(x,ξ)}(T (M)) : (ν(x), X_{h}) = 0, (ξ, X_{v}) = 0}  (2.2)

It means that T_{x}^{(1)} (M) = {η ∈ T_{x} (M) : (η, ν (x)) = 0}, T_{x}^{(2)}(M) = {η ∈ T_{x} (M) : (ξ, η) = 0}. We have the same vertical derivative as in the case of Ω (M) . The horizontal derivative for this case will be denoted by ∇^{∥}, so that

∇^{∥} = (∇ − ν(ν, ∇))|_{π_{1}^{−1}(Γ)}.

Clearly, (ν, ∇^{∥}) = 0.

We now state a similar definition of vertical and horizontal derivatives on a submanifold for semibasic tensor fields. We recommend chapter 3 of [Sh] for more details.

Let T_{s}^{r}(M) denote the bundle of tensor fields of degree (r, s) on M. A section of this bundle is called a tensor field of degree (r, s). Let π_{s}^{r} : T_{s}^{r}(M) → M be the projection. A fiber map u : T (M) → T_{s}^{r}(T M), i.e. π_{s}^{r} ∘ u = π is called a semibasic tensor field of degree (r, s) on the manifold T (M). Denote by ξ the semibasic vector field given by the identity map T (M) → T (M). An arbitrary tensor field u of degree (r, s) on the manifold M, i.e. section u : M → T_{s}^{r}(M) defines by the formula u ∘ π a semibasic tensor field (since π_{s}^{r} ∘ (u ∘ π) = (π_{s}^{r} ∘ u) ∘ π = id ∘ π = π). The map u → u ∘ π identifies tensor fields on M and ξ-constant semibasic tensor fields on T (M). A semibasic tensor field on the submanifold Σ ⊂ T (M) is defined as the fiber map u : Σ → T_{s}^{r}(M), i.e. π_{s}^{r} ∘ u = π|_{Σ}. If u is semibasic tensor field on T (M), then its restriction u|_{Σ} is a semibasic tensor field on Σ. Using the metric g we can identify the bundle T_{s}^{r}(M) with T_{0}^{r+s}(M) and the bundle T_{r+s}^{0}(M) with T_{r+s}^{∗}(M).

A semibasic tensor field u : T (M) → T_{m}^{0}(T M) is naturally identified with the polynomial of degree m in η, u_{(m)} : T (M)×T^{m}(M) → R, u_{m} (x, ξ, η_{(1)}, ..., η_{(m)}) = u_{i_{1}...i_{m}} (x, ξ) η_{(1)}^{i_{1}}...η_{(m)}^{i_{m}}, where the fibers of the vector bundle T (M) × T^{m}(M) = T^{m+1}(M) are given by

T_{x}^{m+1}(M) = \underbrace{T_{x} (M) × ... × T_{x} (M)}_{m+1}.

The horizontal derivative ∇_{h}u_{(m)} and vertical derivative ∇_{v}u_{(m)} are the fiber maps T (M) × T^{m}(M) → T^{∗}(M) = T_{1}^{0}(M), and therefore are semibasic tensor fields on T (M) × T^{m}(M) of degree 1. The corresponding polynomials (∇_{h}u_{(m)})_{(1)}, (∇_{v}u_{(m)})_{(1)} on T (M) × T^{m+1}(M) define semibasic tensor fields ∇u, ∇_{ξ}u of degree (m + 1), which we call the horizontal and vertical derivatives of u. Thus, by definition

(∇u)_{(m+1)} = (∇_{h}u_{(m)})_{(1)}, (∇_{ξ}u)_{(m+1)} = (∇_{v}u_{(m)})_{(1)}.

Straightforward calculations give the representation in local coordinates:

(∇u)_{i_{1}...i_{m+1}} = ˜∇_{i_{m+1}}u_{i_{1}...i_{m}} − Γ_{i_{(m+1)}k}^{j}ξ^{k} \frac{∂u_{i_{1}...i_{m}}}{∂ξ^{j}} , (∇_{ξ}u)_{i_{1}...i_{m+1}} = \frac{∂u_{i_{1}...i_{m}}}{∂ξ^{i_{m+1}}} ,

where ˜∇ denotes the usual covariant derivative on the manifold (M, g). Notice, that for ξ-constant tensor fields, ∇u = ˜∇u and since we identify ξ-constant semibasic tensor fields with tensor fields on M, we will use one notation ∇ for covariant and horizontal derivatives.

As in the case of maps, we define tangent derivatives of semibasic tensor fields on the submanifold Σ ⊂ T (M):

∇^{∥}u = ∇(u ∘ p)|_{Σ}, ∇_{ξ}^{∥}u = ∇_{ξ}(u ∘ p)|_{Σ}.

In the case Σ = Ω(M) we keep the notation ∇ instead of ∇^{∥} and use ∂ instead ∇_{ξ}^{∥}. We mention the following formulas, [Sh] :

∇g = 0, ∇ξ = 0, ∂_{j}ξ^{i} = δ_{j}^{i} − ξ^{i}ξ_{j},  (2.3)
[∇, ∂] = 0, [∂_{i}, ∂_{j}] = ξ_{i}∂_{j} − ξ_{j}∂_{i},  (2.4)
[∇_{i}, ∇_{j}]u = −R_{qij}^{p}∂_{p}u,  (2.5)

where R is the curvature tensor. In the last formula u is a scalar. In the case that Σ = π_{1}^{−1} (Γ) we will use the same notations ∇^{∥}, ∂ as in the case of maps.

3 The geodesic X-ray transform

In this section we study the solvability of the equation I^{∗}w = h and prove Theorem 1.4.

Lemma 3.1. Let V be an open set of a Riemannian manifold (M, g). We can define the ray transform as before. Then the normal operator I^{∗}I is an elliptic pseudodifferential operator of order −1 on V with principal symbol c_{n} |ξ|^{−1} where c_{n} is a constant.

Proof. It is easy to see, that

(I^{∗}If) (x) = ∫_{Ω_{x}} dΩ_{x} ∫_{−τ(x,−ξ)}^{τ(x,ξ)} f (γ (x, ξ, t)) dt = 2 ∫_{Ω_{x}} dΩ_{x} ∫_{0}^{τ(x,ξ)}f (γ (x, ξ, t)) dt.  (3.1)

Before we continue we make a remark concerning notation. We have used up to know the notation γ(x, ξ, t) for a geodesic. But it is known [J] , that a geodesic depends smoothly on the point x and vector ξt ∈ T_{x}(M). Therefore in what follows we will also use sometimes the notation γ(x, ξt) for a geodesic. Since the manifold M is simple and any small enough neighborhood U (in (S, g)) is also simple (an open domain is simple if its closure is simple). For any point x ∈ U there is an open domain D_{x}^{U} ⊂ T_{x} (U) such that exponential map exp_{x} : D_{x}^{U} → U, exp_{x}η = γ(x, η) is a diffeomorphism onto U. Let D_{x}, x ∈ M be the inverse image of M, then exp_{x}(D_{x}) = M and exp_{x}|_{D_{x}} : D_{x} → M is a diffeomorphism.

Now we change variables in (3.1), y = γ(x, ξt). Then t = d_{g} (x, y) and

(I^{∗}If) (x) = ∫_{M} K (x, y) f (y) dy,

where

K (x, y) = 2\frac{det (exp_{x}^{−1})′ (x, y) \sqrt{det g (x)}}{d_{g}^{n−1} (x, y)}.

Notice, that since

γ(x, η) = x + η + O(|η|^{2}),  (3.2)

it follows, that the Jacobian matrix of the exponential map is 1 at 0, and then det(exp_{x}^{−1′})(x, x) = 1/ det (exp_{x})′ (x, 0) = 1. ¿From (3.2) we also conclude that

d^{2} (x, y) = G_{ij} (x, y) (x − y)^{i} (x − y)^{j} , G_{ij} (x, x) = g_{ij} (x) , G_{ij} ∈ C^{∞} (M × M)

Therefore the kernel of I^{∗}I can be written in the form

K (x, y) = \frac{2 det (exp_{x}^{−1})′ (x, y) \sqrt{det g (x)}}{(G_{ij} (x, y) (x − y)^{i} (x − y)^{j})^{(n−1)/2}} .

Thus the kernel K has at the diagonal x = y a singularity of type |x − y|^{−n+1} . The kernel

K_{0} (x, y) = \frac{2\sqrt{det g (x)}}{(g_{ij} (x) (x − y)^{i} (x − y)^{j})^{(n−1)/2}}

has the same singularity. Clearly, the difference K −K_{0} has a singularity of type |x − y|^{−n+2} . Therefore the principal symbols of both operators coincide. The principal symbol of the integral operator, corresponding to the kernel K_{0} coincide with its full symbol and is easily calculated. As a result

σ (I^{∗}I) (x, ξ) = 2\sqrt{det g (x)} ∫ \frac{e^{−i(y,ξ)}}{(g_{ij} (x) y^{i}y^{j})^{(n−1)/2}} dy = c_{n} |ξ|^{−1} .

Let r_{M} denotes the restriction from S onto M.

Theorem 3.1. Let U be a simple neighborhood of the simple manifold M. Then for any function h ∈ H^{s} (M) , s ≥ 0 there exists function f ∈ H^{s−1} (U) , r_{M}I^{∗}If = h.

Proof. Let (M, g) be simple and embedded into a compact Riemannian manifold (S, g) without boundary, of the same dimension. Choose a finite atlas of S, which consist of simple open sets U_{k} with coordinate maps κ_{k} : U_{k} → R^{n}. Let {φ_{k}} be the subordinated partition of unity: φ_{k} ≥ 0, suppφ_{k} ⊂ U_{k}, ∑ φ_{k} = 1. We assume without loss of generality that M ⊂ U_{1} and φ_{1}|_{M} = 1. We consider the operators I_{k}, I_{k}^{∗} for the domain U_{k}, and the pseudodifferential operator on (S, g)

P f = ∑_{k} φ_{k} (I_{k}I_{k}^{∗}) (f|_{U_{k}}) , f ∈ D′ (X) .

Every operator I_{k}I_{k}^{∗} : C_{0}^{∞} (U_{k}) → C^{∞} (U_{k}) is an elliptic pseudodifferential operator of order −1 with principal symbol c_{n} |ξ|^{−1} , ξ ∈ T (U_{k}) . Then P is an elliptic pseudodifferential operator with principal symbol c_{n} |ξ|^{−1} , ξ ∈ T (S) and, therefore, is a Fredholm operator from H^{s}(S) into H^{s+1}(S). We have that Ker P has finite dimension, Ran P is closed and has finite codimension. Notice, that P^{∗} = P (more precisely if P^{s} = P : H^{s} (S) → H^{s+1} (S) , then (P_{s})^{∗} = P_{−s−1}).

For arbitrary s ≥ 0 the operator r_{M} : H^{s} (S) → H^{s} (M) is bounded and r_{M}(H^{s} (S)) = H^{s} (M) . Then the range of r_{M}P : H^{s} (S) → H^{s+1} (M) , s ≥ −1 is closed.

Since M is only covered by U_{1} and φ_{1}|_{M} = 1 we have that r_{M}P f = r_{M}I_{1}^{∗}I_{1} (f|_{U_{1}}). Thus, the range of the operator r_{M}I_{1}^{∗}I_{1} : H^{s} (U_{1}) → H^{s+1} (M) , s ≥ −1 is closed. Now to prove the solvability of the equation

r_{M}I_{1}^{∗}I_{1}f = h ∈ H^{s+1} (M) , s ≥ −1

in H^{s} (U_{1}) is sufficient to show that the kernel of the adjoint (r_{M}I_{1}^{∗}I_{1})^{∗} : (H^{(s+1)} (M))^{∗} → (H^{s} (U_{1}))^{∗} is zero.

Let ⟨, ⟩_{M} and ⟨, ⟩ be dualities between H^{s}(M) and (H^{s})^{∗} (M) or H^{s}(S) and H^{−s}(S) respectively. The dual space (H^{s} (M))^{∗} , s ≥ 0 can be identified with the subspace of H^{−s} (S) :

(H^{s} (M))^{∗} = H^{−s} (M) = {u ∈ H^{−s} (S) : suppu ⊂ M} .

For any f ∈ H^{s} (U_{1}) , u ∈ H^{−(1+s)} (M) we have

⟨r_{M}I_{1}^{∗}I_{1}f, u⟩_{M} = ⟨P_{s}f̃, u⟩ = ⟨f̃, P_{−s−1}u⟩,

where f̃ is an arbitrary continuation of f on the manifold S. On the other hand

⟨r_{M}I_{1}^{∗}I_{1}f, u⟩_{M} = ⟨f, (r_{M}I_{1}^{∗}I_{1})^{∗} u⟩_{M} .

Since f̃ is arbitrary, then equality ⟨f̃, P_{−s−1}u⟩ = ⟨f, (r_{M}I_{1}^{∗}I_{1})^{∗} u⟩_{M} implies (r_{M}I_{1}^{∗}I_{1})^{∗} = r_{U_{1}}P_{−s−1} = r_{U_{1}}I_{1}^{∗}I_{1}.

Because of ellipticity the equality r_{U_{1}}P u = 0 implies smoothness u|_{U_{1}}, and then u ∈ H^{−s−1} (M) implies of u ∈ C_{0}^{∞}(U_{1}). Since r_{U_{1}}P u = I_{1}^{∗}I_{1}u, then I_{1}^{∗}I_{1}u = 0 ⟹ I_{1}u = 0 ⟹ u = 0.

Now we are ready to prove Theorem 1.4.

Proof. Let I, I_{1} be the geodesic X-ray transforms on M and U_{1} respectively. From Theorem 3.1 it follows that for any h ∈ C^{∞}(M) there exists f ∈ C^{∞}(\overline{U}), such that r_{M}I_{1}^{∗}I_{1}f = h. Then u^{f} ∈ C^{∞}(Ω(Ṅ)) where Ṅ denotes the interior of N. Let w = 2u_{+}^{f}|_{∂_{+}Ω(M)}, where u_{+}^{f} is the even part with respect to ξ. Then it easy to see, that w_{ψ} = 2u_{+}^{f}|_{Ω(M)} and I^{∗}w = h. The function w ∈ C_{α}^{∞}(∂_{+}Ω(M)) since w_{ψ} ∈ C^{∞}(Ω(M)).

4 Scattering relation and folds

In this section we prove Lemma 1.1.

As indicated before we embed (M, g) into a compact manifold (S, g) with no boundary. Let (N, g) be an arbitrary neighborhood in (S, g) of the manifold (M, g), such that any geodesic γ(x, ξ, t), (x, ξ) ∈ Ω(N) intersects the boundary ∂N transversally. Then the length of the geodesic ray τ is a smooth function on Ω(Ṅ) and the map ψ : ∂Ω(M) → ∂_{+}Ω(N), defined by

ϕ(x, ξ) = φ_{τ(x,ξ)}(x, ξ), (x, ξ) ∈ ∂Ω(M),  (4.1)

is smooth as well. Moreover it turns out ϕ is a fold map with fold ∂_{0}Ω(M). This fact will be proved in the next Theorem. Once this is proven Lemma 1.1 follows from [H] Theorem C.4.4. From the assumption A_{+}w ∈ C^{∞}(∂Ω(M)) we deduce the existence of a smooth function v on a neighborhood of the range ϕ(∂Ω(M)) such that w = v ∘ϕ. Consider function w_{ψ} = w ∘ α ∘ ψ. Change notation ψ to ψ_{M}, keeping w_{ψ}. Denote by ψ_{N} the map, analogical to ψ_{M},

ψ_{N} (x, ξ) = φ_{τ(x,ξ)} (x, ξ) , (x, ξ) ∈ Ω (N) .

Then w_{ψ} = v ∘ ϕ ∘ α ∘ ψ_{M}. It easy to see, that ϕ ∘ α ∘ ψ_{M} = ψ_{N}|_{Ω(M)}. Since the map ψ_{N} is smooth on Ω (M) , then w_{ψ} ∈ C^{∞} (Ω (M)), i.e. w ∈ C_{α}^{∞}(∂_{+}Ω(M)). Thus Lemma 1.1 is proven once we show that ϕ is a fold.

Theorem 4.1. Let (M, g) be a strictly convex, non-trapping manifold and N an arbitrary neighborhood of M, such, that any geodesic γ(x, ξ, t), (x, ξ) ∈ Ω(Ṅ) intersects the boundary ∂N transversally. Then the map ϕ, defined by (4.1) is a fold with fold ∂_{0}Ω(M).

First we recall the definition of a Whitney fold.

Definition 4.1. Let M, N be C^{∞} manifolds of the same dimension and let f : M ⟶ N be a C^{∞} map with f(m) = n. The function f is a Whitney fold (with fold L) at m if f drops rank by one simply at m, so that {x; df(x) is singular } is a smooth hypersurface near m and ker(df(m)) is transverse to T_{m}L.

We rewrite the definition of a fold below in a form that we will use.

Let γ : (a, b) → M, 0 ∈ (a, b) be an arbitrary curve in M. Then f induces the curve γ_{1} : (a, b) → N in the manifold N, γ_{1} = f ∘ γ. The derivative f′(m) of the map f at the point m maps the tangent vector ξ = γ̇(0) ∈ T_{m}(M) into the tangent vector η = γ̇_{1}(0) = f′(m)ξ ∈ T_{f(m}(N). The (acceleration) vectors γ̈(0), γ̈_{1}(0) belong to the spaces T_{ξ}(T_{m}(M)) and T_{η}(T_{f(m)}(N)) respectively, which are tangent to the vector spaces T_{m}(M) and T_{f(m)}(N) and may be identified with them. Fix a point m ∈ M and nonzero vector ξ ∈ Kerf′(m). Let us consider the map L_{ξ} : T_{m} → T_{f(m)}, L_{ξ}(γ̈(0)) = γ̈_{1}(0). A calculation in local coordinates gives that the range of the map L_{ξ} is an affine subspaces in T_{f(m)}(N), RanL_{ξ} = a(ξ) + Ranf′(m), where a(ξ) is a vector, depending on ξ. Moreover the function

Hessf(m)(ξ, Y) = ⟨L_{ξ}(X), Y ⟩, Y ∈ (Ranf′(m))^{⊥}

does not depend on X. Here ⟨, ⟩ corresponds to the duality of the spaces T_{n}(N) and T_{n}^{∗}(N). It follows immediately from the that the definition of a fold is equivalent to

Definition 4.2. The map f is a fold at the point m if

(i) dimKer f′(m) = 1.

(ii) Hessf(m)(ξ, Y) ≠ 0.

Remark 4.1. Condition 4.2 (i) implies that dimCoker f′(m) = 1 and Ran L_{ξ} is an hyperplane. and condition 4.2(ii) implies that the hyperplane RanL_{ξ} does not pass trough the origin.

The following proposition is immediate.

Proposition 4.1. Let f : M → N be a smooth map, dim M = dim N. Let e : M_{0} → M be an hypersurface and the map f ∘ e : M_{0} → f (M_{0}) is non-singular. Then dim Ker f′ (x) ≤ 1 for any point x ∈ M_{0}.

Now we start proving Theorem 4.1.

Proof. We first prove Condition 4.2(i). The smooth map ϕ|_{∂_{0}Ω(M)} : ∂_{0}Ω(M) → ϕ(∂_{0}Ω(M)) is bijective and therefore dimKer ϕ′(x_{0}, ξ_{0}) ≤ 1 for any point (x_{0}, ξ_{0}) ∈ ∂_{0}Ω(M). We prove that the vector X_{0} ∈ T_{(x_{0},ξ_{0})}(∂Ω(M)), X_{0h} = ξ_{0}, X_{0v} = 0, where X_{0h}, X_{0v} are its horizontal and vertical components (see section 2) belongs to Ker ϕ′(x_{0}, ξ_{0}). Represent ϕ = ψ_{N} ∘ e, where e : ∂Ω(M) → Ω(N) is the embedding. Then ϕ′(x, ξ)X = ψ′_{N}(x, ξ)X (we identify here the vector X with its embedding e′(x, ξ)X ∈ T_{(x,ξ)}(∂Ω(M))). Then X_{0} ∈ Ker ϕ′(x_{0}, ξ_{0}) iff X_{0} ∈ T_{(x_{0},ξ_{0})}(∂Ω(M)) ⋂ Kerψ′_{N}(x_{0}, ξ_{0}). The fact that X_{0} ∈ T_{(x_{0},ξ_{0})} (∂Ω (M)) follows from equality (ν (x_{0}) , ξ_{0}) = 0 (see (2.2)). The equality ψ′_{N}(x_{0}, ξ_{0})X_{0} = (∇ψ_{N}(x_{0}, ξ_{0}), ξ_{0}) = 0 is obvious, since for any point (x, ξ) ∈ Ω(N) the point ψ_{N}(x, ξ) is left fixed by geodesic flow and (∇ψ_{N}(x, ξ), ξ) = Hψ_{N}(x, ξ) = 0. Thus, the vector X_{0} = (ξ_{0}, 0) ∈ Ker ϕ′(x_{0}, ξ_{0}) and the first condition in the definition of a fold has been verified.

Let us prove the second condition in the definition of a fold by contradiction. Assume that Hess ϕ(x_{0}, ξ_{0})(X_{0}, Y^{0}) = 0, where Y^{0} ∈ Ker(ϕ′(x_{0}, ξ_{0}))^{∗}. We will show then Y^{0} = 0. Denote (y, η) = ϕ(x, ξ). Then in local coordinates we have for the covector Y^{0} = (Y_{1}^{0}, ..., Y_{2n}^{0}):

Y_{i}^{0}∇_{k}^{∥}y^{i}(x_{0}, ξ_{0}) + Y_{n+i}^{0}∇_{k}^{∥}η^{i}(x_{0}, ξ_{0}) = 0,  (4.2)
Y_{i}^{0}∂_{k}y^{i}(x_{0}, ξ_{0}) + Y_{n+i}^{0}∂_{k}η^{i}(x_{0}, ξ_{0}) = 0.  (4.3)

We write down these equations using the horizontal and vertical parts of the covector Y^{0} (see section 2) and the Jacobi fields A_{(k)} (x_{0}, ξ_{0}, t) , B_{(k)} (x_{0}, ξ_{0}, t) , k = 1, ..., n on the geodesic γ(x_{0}, ξ_{0}, t) :

A_{(k)} = ∇_{k}^{∥}γ + γ̇∇_{k}^{∥}τ, B_{k} = ∂_{k}γ + γ̇∂_{k}τ.

Then the equations (4.2),(4.3) can be written in the form

[(A_{(k)}, Y_{h}^{0}) + (DA_{(k)}, Y_{v}^{0})]|_{t=τ} = 0,
[(B_{(k)}, Y_{h}^{0}) + (DB_{(k)}, Y_{v}^{0})]|_{t=τ} = 0,

where D means covariant derivative along geodesic. Without lost of generality one can take Y_{v}^{0} = Z^{c}|_{t=τ}, Y_{h}^{0} = −DZ^{c}|_{t=τ}, where Z^{c} is some Jacobi field on γ(x_{0}, ξ_{0}, t). It is known that for any Jacobi fields X, Y, the Wronskian {X, Y } = (DX, Y) − (X, DY) is constant. Then we get

{Z^{c}, A_{(k)}}|_{t=0} = 0, {Z^{c}, B_{(k)}}|_{t=0} = 0.

The initial data is given by

A_{(k)}^{j}|_{t=0} = δ_{k}^{j} − ν_{0}^{i}ν_{0k} + ξ_{0}^{j}∇_{k}^{∥}τ, DA_{(k)}|_{t=0} = 0,
B_{(k)}|_{t=0} = ξ_{0}∂_{k}τ, (DB_{(k)})^{j}|_{t=0} = δ_{k}^{j} − ξ_{0}^{j}ξ_{0k},

where ν_{0} = ν(x_{0}). We have

Z^{c}|_{t=0} = aξ_{0} + c∂τ, DZ^{c}|_{t=0} = bν_{0} − c∇^{∥}τ,

where a, b, c are some constants, depending on the point (x_{0}, ξ_{0}). From these general solutions we find one satisfying the condition that Y^{0} ∈ T_{ϕ(x,ξ)}(∂Ω(N)), or the conditions (2.2),

(Y_{h}^{0}, ν_{N}(y)) = −(DZ^{c}|_{t=τ}, ν_{N}(y)) = 0,  (4.4)
(Y_{v}^{0}, η) = (Z^{c}|_{t=τ}, η) = 0,  (4.5)

where ν_{N} is the normal to ∂N. Consider at first the condition (4.5). Using the Jacobi field J = (τ − t)γ̇ it can be rewritten in the form

{Z^{c}, J}|_{t=τ} = {Z^{c}, J}|_{t=0} = 0,

and, therefore, a = −cτ. Thus,

Z^{c}|_{t=0} = c(∂τ − ξ_{0}τ), DZ^{c}|_{t=0} = bν_{0} − c∇^{∥}τ.

Now consider our hypothesis that Hess ϕ_{(x_{0},ξ_{0})}(X_{0}, Y^{0}) = 0. A straightforward calculation shows that in local coordinates we have

Hess ϕ_{(x_{0},ξ_{0})}(X_{0}, Y^{0}) = −B(x_{0}, ξ_{0}){Z, P }|_{t=τ} = −B(x_{0}, ξ_{0}){Z^{c}, P }|_{t=0},

where B = −ξ^{i}ξ^{j}∇_{i}^{∥}ν_{j} is the second fundamental form of the boundary ∂M and

P = (ν, ∇γ) + (ν, ∇τ)γ̇

is a Jacobi field. Since B > 0, our hypothesis gives that {Z, P }|_{t=0} = 0. Because of the initial data P (x_{0}, ξ_{0}, 0) = ν_{0} + (ν_{0}, ∇τ)ξ, DP (x_{0}, ξ_{0}, 0) = 0 we obtain b = −c(ν_{0}, ∇τ) and

Z^{c}|_{t=0} = c(∂τ − ξ_{0}τ), DZ^{c}|_{t=0} = −c∇τ.

Now we prove that for any point (x, ξ) ∈ Ω(Ṅ) the vector field

Z (x, ξ, t) = (∂τ − ξτ) (φ_{t} (x, ξ)) , t ∈ [−τ (x, −ξ) , τ (x, ξ)]  (4.6)

on the geodesic γ (x, ξ, t) is a Jacobi field and

DZ (x, ξ, t) = (∇τ) (φ_{t} (x, ξ)) .  (4.7)

Applying to the equation Hτ = −1 the operator H∂ and using (2.3)-(2.5) we get H^{2}X + RX = 0, (RX)^{i} (x, ξ) = R_{jkl}^{i} (x) ξ^{j}X^{k}ξ^{l}, where the semibasic vector field X = ∂τ − ξτ. The operators H and D are related by:

(HX) ∘ φ_{t} = D(X ∘ φ_{t}),  (4.8)

where X is an arbitrary semibasic vector field. Then (4.6) is a Jacobi field. Since from Hτ = −1 it follows H(∂τ − ξτ) = −∇τ, we obtain (4.7) from (4.8).

Thus, Z^{c}(x_{0}, ξ_{0}, t) = cZ(x_{0}, ξ_{0}, t) and condition (4.4) gives

0 = (DZ|_{t=τ}, ν_{N}(y)) = −c(∇τ(y, η), ν_{N}(y)).  (4.9)

Now we show that

∇τ|_{∂_{−}Ω(N)} = −\frac{ν_{N}}{(ν_{N}, ξ)}.  (4.10)

Let h be a smooth function on N, such that h|_{∂M} = 0, ∇h|_{M} ≠ 0. Then since h(x, ξ, τ(x, ξ)) = 0, (x, ξ) ∈ Ω(N), we have that

ν_{i}^{N}(y)∇γ_{(k)}^{i}(x, ξ, τ(x, ξ)) + (ν_{N} (y) , η) ∇_{k}τ(x, ξ) = 0,

where (y, η) = φ_{τ(x,ξ)}(x, ξ) ∈ ∂_{−}Ω(N). Change here (x, ξ) to φ_{t}(x, ξ), t ∈ [0, τ (x, ξ)] . Then the point (y, η) does not change. Using the initial data ∇γ_{(k)}^{i}(x, ξ, 0) = δ_{k}^{i} we get in the limit ν_{N}(y) + (ν_{N}(y), η)∇τ(y, η) = 0. Thus, (4.10) is proved and we conclude from (4.9) that c = 0.

Thus, Hess ϕ_{(x_{0},ξ_{0})}(X_{0}, Y^{0}) = 0 iff Ker (ϕ′(x_{0}, ξ_{0}))^{∗} = {0}. We have finished the proof of the Theorem.

5 The Hilbert transform and geodesic flow

In this section we prove Theorem 1.5 in the introduction.

Let H be the Hilbert transform as defined in (1.3)

We have that H is a unitary operator in the space L_{0}^{2}(Ω_{x}) = {u ∈ L^{2}(Ω_{x}) : u_{0} = 0},

(u, v) = (Hu, Hv), ∀u, v ∈ L_{0}^{2}(Ω_{x}),
H^{2}(u) = −u, ∀u ∈ L_{0}^{2}(Ω_{x}).

Clearly, all these properties remain the same if we change Ω_{x} to Ω(M).

In order to prove Theorem 1.4 we need the following commutator formula which is valid for Riemannian manifolds of any dimension

Lemma 5.1. Let u be a smooth function on the manifold Ω^{2}(M) = ⋃_{x∈M} Ω_{x}^{2}, Ω_{x}^{2} = {(x, ξ, η) : ξ, η ∈ Ω_{x}}. Then

∇ ∫_{Ω_{x}} u(x, ξ, η)dΩ_{x} (η) = ∫_{Ω_{x}} ∇^{(2)}u(x, ξ, η)dΩ_{x} (η) ,  (5.1)

where ∇^{(2)} under the integral sign in (5.1) denotes the horizontal derivative on Ω^{2}(M),

∇_{j}^{(2)}u(x, ξ, η) = (\frac{∂}{∂x^{j}} − Γ_{jk}^{i}ξ^{k}∂_{i(ξ)} − Γ_{jk}^{i}η^{k}∂_{i(η)})u(x, ξ, η).

Notice that the horizontal tangential derivative can be defined on T (M) × T (M) in a similar fashion to the case of T (M) in section 2.

Proof. Let φ ∈ C_{0}^{∞} (R^{+}) be arbitrary function. We define the function v on T^{2} (M) by

v(x, ξ, η) = φ (|η|) u(x, ξ/ |ξ| , η/ |η|)

Let us consider the integral

S(x, ξ) = ∫_{T_{x}(M)} v(x, ξ, η)dT_{x} (η) .

Identifying T_{x}(M) with R^{n} we have

S(x, ξ) = ∫_{R^{n}} v(x, ξ, η)\sqrt{det g (x)}dη.

Then

∇_{j}S = \frac{∂S}{∂x^{j}} − Γ_{jk}^{i}ξ^{k} \frac{∂S}{∂ξ^{i}} = ∫_{R^{n}} (\frac{∂v}{∂x^{j}} − Γ_{jk}^{i}ξ^{k} \frac{∂v}{∂ξ^{i}})\sqrt{det g}dη + ∫_{R^{n}} v \frac{∂ ln \sqrt{det g (x)}}{∂x^{j}} \sqrt{det g}dη.

Since ∂ ln \sqrt{det g}/dx^{j} = Γ_{jk}^{k} we rewrite the last integral in the form

∫_{R^{n}} v \frac{∂}{∂η^{k}} (Γ_{jl}^{k}η^{l}) \sqrt{det g}dη.

Then

∇_{j}S = ∫_{R^{n}} (\frac{∂v}{∂x^{j}} − Γ_{jk}^{i}ξ^{k} \frac{∂v}{∂ξ^{i}} − Γ_{jl}^{k}η^{l} \frac{∂v}{∂η^{k}})\sqrt{det g}dη.

Since

(\frac{∂}{∂x^{j}} − Γ_{jk}^{i}ξ^{k} \frac{∂}{∂ξ^{i}} − Γ_{jl}^{k}η^{l} \frac{∂}{∂η^{k}}) |η| = 0,

then after changing to spherical coordinates we obtain

∇S(x, ξ) = ∫_{0}^{∞}φ (t) t^{n−1}dt ∫_{Ω_{x}} ∇u(x, ξ, η)dΩ_{x} (η) .  (5.2)

Now S in spherical coordinates is given by

S(x, ξ) = ∫_{0}^{∞}φ (t) t^{n−1}dt ∫_{Ω_{x}} u(x, ξ, η)dΩ_{x} (η)  (5.3)

We conclude (5.1) using (5.2),(5.3).

Now we prove Theorem 1.4. Proof. A straightforward calculation gives that

∇\frac{1 + (ξ, η)}{(ξ_{⊥}, η)} = 0

and therefore we have

∇Hu(x, ξ) = \frac{1}{2π} ∫_{Ω_{x}} \frac{1 + (ξ, η)}{(ξ_{⊥}, η)}∇u(x, η)dΩ_{x} (η) .

For any pair vectors ξ, η ∈ Ω_{x} we have

η = (ξ, η)ξ + (ξ_{⊥}, η)ξ_{⊥}, η_{⊥} = −(ξ_{⊥}, η)ξ + (ξ, η)ξ_{⊥}, (ξ, η)^{2} + (ξ_{⊥}, η)^{2} = 1.

Then

η \frac{1 + (ξ, η)}{(ξ_{⊥}, η)} = ξ \frac{(ξ, η) + (ξ, η)^{2}}{(ξ_{⊥}, η)} + ξ_{⊥}(1 + (ξ, η)) =

= ξ \frac{(ξ, η) + 1}{(ξ_{⊥}, η)} − ξ (ξ_{⊥}, η) + ξ_{⊥}(ξ, η) + ξ_{⊥} = ξ \frac{1 + (ξ, η)}{(ξ_{⊥}, η)} + ξ_{⊥} + η_{⊥}.

Thus

HHu = HHu + H_{⊥}u_{0} + (H_{⊥}u)_{0}

and Theorem 1.4 is proved.

References

[An] Yu. E. Anikonov, Some Methods for the Study of Multidimensional Inverse Problems , Nauka, Sibirsk Otdel., Novosibirsk (1978).

[BCG] G. Besson, G. Courtois, and S. Gallot Entropies et rigidités des espaces localement symétriques de courbure strictment négative, Geom. Funct. Anal., 5(1995), 731-799.

[BG] I.N. Bernstein and M.L. Gerver, Conditions on distinguishability of metrics by hodographs. Methods and Algorithms of Interpretation of Seismological Information, Computerized Seismology 13, Nauka, Moscow, 50–73 (in Russian.)

[B] G. Beylkin, Stability and uniqueness of the solution of the inverse kinematic problem in the multidimensional case, J. Soviet Math. 21(1983), 251–254.

[C] C. Croke, Rigidity and the distance between boundary points, J. Differential Geom., 33(1991), no. 2, 445–464.

[C1] C. Croke, Rigidity for surfaces of non-positive curvature, Comment. Math. Helv., 65(1990), 150-169.

[E] H. Eliasson, Geometry of manifolds of maps, J. Diff. Geom., 1(1967), 169-194.

[GN] M. L. Gerver and N. S. Nadirashvili, An isometricity conditions for Riemannian metrics in a disk, Soviet Math. Dokl. 29 (1984), 199–203.

[Gr] M. Gromov, Filling Riemannian manifolds, J. Differential Geometry 33(1991), 445–464.

[G] V. Guillemin, Sojourn times and asymptotic properties of the scattering matrix. Proceedings of the Oji Seminar on Algebraic Analysis and the RIMS Symposium on Algebraic Analysis (Kyoto Univ., Kyoto, 1976). Publ. Res. Inst. Math. Sci. 12(1976/77), supplement, 69–88.

[H] L. Hörmander, The Analysis of liner partial differential operators III, Springer-Verlag, Berlin-Heildelberg-New York-Tokyo, 1985.

[K] W. Klingenberg, Riemannian geometry, second edition, de Gruyter Studies in Mathematics, Berlin-New York, 1995.

[J] J. Jost, Riemannian geometry and geometric analysis, Third edition, Universitext, Springer, 2002.

[LaU] M. Lassas and G. Uhlmann, On determining a Riemannian manifold from the Dirichlet-to-Neumann map, Annales Scientifiques de L’ Ecole Normale Superieure, 34(2001), 771-787.

[LeU] J. Lee and G. Uhlmann, Determining anisotropic real-analytic conductivities by boundary measurements, Comm. Pure Appl. Math., 42(1989), 1097-1112.

[M] R. Michel, Sur la rigidité imposée par la longueur des géodésiques, Invent. Math. 65(1981), 71-83.

[Mu] R. G. Mukhometov, The reconstruction problem of a two-dimensional Riemannian metric, and integral geometry (Russian), Dokl. Akad. Nauk SSSR 232(1977), no. 1, 32–35.

[Mu1] R. G. Mukhometov, On one problem of reconstruction of Riemannian metric (Russian), Siberian Math. Journal 22(1981), no. 3, 119–135.

[MR] R. G. Mukhometov and V. G. Romanov, On the problem of finding an isotropic Riemannian metric in an n-dimensional space (Russian), Dokl. Akad. Nauk SSSR 243(1978), no. 1, 41–44.

[O] J. P. Otal, Sur les longuer des géodésiques d’une métrique a courbure négative dans le disque, Comment. Math. Helv. 65(1990), 334–347.

[R] V. G. Romanov, Inverse Problems of Mathematical Physics, VNU Science Press, Utrech, the Netherlands, 1987.

[Sh] V. A. Sharafutdinov, Integral geometry of tensor fields, VSP, Utrech, the Netherlands (1994).