Reconstructions from boundary measurements on admissible manifolds

We prove that a potential $q$ can be reconstructed from the Dirichlet-to-Neumann map for the Schrodinger operator $-\Delta_g + q$ in a fixed admissible 3-dimensional Riemannian manifold $(M,g)$. We also show that an admissible metric $g$ in a fixed conformal class can be constructed from the Dirichlet-to-Neumann map for $\Delta_g$. This is a constructive version of earlier uniqueness results by Dos Santos Ferreira et al. on admissible manifolds, and extends the reconstruction procedure of Nachman in Euclidean space. The main points are the derivation of a boundary integral equation characterizing the boundary values of complex geometrical optics solutions, and the development of associated layer potentials adapted to a cylindrical geometry.


Introduction
This paper is concerned with the problem of reconstructing material parameters of a medium from boundary measurements. A typical question of this type is Calderón's inverse conductivity problem [6], which consists in recovering the conductivity of a body from voltage to current measurements at the boundary. For bounded domains in Euclidean space in dimensions n ≥ 3, it was proved in [30] that a smooth positive scalar conductivity σ is uniquely determined by the Dirichlet-to-Neumann map (DN map) Λ σ representing the boundary measurements. This uniqueness result was then extended to a reconstruction procedure in [21] and independently in [24], see also [13]. In two dimensions, uniqueness and reconstruction for this problem was proved even for bounded measurable conductivities in [2], [3].
In this paper we consider Calderón's inverse problem and related questions in anisotropic media, where the conductivity depends on direction. This corresponds to replacing the scalar conductivity σ by a smooth symmetric positive definite matrix. The question then is to recover the matrix σ from the DN map Λ σ , up to the natural obstruction given by diffeomorphisms which fix the boundary. If n = 2, it is proved in [1] that any bounded measurable matrix conductivity σ is determined by Λ σ up to diffeomorphism. For constructive results see [15] and the references therein.
In three and higher dimensions the anisotropic Calderón problem is open even for smooth matrix conductivities. We refer to [7] for a more thorough discussion and references to known results. It was observed in [19] that the anisotropic Calderón problem is closely related to certain inverse problems for the Laplace-Beltrami operator on a Riemannian manifold, which we set out to define. Statement of main results. Let (M, g) be a compact oriented Riemannian manifold with C ∞ boundary, and let ∆ g be the Laplace-Beltrami operator. In local coordinates ∆ g u = |g| −1/2 n j,k=1 ∂ ∂x j |g| 1/2 g jk ∂u ∂x k where g = (g jk ) is the metric in local coordinates, (g jk ) is the inverse matrix of (g jk ), and |g| = det(g jk ). Consider the Dirichlet problem For any f ∈ H 3/2 (∂M ) there is a unique solution u ∈ H 2 (M ), and the DN map is defined by where the normal derivative is given by g jk ∂u ∂x j ν k .
Here ν k = n k,l=1 g kl ν l , and (ν 1 , . . . , ν n ) is the coordinate expression for the unit outer normal vector ν on ∂M .
Our first result states that the map Λ g constructively determines g within a known conformal class of admissible metrics (as defined below). The next question concerns an inverse problem for the Schrödinger equation in (M, g). If q is a smooth function on M , we consider the Dirichlet problem (−∆ g + q)u = 0 in M, u = f on ∂M. We make the standing assumption that 0 is not a Dirichlet eigenvalue of −∆ g + q in M . This means that for any f ∈ H 3/2 (∂M ) the equation has a unique solution u ∈ H 2 (M ), and the DN map can be defined by The second main result is as follows.
Theorem 1.2. Let (M, g) be a given admissible 3-dimensional manifold. If q is a smooth function on M , then from the knowledge of Λ g,q one can constructively determine q.
To complete the statement of the main results, let us give the definition of admissible manifolds. These arose in [7] as the first class of manifolds beyond real-analytic or Einstein ones for which one can prove uniqueness results for the anisotropic inverse problems described above.

Definition.
A compact oriented Riemannian manifold (M, g) with smooth boundary is admissible if dim(M ) ≥ 3 and if (M, g) ⊂⊂ (T int , g) where T = R × M 0 is a cylinder with metric g = c(e ⊕ g 0 ) (here c is a smooth positive function and e is the Euclidean metric on R), and (M 0 , g 0 ) is an (n − 1)-dimensional simple manifold.
Thus, up to a conformal factor, an admissible manifold is embedded in a cylinder (R × M 0 , e ⊕ g 0 ) and therefore has a Euclidean direction. The transversal manifold (M 0 , g 0 ) needs further to be simple: Definition. A compact manifold (M 0 , g 0 ) with boundary is simple if for any p ∈ M 0 the exponential map exp p is a diffeomorphism from its maximal domain in T p M 0 onto M 0 , and if ∂M 0 is strictly convex (meaning that the second fundamental form is positive definite).
Examples of admissible manifolds include subdomains of the model spaces (Euclidean space, sphere minus a point, hyperbolic space), sufficently small subdomains of any conformally flat manifold, and domains in R n equipped with a metric of the form where c is a positive smooth function and where g 0 is a simple metric in the x ′ variables.
The unique determination results corresponding to Theorems 1.1 and 1.2 were proved in [7], where also earlier work is discussed and further references are given. There are several recent results that are concerned with the twodimensional case. If M is a domain in R 2 , [5] proved the uniqueness result corresponding to Theorem 1.2 and briefly discussed a reconstruction procedure. A proof with constructive character was given in [11] for arbitrary Riemann surfaces (M, g) with boundary (also for the magnetic Schrödinger operator), based on earlier nonconstructive proofs in [10], [17]. Various constructive results for the two-dimensional case, also on Riemann surfaces, appear in [12], [14], [15], [16], and for the three-dimensional case an improved reconstruction result is given in [25].
Outline of proof. As mentioned, the uniqueness results corresponding to Theorems 1.1 and 1.2 were proved in [7], but the proofs were not constructive. The main point of the present paper is to give constructive proofs, following the well-known reconstruction procedure of Nachman [21] in the case where M is a bounded domain in R n and g is the Euclidean metric. We do not make any claims about the practicality of the reconstruction procedure, but we do prove that all the steps in the corresponding uniqueness proofs in [7] can be carried out in a constructive way.
The argument in [21] involves complex geometrical optics (CGO) solutions u = e −ζ·x (1+ r) to the Schrödinger equation (−∆ + q)u = 0 in R n , and relies in a crucial way on a uniqueness notion for these solutions upon fixing decay at infinity. One has several equivalent ways of characterizing these solutions, and in particular it is possible to recover the boundary value u| ∂M as the unique solution to a boundary integral equation on ∂M involving Λ g,q and other known quantities.
CGO solutions on admissible Riemannian manifolds were constructed in [7] by Carleman estimates. The solutions were given in a compact manifold, and the construction did not involve a notion of uniqueness. The paper [18] introduced a direct Fourier analytic construction of CGO solutions, valid in the cylinder T and with a uniqueness notion obtained by fixing a decay condition in the Euclidean variable and Dirichlet boundary values on ∂T . We shall use the solutions constructed in [18] to prove Theorem 1.2 (which implies Theorem 1.1 after a simple reduction).
We next sketch the proof of Theorem 1.2. Let (M, g) ⊂⊂ (T int , g) be an admissible manifold, and assume that g = e ⊕ g 0 where (M 0 , g 0 ) is simple. Here we suppose, for simplicity, that c = 1. We assume that Λ g,q and (M, g) are known (thus also Λ g,0 is known), and use the basic integral identity which is valid for any solutions u, v ∈ H 2 (M ) of (−∆ g + q)u = 0 and −∆ g v = 0 in M . We take u and v to be suitable CGO solutions such that u| ∂M may be obtained from Λ g,q as the unique solution of a boundary integral equation, and v| ∂M is explicitly given. Then the left hand side of (1.1) is known. Taking the limit as τ → ∞ and varying certain parameters in the solutions, we recover in this way the quantity for any θ ∈ S n−2 , where λ is any nonzero real number and (r, θ) are polar normal coordinates in M 0 with center on ∂M 0 . We have extended q into T M as a function in C ∞ c (T int ). Now, since (r, θ) are polar normal coordinates, the curves γ : r → (r, θ) are unit speed geodesics in (M 0 , g 0 ) for any fixed θ. Denoting the quantity in brackets in (1.2) by f λ (r, θ), it follows that we have recovered γ e −2λr f λ (γ(r)) dr for any maximal geodesic γ going from ∂M 0 into M 0 , and for any λ = 0. This is the attenuated geodesic ray transform of f λ with constant attenuation −2λ, see [7], [29]. For any λ such that this transform can be inverted, we recover f λ which is just the one-dimensional Fourier transform It was proved in [7] and [8] that the attenuated ray transform is invertible for small |λ|, thus giving information on F x 1 {q( · , x ′ )} for small frequencies. This determines the compactly supported function q( · , x ′ ) uniquely by the Paley-Wiener theorem. To make this step more constructive one would like to invert the attenuated ray transform for all λ, which would yield q( · , x ′ ) by taking the inverse Fourier transform. Up to now the argument has been valid for dim(M ) ≥ 3. However, if dim(M ) = 3 then (M 0 , g 0 ) is a 2D simple manifold, and the recent result [28] shows that the attenuated ray transform is invertible for any attenuation. We can use the inversion procedure in [28] to conclude the reconstruction of q from Λ g,q if (M, g) is 3-dimensional.
Here u 0 is an explicit function on ∂M depending on various parameters, γ is the trace operator H 2 (M ) → H 3/2 (∂M ), and S τ is a special single layer potential depending on a large parameter τ and adapted to the CGO solutions and the geometry of the cylinder (T, g) = (R × M 0 , e ⊕ g 0 ). In fact, we have where the integral kernel K τ (x, y) is explicitly determined by τ and by the Dirichlet eigenvalues and eigenfunctions of the Laplace-Beltrami operator on (M 0 , g 0 ). We establish basic properties of the single layer operator in Section 2. We prove that for suitable choices of u 0 and for |τ | large, the equation (1.3) has a unique solution f ∈ H 3/2 (∂M ), and one has f = u| ∂M where u is the corresponding CGO solution. Since the operator on the left hand side of (1.3) is determined by the boundary measurements and since u 0 is explicit, we can indeed determine the boundary values of CGO solutions by solving this Fredholm integral equation.
This approach is analogous to [21] which considers the Euclidean case, except that the uniqueness notion for CGO solutions is obtained from decay conditions and Dirichlet boundary values on the cylinder (T, g) instead of a decay condition at infinity in R n . Recently in [23] another constructive approach appeared. There the boundary integral equation is obtained via Carleman estimates in M , and no extension of M to a larger set is needed. It is presumable that a similar approach would work in our case. However, the single layer potential obtained from Carleman estimates is perhaps not so explicit as the operator S τ introduced above.
We remark here that it would be interesting to establish reconstruction results corresponding to Theorems 1.1 and 1.2 for the magnetic Schrödinger equation or for the time-harmonic Maxwell equations. Uniqueness results for these equations are proved in [7], [18], and constructive results in the Euclidean case appear in [26], [27].
Structure of paper. Section 1 is the introduction. The basic properties of the single layer operator S τ and related Faddeev Green functions are considered in Section 2. In Section 3 we introduce several equivalent ways of characterizing CGO solutions, including the required boundary integral equation. The results in Sections 2 and 3 are in fact valid for any transversal manifold (M 0 , g 0 ) with smooth boundary (not necessarily simple). The proofs of Theorems 1.1 and 1.2 are given in Section 4.
Acknowledgements. C.E.K. is partly supported by the NSF grant DMS-0968472. M.S. is supported in part by the Academy of Finland. G.U. is partly supported by NSF, a Chancellor Professorship at UC Berkeley and a Senior Clay Award.

Boundary layer potentials
Notation and function spaces. In this section we assume that (M, g) is a compact manifold with smooth boundary, having dimension n ≥ 3, and that (M, g) ⊂⊂ (T int , g) where T = R × M 0 and g = e ⊕ g 0 , and (M 0 , g 0 ) is any compact (n − 1)-dimensional manifold with boundary (no restrictions on the metric g 0 ). Points of T are written as x = (x 1 , x ′ ) where x 1 is the Euclidean variable and x ′ is a point in the transversal manifold M 0 .
We write · , · for the inner product of tangent vectors, 1-forms, and other tensors, and | · | for the norm. The volume element in (T, g) is with dV g 0 the volume element in (M 0 , g 0 ). We also write Γ = ∂M ⊂⊂ T , and denote by dS the volume element on Γ.
Let L 2 (T ) = L 2 (T, dV ) be the standard L 2 space in T , and let H s (T ) be the corresponding L 2 Sobolev spaces. Since M 0 is compact, we define We define, in the L 2 (T ) duality, H −1 (T ) = (H 1 0 (T )) * . On Γ = ∂M we consider the usual space L 2 (Γ) = L 2 (Γ, dS) and the corresponding Sobolev spaces H s (Γ).
The Parseval identity implies that Standard single layer operator. Let K 0 be the usual inverse of the Dirichlet Laplacian on T , defined as follows: since any u ∈ H 1 0 (T ) satisfies the Poincaré inequality We also consider the trace operator which restricts functions to Γ, Definition. The standard single layer operator on T is the map The next result gives the basic jump and mapping properties of S 0 . Here we write M − = M int , M + = T M , and γ ∓ u = (u| M ∓ )| Γ for the restriction of u to Γ from the interior or exterior. If u is a function on T Γ such that u| M ∓ are H 1 in M ∓ and satisfy −∆ g u = 0 in M ∓ , we define the normal derivatives from the interior or exterior weakly as elements of H −1/2 (Γ) by where e h ∈ H 1 (T ) is any function with e h | Γ = h and e h | ∂T = 0. The jumps on Γ are defined by Proof. Harmonicity and the first jump property are direct consequences of the properties of K 0 . The definitions also imply that for h ∈ H 1/2 (Γ), This shows the second jump property. If f ∈ H k+1/2 (Γ) with k ≥ 0, then u satisfies The transmission property [20,Theorem 4.20] implies that u| M ± are H k+2 near Γ. Then the properties for u follow from standard interior and boundary regularity for −∆ g and from interpolation.
Corollary 2.2. The trace single layer potential satisfies for s ≥ −1/2 , the Schwartz kernel theorem shows that there is a distributional integral kernel K 0 (x, y). (We restrict to T int to avoid having to talk about distributions on manifolds with boundary.) Then formally We will need some basic properties of the integral kernel. These are easily obtained by comparing to the Green function on compact manifolds [4], [31].
Lemma 2.3. The kernel K 0 is smooth in T int ×T int away from the diagonal, and it satisfies (with d the Riemannian distance) Also, K 0 (x, y) = K 0 (y, x), and for any x ∈ T int one has ∆ g (K 0 (x, · )) = 0 in T int {x}.
Let U ⊂⊂ W ⊂⊂ T int where U is open and (W , g) is a compact manifold with smooth boundary, and let G(x, y) be the Dirichlet Green function for the Laplacian in (W , g) [4, Section 4.2]. We will prove that U). Since the function G(x, y) has the stated properties by [4,Theorem 4.17], they also follow for K 0 by using smoothness of R and simple arguments.
Consider ϕ ∈ C ∞ c (U ) and let v = u 0 − u 1 where u 0 = K 0 ϕ and , and ∆ g v = 0 in W . By elliptic regularity and Sobolev embedding,

This may be rewritten as
τ -dependent single layer potential. In [18] (see also Proposition 3.1 below) it is shown that for any τ ∈ R with |τ | ≥ 1 and τ 2 / ∈ Spec(−∆ g 0 ), and given δ > 1/2, one has a bounded linear operator . If τ is as above, we define another operator From the properties of G τ , it immediately follows that K τ is an inverse for the Laplacian: one has −∆ g K τ = Id on L 2 c (T ). The map K τ is a Faddeev type Green operator on T . It differs from the standard Green operator K 0 by having exponential factors in its integral kernel. Also, K τ − K 0 maps L 2 c (T ) into C ∞ (T ) since −∆ g (K τ − K 0 )f = 0 for all f ∈ L 2 c (T ). This suggests the following result. We will do the proof with some care to ensure that the boundary ∂T does not pose a problem.
Proof. We prove the result for τ ≥ 1, the case τ ≤ −1 being similar. The expression for G τ in [18,Proposition 4.1] shows that for any f ∈ L 2 δ (T ) with δ > 1/2 we have This implies that for f ∈ L 2 c (T ) one has with convergence in L 2 loc (T ). The second identity follows by solving the equation −∆ g u = f in T , f ∈ L 2 (T ), u| ∂T = 0, by taking the Fourier transform in x 1 and expanding in terms of Dirichlet eigenfunctions in x ′ . Note that K 0 is obtained from K τ by setting τ = 0 formally. We dη.

It follows that
We may interpret the right hand side as a limit of integrals over closed rectangular contours since F (±R + it) where 0 ≤ t ≤ τ decays as R → ∞. Now F (z) is analytic in C {±ia} with simple poles at ±ia, so (2.1) follows from the residue theorem.
Since R τ u| ∂T = 0 for any u, the preceding result implies that Definition. If |τ | ≥ 1 and τ 2 / ∈ Spec(−∆ g 0 ), we define the τ -dependent single layer potential loc,0 (T ). The basic properties of S τ follow immediately from the previous results.

Boundary integral equation
We now describe four equivalent problems for characterizing the CGO solutions u: a differential equation (DE), integral equation (IE), exterior problem (EP), and boundary integral equation (BE). In this section we assume that (M, g) ⊂⊂ (T int , g) is a compact manifold with smooth boundary and T = R×M 0 , g = e⊕g 0 , and (M 0 , g 0 ) is any compact (n−1)-dimensional manifold with smooth boundary. We use the notations in Section 2.
As a first step, we quote the basic existence and uniqueness result concerning G τ from [18, Proposition 4.3 and subsequent remark].
Proposition 3.1. Let δ > 1/2 and λ = 0, and suppose that q ∈ L ∞ (T ) is compactly supported. There exists τ 0 ≥ 1 (if q = 0 then τ 0 = 1) such that whenever |τ | ≥ τ 0 and τ 2 / ∈ Spec(−∆ g 0 ), then for any Finally, w satisfies the estimates In the following result we extend q ∈ L ∞ (M ) by zero into T , and let τ 0 be as in Proposition 3.1. The result considers CGO solutions of the form where u 0 is any harmonic function in H 2 loc (T ), and r ∈ H 1 −∞,0 (T ). We will only fix the choice of the free solution u 0 in the next section.
Proposition 3.2. Let q ∈ L ∞ (M ) be such that 0 is not a Dirichlet eigenvalue of −∆ g + q in M , and let |τ | ≥ τ 0 and τ 2 / ∈ Spec(−∆ g 0 ). Further, let u 0 ∈ H 2 loc (T ) be such that ∆ g u 0 = 0 in T . Consider the following problems: Each of these problems has a unique solution. The right hand side is in L 2 c (T ), so by Proposition 3.1 there is a unique solution r ∈ H 1 −∞,0 (T ). This proves that (DE) has a unique solution. It remains to prove that all four problems are equivalent in the sense described above.
Since q is compactly supported in T also v = −q(r + e τ x 1 u 0 ) is compactly supported. Thus we may apply G τ to both sides of the last identity to obtain r + G τ (qr) = −G τ (e τ x 1 qu 0 ).
Multiplying by e −τ x 1 and adding u 0 to both sides gives (IE).
(DE) =⇒ (EP): Letũ solve (DE), and define u =ũ| T M . Clearly properties i), ii) and iii) of (EP) are valid. We need to show iv). Sinceũ solves the equation (−∆ g + q)ũ = 0 in M , we have , and let f = u| Γ . We fix a point x ∈ T int M and let v(y) = K τ (x, y) where y ∈ M . This is a smooth function in M by Lemma 2.5. Now Green's theorem implies By (DE) we have ∆ g u = qu and ∂ ν u| Γ = Λ g,q f . Using the properties in Lemma 2.5 we obtain which is valid for x ∈ T int M . The function v is harmonic in M , hence ∂ ν v| Γ = Λ g,0 (v| Γ ). The symmetry of Λ g,0 implies We obtain 2) Adding u to both sides, using the fact that u solves (IE), and taking traces on Γ gives (BE).
To prove (EP) iii) it is sufficient to show that for any h ∈ H 1/2 (Γ), Formally one has e τ x 1 S τ h = G τ e τ x 1 γ * h where G τ maps L 2 c (T ) to H 1 −∞,0 (T ). However, we have not proved that G τ has good mapping properties on negative order Sobolev spaces. Thus, the proof proceeds differently and involves an extension w of e τ x 1 S τ h| T M into T such that w = G τ ψ for some ψ ∈ L 2 c (T ). This will imply (3.3) by the mapping properties of G τ . Define is a function chosen so that e −τ x 1 w ∈ H 2 loc (T ). Clearly we need that F | Γ = 0, and since . We can take F to be any function in H 2 (M ) with this Cauchy data, and then e −τ x 1 w and also w is in H 2 loc (T ). We now observe that Since w ∈ H 2 loc (T ) this implies that e τ x 1 (−∆ g )e −τ x 1 w = ψ where ψ ∈ L 2 c (T ). Consequently w = G τ ψ ∈ H 1 −∞,0 (T ) and we have proved (3.3).
Finally, let us verify that the boundary integral equation (BE) in Proposition 3.2 is indeed Fredholm. Proof. Let f ∈ H 3/2 (Γ), and let u = P q f where P q : The exact same argument leading to (3.2) in the proof of Proposition 3.2 shows that where E : L 2 (M ) → L 2 (T ) is extension by zero and J : H 2 (M ) → L 2 (M ) is the natural inclusion. Taking traces on Γ, we obtain the factorization γS τ (Λ g,q − Λ g,0 ) = γK τ qEJP q .
The result follows since on the right hand side J is compact and all other operators are bounded.

Proofs of the main results
In Sections 2 and 3 we considered layer potentials and equivalent problems characterizing CGO solutions in the case where (M, g) ⊂⊂ (T int , g) where T = R×M 0 , g = e⊕g 0 , and (M 0 , g 0 ) can be any compact (n−1)-dimensional manifold with boundary. Now we specialize to the case where (M 0 , g 0 ) is simple and prove Theorems 1.1 and 1.2.
The first step is to fix the harmonic functions u 0 used in Proposition 3.2. We first choose a simple manifold (M 0 , g 0 ) such that (M 0 , g 0 ) ⊂⊂ (M 0 , g 0 ). Below, we will write (r, θ) for the polar normal coordinates in (M 0 , g 0 ) with center at a given point p ∈M 0 M 0 (these exist globally because the manifold is simple), and we write, following [7, Section 5], where λ is a fixed nonzero real number and b ∈ C ∞ (S n−2 ) is a fixed function. Note thatã ∈ C ∞ (T ) since the coordinates (r, θ) are smooth in M 0 . Proof. If u 0 is of the required form, then ∆ g u 0 = 0 is equivalent with Here we have extended · , · as a complex bilinear form to complex valued 1-forms. As in [7, Section 5], we see that dΦ, dΦ = 0 and also that (2 dΦ, d · + ∆ g Φ)(|g| −1/4 e iλ(x 1 +ir) b(θ)) = 0 (this was the reason for the choice ofã). Consequently Then for any δ > 1/2 one has f ∈ L 2 −δ (T ), the norm f L 2 −δ (T ) is independent of τ , and the Fourier transform F x 1 f ( · , x ′ ) is supported in {|ξ 1 | ≥ |λ|}. By Proposition 3.1 we have a solution r 0 = G τ f of (4.1), which gives the required solution u 0 .
We can now prove the main theorems.
Proof of Theorem 1.2. We first consider the case, as in the beginning of this section, where (M, g) is an admissible manifold with conformal factor c = 1. Suppose that the manifold (M, g), and consequently also (M 0 , g 0 ), and the map Λ g,q are known. We wish to determine q from this knowledge.
First note the basic integral identity (see [7, Lemma 6.1]) which is valid for any u j ∈ H 2 (M ) with (−∆ + q)u 1 = 0 in M , ∆ g u 2 = 0 in M , and u j | ∂M = f j . We consider CGO solutions in T of the form where u 0,j are harmonic functions provided by Proposition 4.1 having the form Here τ ≥ τ 0 and τ 2 / ∈ Spec(−∆ g 0 ), (r, θ) are polar normal coordinates in (M 0 , g 0 ) with center at p ∈M 0 M 0 , λ = 0, b is a smooth function in S n−2 , and ψ j are explicit functions with G ±τ ψ j L 2 (M ) = O(|τ | −1 ).
The point is that u 0,j are explicit functions which can be constructed from the knowledge of (M, g), and also f 2 = u 0,2 | Γ is known. By Proposition 3.2 there is a unique CGO solution u 1 of the above form, and the boundary value f 1 = u 1 | Γ is the unique solution in H 3/2 (Γ) of the boundary integral equation (Id + γS τ (Λ g,q − Λ g,0 ))f 1 = u 0,1 on Γ.
Since the operator on the left and the function on the right are known from our data, we can construct f 1 as the unique solution of this Fredholm integral equation. Then the left hand side of (4.2) is known, and consequently we can determine from our data the integrals for any u 1 and u 2 as above.
At this point it is convenient to extend q into T as a function in C ∞ c (T int ). This may be done by recovering the Taylor series of q on ∂M via boundary determination [7, Section 8] (this procedure is constructive), and by extending q to a function in C ∞ c (T int ) so that q| T M is known. Using that dV = |g| 1/2 dx 1 dr dθ, the last integral becomes Denoting the quantity in brackets by f λ (r, θ) and by varying the smooth function b, we determine the integrals ∞ 0 e −2λr f λ (r, θ) dr for all θ ∈ S n−2 .
These integrals are known for any nonzero real number λ and for any point p ∈M 0 M 0 which is the center of the polar normal coordinates (r, θ) inM 0 . Noting that r → (r, θ) is the unit speed geodesic in (M 0 , g 0 ) starting at p in direction θ, and letting p approach ∂M 0 , we can recover the integrals T 0 e −2λr f λ (γ(r)) dr (4.4) for any geodesic γ : [0, T ] → M 0 where γ(0), γ(T ) ∈ ∂M 0 and γ(t) for 0 < t < T lies in M int 0 . This is the attenuated geodesic ray transform of f λ in (M 0 , g 0 ), with constant attenuation −2λ. Now, assuming dim(M ) = 3 so (M 0 , g 0 ) is 2-dimensional, we invoke the invertibility result for the attenuated ray transform [28] which allows to recover the function f λ in M 0 from the integrals (4.4) for any λ. Thus, we have determined the integrals ∞ −∞ e 2iλx 1 q(x 1 , x ′ ) dx 1 for any λ = 0 and for any x ′ ∈ M 0 . This determines q in M by inverting the one-dimensional Fourier transform.
We have proved the theorem in the case where (M, g) is an admissible manifold and with conformal factor c = 1. For general conformal factors, suppose that (M, g) is admissible and g = cg whereg = e ⊕ g 0 . Define alsõ q = c(q − q c ) where q c = c n−2 4 ∆ cg (c − n−2 4 ). The identity c n+2 4 (−∆ cg + q)(c − n−2 4 u) = (−∆g +q)u implies that, since νg = c 1/2 ν g , Thus, from the knowledge of Λ g,q and (M, g) we can determine Λg ,q . The proof above then shows that one can reconstructq, from which q is easily determined.
Proof of Theorem 1.1. Let (M,g) be admissible and known and suppose that Λ cg = Λ cg,0 is known. By boundary determination [7] we can determine c| ∂M and ∂ νg c| ∂M . The identity (4.5) shows that Λg ,q f = c ). This shows that Λ cg determines Λg ,q , and Theorem 1.2 implies that we can recoverq.
We write w = log c − n−2 This nonlinear Dirichlet problem has a unique solution by the maximum principle [9] and we have already recovered the right hand sideq and the boundary value log c − n−2 4 | ∂M , so we may construct w in M by solving the problem. This determines c in M .