The geodesic X-ray transform with a $GL(n,\mathbb{C})$-connection

We derive reconstruction formulas for a family of geodesic ray transforms with connection, defined on simple Riemannian surfaces. Such formulas provide injectivity of such all transforms in a neighbourhood of constant curvature metrics and non-unitary connections with curvature close to zero. If certain Fredholm equations are injective in the absence of connection, then for any smooth enough connection multiplied by a complex parameter, the corresponding transform is injective for all values of that parameter outside a discrete set. Range characterizations are also provided, as well as numerical illustrations.


Introduction
This paper is concerned with the attenuated X-ray transform on a non-trapping surface. We shall consider attenuations determined by a n × n matrix A of complex-valued 1-forms (a GL(n, C)connection).
Consider (M, g) a compact oriented Riemannian surface with smooth boundary. We let SM = {(x, v) ∈ T M : |v| g = 1} be the unit tangent bundle with geodesic flow ϕ t : SM → SM , defined on the domain where τ (x, v) is the first time at which the geodesic γ (x,v) with initial conditions (x, v) hits the boundary ∂M . Recall that ϕ t is defined as ϕ t (x, v) := (γ (x,v) (t),γ (x,v) (t)), with infinitesimal generator X (x,v) = dϕt dt (x, v)| t=0 . The manifold is said to be non-trapping if τ (x, v) < ∞ for all (x, v) ∈ SM . In this paper we consider non-trapping surfaces where ∂M is strictly convex, meaning that the second fundamental form of ∂M ⊂ M is positive definite. This is already enough to imply that M is a disk (cf. [22,Proposition 2.4]). If in addition (M, g) has no conjugate points we say that the surface is simple.
Given A, consider the matrix weight w : SM → GL(n, C) that arises as a solution of the transport equation on SM : Xw = wA, w| ∂ + (SM ) = id, where ∂ + (SM ) denotes the set of (x, v) ∈ ∂(SM ) such that v points inside M , i.e. ν(x), v ≤ 0 where ν is the outer unit normal at ∂M . We define the attenuated X-ray transform associated with the connection A, I A : C ∞ (SM, C n ) → C ∞ (∂ + (SM ), C n ) as: If f ∈ C ∞ (M, C n ) we shall set I A,0 (f ) := I A (f • π) where π : SM → M is the footpoint projection.
A question of fundamental importance in the subject, is whether I A,0 is injective. In [21], the authors prove that in the case of a unitary connection and a Higgs field, the corresponding ray transform on a simple surface is injective. In [23], the same authors provide a range characterization for the attenuated ray transform. The salient feature here is that the connection need not be unitary (or, equivalently, Hermitian) in the sense that A = −A * .
To put the X-ray transform (3) into perspective, consider first a general matrix weight w : SM → GL(n, C). For each fixed x ∈ M , the quantity w −1 Xw(x, v) may be expanded in the velocities v as w −1 Xw(x, v) = Φ(x) + A(x, v) + higher order terms in v.
Hence the transport equation (2) tells that the X-ray transform with connection picks up precisely the term in the expansion above that gives linear dependence in velocities. If we were to pick just Φ(x) we would have a Higgs field or potential and for n = 1 and g flat, this reduces to the usual attenuated ray transform that is so prominent in SPECT (single photon emission computed tomography). Inversion formulas and range characterization for this very important case were obtained in [2,17,18]. Even for n = 1, if the weight w is allowed to be arbitrary, the attenuated ray transform in 2D may not be injective [4], but one may speculate that if the expansion in (4) is finite, injectivity may persist. In dimensions ≥ 3, the game changes and very general injectivity results have been obtained in [24]. Besides the motivation coming from medical imaging and SPECT, there is another reason for considering the problem of injectivity of I A,0 and it has to do with the non-linear inverse problem of recovering A from its scattering data, or non-abelian X-ray transform. It would be impossible to do justice to the literature on the topic here, but we refer to [5,19,21,24] and references therein; the last two references also have a discussion about a pseudo-linearization procedure that allows to connect the linear and non-linear problems.
Our approach in order to invert such transforms explicitely is a generalization of inversion formulas derived in [26] and further analyzed in [10] for geodesic ray transforms on simple surfaces. The first author then provided generalizations of such approaches in the case of symmetric differentials on simple surfaces [12] and recently provided inversion formulas for geodesic X-ray transforms with scalar Higgs-field type attenuations [13] (that is, w −1 Xw = Φ(x) and n = 1).
In [5] it is proved that I A,0 is injective for an arbitrary GL(n, C)-connection when (M, g) is a domain in R 2 ; this uses a delicate theorem about existence of holomorphic integrating factors established in [6,Theorem 5]. Generic injectivity for the case of simple manifolds, including the case when both g and A are real analytic is proved in [32]. In [25], injectivity of I A,0 is proved for an arbitrary GL(n, C)-connection whenever (M, g) is a negatively curved simple manifold. In spite of all this progress the following question remains open: Question: Let (M, g) be a simple surface and A a GL(n, C)-connection. Is I A,0 injective? Note that the question has a positive answer for n = 1, this follows essentially from the methods in [28], cf. Proposition 22 below. Theorem 5 below provides several new instances in which I A,0 is proved to be injective, and we shall also provide range characterizations. Numerical simulations illustrating the effectiveness of our approach will appear in future work. We now proceed to state our results in detail.
for f ∈ C ∞ 0 (M, C n ) (i.e., a smooth function vanishing at the boundary), extendible to H 1 0 (M, C n ) by continuity. Note that I A,⊥ can also be defined on functions which do not vanish at ∂M , and the difference will be studied in much detail in Section 5.
Upon defining the operators , we first derive the following formulas, true on any non-trapping Riemannian surface with strictly convex boundary: Theorem 1. Let (M, g) be a non-trapping Riemannian surface with boundary. Then the following equations hold: Formulas (5)-(6) take the form of filtered-backprojection algorithms, where the operator B A,+ HQ A,− (defined in Section 3.3, see (19) and (21)) can be viewed as a filter in data space, while the operators I * −A * ,0 , I * −A * ,⊥ , formal adjoints of I −A * ,0 : L 2 (M, C n ) → L 2 µ (∂ + (SM ), C n ) and I −A * ,⊥ : H 1 0 (M, C n ) → L 2 µ (∂ + (SM ), C n ) respectively, are sometimes referred to as backprojection operators.
Remark 2. While the transform I A,⊥ can be defined for smooth functions with non-zero boundary values, Equation (6) no longer holds in this augmented space, as is illustrated on the Euclidean transform I ⊥ wihout connection in [15,Proposition 5]. There, , it is then shown that formula (6) applied to I ⊥ f recovers f 0 + 1 4 f ∂ and not f 0 + f ∂ . If, in addition, (M, g) is simple, then the operators W A and W A,⊥ extend as compact operators W A , W A,⊥ : L 2 (M, C n ) → L 2 (M, C n ), see Lemma 16 below. In particular, equations (5) and (6) are Fredholm equations, invertible up to a finite-dimensional kernel. In fact, we can prove something with stronger implications: Theorem 3. For any analytic C 1 (M, (Λ 1 ) n×n )-valued family of connections λ → A λ , the corresponding L 2 (M, C n ) → L 2 (M, C n )-valued families of operators λ → W A λ and λ → W A λ ,⊥ are analytic.
By analytic Fredholm theory (see, e.g., [27, Thm. VI.14]), Theorem 3 implies that for is invertible for some value λ 0 , then this remains true for all complex values λ outside a discrete set, which from the Fredholm equations implies that I A λ ,0 is injective for all such values (note that if I A,0 (f ) = 0, then f ∈ C ∞ 0 (M, C n ), cf. Proposition 27 below). For similar purposes of generic uniqueness in inverse problems, prior uses of analytic Fredholm theory have appeared for instance in [30] in the case of the radiative transport equation, and in [31] in the case of Calderón's inverse conductivity problem.
We then focus on obtaining estimates for the error operators W A , W A,⊥ , whose study starts in [26,10] in the case without connection (call W ≡ W 0 the corresponding operator). Obtaining transparent estimates is not obvious, as the constants derived in [10] are not well-controlled by intrinsic geometric quantities. Such estimates have recently been obtained in [8] on surfaces with negative curvature, allowing non-trivial trapping. In an attempt to quantify simplicity and obtain more transparent error estimates, we recall that the absence of conjugate points is equivalent to the non-vanishing of the following function b : Since additionally, lim t→0 + |b(x,v,t)| t = 1 for every (x, v) ∈ SM , and since D is compact, the following claim is obvious If (M, g) is simple, there exist positive constants C 1 (M, g) and C 2 (M, g) in which case we say that (M, g) is a simple Riemannian surface with constants C 1 , C 2 . A finer analysis of the Schwartz kernels of the error operators then allows to prove the following theorem. In the statement, for an n × n matrix M , we denote M := (tr (B * B)) 1 2 its Frobenius norm.
Theorem 4. Let (M, g) be a simple Riemannian surface with constants C 1 , C 2 as in (7) and Gaussian curvature κ(x). Given the C 1 connection A with curvature F A , let us denote As consequences of Theorems 3 and 4, we obtain the following main conclusions.
Theorem 5. Let (M, g) be a simple surface and A a C 1 connection. Then the following conclusions hold: (i) If κ is constant and A is flat, the operators W A and W A,⊥ vanish identically and Theorem 1 implies that the transforms I A,0 , I A,⊥ , I −A * ,0 and I −A * ,⊥ are all injective, with explicit, one-shot inversion formulas.
(ii) Injectivity still holds if (n, are such that the right hand side of (8) is less than 1, with a Neumann series type inversion.
In the statement of Theorem 5, injectivity of I −A * ,0 and I −A * ,⊥ comes from the fact that one may consider the Fredholm equations (5)-(6) corresponding to I −A * ,0 and I −A * ,⊥ , and since we prove in Lemma 13 that W A and W −A * ,⊥ are L 2 (M, C n ) → L 2 (M, C n ) adjoints, then invertibility of Id+W 2 A is equivalent to invertibility of Id+W 2 −A * ,⊥ . It is conjectured that Id+W 2 is injective on any simple surface.
Finally, we provide a range characterization for the operators I A,0 and I A,⊥ whenever the operator I −A * ,0 is injective. In order to obtain such a characterization, we must first establish a series of results building equivalence of injectivities between transforms with different connections. This takes us to formulating a few key results. In what follows, we denote by I A,m := I A | Ωm , where Ω m is defined as Ω m = Ker(V − imId) ∩ C ∞ (SM, C n ) and V is the vertical vector field (see Section 3).
• If I A,0 is injective, then so is I A+ωIn,0 for any scalar one-form ω. See Proposition 22.
• If I A,0 is injective, then I A,m is injective for any m ∈ Z. See Proposition 23.
• I A,0 is injective if and only if I A,⊥ is injective. In particular, all conclusions above hold if we only assume I A,⊥ injective instead. See Propositions 26 and 23.
For the range characterization, we extend I A,⊥ to all functions in C ∞ (M, C n ) and not just those vanishing at the boundary. One may define the formal operators P A,± := B A,− H ± Q A,+ , where B A,− and Q A,+ are defined in Section 3.3 and H ± denote odd and even fiberwise Hilbert transforms (see Section 3). The operators P A,± , defined in the smooth setting on a space denoted S ∞ A (∂ + (SM ), C n ) (see (20)), are boundary operators which only depend on the scattering relation and the scattering data C A , and they allow to describe the ranges of I A,0 and I A,⊥ as follows.
Theorem 6 (Range characterization of I A,0 and I A,⊥ ). Suppose that (M, g) is a simple surface and I −A * ,0 is injective, and let I ∈ C ∞ (∂ + (SM ), C n ). Then the following claims hold. w ∈ S ∞ A (∂ + (SM ), C n ) such that I = P A,+ w. Such range characterizations were previously established in [26] in the case without connection, in [23,1] in the case of unitary connections and Higgs fields, and recently in the case of the attenuated transform [3]. The range characterization for I 0 was recently proved by the first author to be the generalization of the classical moment conditions for compactly supported functions in the Euclidean case, see [14, Outline. The remainder of the article is organized as follows. We recall generalities on the geometry of the unit circle bundle, transport equations with connection, with additional remarks on the symmetries in the data space L 2 µ (∂ + (SM ), C n ) in Section 3. In Section 4, we prove Theorem 1 and study the error operators W A , W A,⊥ in detail, including the proof of Theorem 4. In Section 5, injectivity of ray transforms corresponding to different connections or different harmonic levels are inter-related, and the relation between the transform I A over one-forms and the transform I A,⊥ is refined. Finally, based on additional preparatory results from Section 5 (namely, Proposition 22), Section 6 presents the range characterization and the proof of Theorem 6.

Setting and notation
Throughout this section we will assume that (M, g) is a non-trapping surface with strictly convex boundary. As a consequence, it is simply connected (hence orientable).
Geometry of the unit tangent bundle. We briefly recall standard notation for the unit sphere bundle SM , see e.g. [21] for more detail. The vector field X ∈ T (SM ) can be completed into a global framing {X, X ⊥ , V } of T (SM ) with structure equations The Sasaki metric on T (SM ) is then the unique metric making this frame orthonormal, with volume form which we denote dΣ 3 . This measure gives rise to an inner product space L 2 (SM, C n ), where the circle action on tangent fibers induces the orthogonal decomposition Upon defining Ω k = C ∞ (SM, C n ) ∩ H k , a function u ∈ C ∞ (SM, C n ) decomposes uniquely as u = k∈Z u k where each u k belongs to Ω k . If u ∈ L 2 (SM, C n ), then each u k belongs to where q denotes a non-vanishing element of C ∞ (SM, C) ∩ ker(V − iId), whose existence is guaranteed by simple connectedness.
Scattering relation. For (x, v) ∈ SM , let us denote both endpoints of the geodesic passing through (x, v). Let α : Transport equations on the unit tangent bundle. As in the Introduction, for f : SM → C n and A a GL(n, C)-connection, we define u f A to be the unique solution to the transport problem Let us denote U A : SM → GL(n, C) the unique matrix solution W to the problem From this solution, we define the scattering data We also define the attenuation function E A : D → GL(n, C) as and in terms of which many kernels will be expressed below. For h defined on ∂ + (SM ), define h ψ,A the unique solution u to the transport problem With the definition of U A , we have, for (x, v) ∈ SM , or, equivalently for the last one, As the matrix A is not necessarily skew-Hermitian, the connections A and −A * are distinct, though we will see below that it is helpful to consider the transforms associated to both jointly. The first important identity to notice is since both functions coincide with the unique solution W to the transport problem Decomposition of X + A and the Guillemin-Kazhdan operators. We may decompose 2 (X ± iX ⊥ ) are the Guillemin-Kazhdan operators, see [9].
In this paper we work exclusively with the case in which M is a disk, hence we can consider global isothermal coordinates (x, y) on M such that the metric can be written as ds 2 = e 2λ (dx 2 + dy 2 ) where λ is a smooth real-valued function of (x, y). This gives coordinates (x, y, θ) on SM where θ is the angle between a unit vector v and ∂/∂x. Then Ω k consists of all functions u = h(x, y)e ikθ where h ∈ C ∞ (M, C n ). In these coordinates, a connection A = A z dz + Azdz (with z = x + iy) takes the form A(x, y, θ) = e −λ (A z (x, y)e iθ + Az(x, y)e −iθ ), and we can give an explicit description of the operators µ ± acting on Ω k . For µ − we have (cf. [20,Equation (24)]): From this expression we may derive the following lemma which will be used later on: Proof. We only prove the claim for µ − , the one for µ + is proved similarly. If we write f = ge i(k−1)θ , using (13) we see that we only need to find h ∈ C ∞ (M, C n ) such that But it is well known that there exists a smooth F : M → GL(n, C) such that∂F +AzF = 0, hence the solvability of (14) reduces immediately to the standard solvability result for the Cauchy-Riemann operator, namely, given a smooth b, there is a such that∂a = b. The existence of F above follows right away from the fact that a holomorphic vector bundle over the disk is holomorphically trivial [7, Theorems 30.1 and 30.4], see also [6,16] for alternative proofs.
Hilbert transform and commutator formulas. An important operator for what follows is the fiberwise Hilbert transform H : L 2 (SM, C n ) → L 2 (SM, C n ), diagonal on the harmonic decomposition in the fiber, and such that H| H k = −i sign(k)Id| H k , with the convention sign(0) = 0.
Using the splitting X + A = µ + + µ − , it is immediate to derive the commutator formulas (see [21, as well as the following identities, obtained by computing [H 2 , X + A] in two ways:

Decompositions of the data space
Also define the antipodal scattering relation to be the mapping α a is an involution and α a (∂ ± (SM )) ⊂ ∂ ± (SM ). It is straightforward to see that the function In particular, this implies the relation Writing this for U A (x, v), we obtain the identity We will use this to characterize the symmetries of the ray transforms over even and odd integrands. In particular, the identity (17) means that Note also the obvious identities Proof. We only treat the case of f even, the odd case being similar. We write (17). The proof is complete.
This motivates a decomposition of This decomposition is unique and given explicitely by Such symmetries, via extension as first integrals of X + A, generate even and odd functions on SM : hence the proof.
In general, the decomposition V A,+ ⊕ V A,− is not orthogonal in L 2 µ (∂ + SM ). In this context of non-Hermitian connections, the more natural relation is the following.

In particular, if the connection is unitary
Proof. It is enough to prove the first equality, as the second follows by considering the connection obtained by fulfilling the conditions of V A,+ and V −A * ,− and using that C −A * = (C * A ) −1 by virtue of (12).
We then write Plugging this last expression into the first equality and applying Santalo's formula, we arrive at The lemma is proved.

Boundary operators
Extending notation from [23], we define for w ∈ C(∂ + (SM ), C n ) Note that Q + w ∈ C(∂(SM ), C n ). We also define where the second equality is established in [23,Lemma 5.1]. This is due in part to the fact that (note the sign difference with [23]). B A,− appears naturally in the fundamental theorem of calculus along a geodesic: for (x, v) ∈ ∂ + (SM ), We state without proof the following straightforward claims.

Inversion formulas and control of the error operators
We will see below that it is somehow natural to consider the inversion of operators I A,0 , I A,⊥ , I −A * ,0 and I −A * ,⊥ together.

Adjoints
For f ∈ C ∞ (M, C n ) and h ∈ C ∞ (∂ + (SM ), C n ), we compute (denote ·, · the Hermitian product on C n ), So we deduce that A similar argument with f ∈ C ∞ 0 (SM ), and using the fact that X * ⊥ = −X ⊥ when either function in the L 2 (SM, C n ) inner product vanishes at the boundary, yields that A direct use of Lemma 9 and inspection on symmetries yields the following

Fredholm equations for I A,0 and I A,⊥ -proof of Theorem 1
As discussed in the introduction, let us define the operators We now prove Theorem 1 before studying the operators W A and W A,⊥ further. For the proof below, let us make the comment that, in terms of solutions of elementary transport problems of the form u f A and h A,ψ defined in Section 3, the solution to the problem Proof of Theorem 1. Here and below, for a function u(x, v) defined on SM , we write the even/odd decomposition with respect to v as u (5)). Start from the equation where the last step comes from the fact that π 0 (X ⊥ − A V )Hu f A,+ = 0. It now remains to write a transport problem for Hu f A,− , for which we use the formula for [H, X + A]: which upon projecting onto even harmonics, yields This equation gives us Plugging this expression of Hu f A,− back into (25) yields the formula Finally, inspection of symmetries shows that B A,− HQ A,− I A,0 f ∈ V A,+ , so that using Lemma 9 and the expression of I * −A * ,⊥ , it is annihilated by I * −A * ,⊥ . As a conclusion, we arrive at (5).
Inversion of I A,⊥ (Proof of (6)). Start from the equation Applying the Hilbert transform to the transport equation above, using the commutators and projecting onto odd harmonics, we obtain From this equation, we get that the function (Hu where Upon applying π 0 (fiber average) to (28), we obtain As in the inversion of I A,0 , we notice that B A,− HQ A,− I A,⊥ f ∈ V A,− and as such is annihilated by I * −A * ,0 . As a conclusion, the reconstruction formula, in its final form, looks like (6).

Properties of the error operators
Unlike the geodesic case studied in [26], W A and W A,⊥ are not always L 2 (M )-adjoints. Consider the equations (5) and (6) corresponding to the connection −A * . They give, Inspecting the right hand sides suggests that, for instance, taking the adjoint equation to (5) would yield (30). A partial answer to this heuristic guess is to establish: Lemma 13. The operators W A and W −A * ,⊥ are L 2 (M, C n ) → L 2 (M, C n ) adjoints. As a consequence, so are W −A * and W A,⊥ . In particular, if A = −A * , then W A and W A,⊥ are adjoints.
Proof. It is enough to check it for f, g ∈ C ∞ 0 (M ). We compute The crossed term is zero because I A,0 f ∈ V A,+ while I −A * ,⊥ g ∈ V −A * ,− and both spaces are orthogonal by virtue of Lemma 10. The lemma is proved.
The next result establishes that in the case where the metric is simple, the reconstruction formulas (5), (6), (29) and (30) are in fact Fredholm equations, as the operators W A,0 and W A,⊥ are compact. In order to prove this, we need to make explicit their Schwartz kernels, which in turns requires some recalls about Jacobi fields.
Kernels of W A and W A,⊥ . We now make explicit the kernels of the operators W A and W A,⊥ defined in (24), by showing the following Lemma 14. The operators W A , W A,⊥ take the form with respective kernels, in exponential coordinates, given by The proof of Lemma 14 makes use of the following property, whose proof we relegate to the appendix: Proof of Lemma 14. Proof of (31). Using the definition (24), with E A defined in (11). In this expression, the only term which differentiates f is given by ))), which we rewrite as The corresponding term can then be rewritten as In the last right-hand-side, the first term vanishes identically and the second cancels out the boundary term in (33) thanks to Lemma 15. We then arrive at an expression for w A as which yields (31) after applying a product rule.
Proof of (32). On to W A,⊥ , we write We rewrite the only term which differentiates f as Integrating this term by parts on S x in the expression of W A,⊥ h(x), this creates a boundary term of the form which vanishes since by assumption h ∈ C ∞ 0 (M ). For the remaining term, we obtain the expression for w A,⊥ as hence (32).
The operators W A and W A,⊥ are compact. In what follows, for a C n×n matrix B, we denote B = (tr (B * B)) 1 2 its Frobenius norm. For an operator of the form where we have defined W(x, y) : x (y)) , we may obtain an estimate on the L 2 (M, C n ) → L 2 (M, C n ) norm of W by computing whenever the right hand side is finite, and where · ρ denotes the spectral norm on C n×n . Using that · ρ ≤ · and changing variable y = Exp x (v, t), we arrive at the following estimate, to be used below which implies both continuity and compactness of W whenever the right hand side is finite.

Lemma 16. The operators W A and W A,⊥ (and by duality via Lemma 13, W −A
Proof. The proof mainly consists in looking at the behavior of w A and w A,⊥ defined in (31) and (32) near t = 0. Near t = 0, the following expansions hold: Together with the fact that the functions V b 2 b 2 2 and V b 1 b 2 vanish as t → 0 (see for instance [12,26]), this allows to deduce that This means in particular that the function k(x,v,t) b 2 (x,v,t) where k ∈ {w A , w A,⊥ } is bounded near t = 0. Since it inherits the regularity of b 1 , b 2 , A outside t = 0 and b 2 does not vanish outside {t = 0} because (M, g) is simple, the only problem was at t = 0. For each operator, changing variables y(v, t) = γ x,v (t) with change of volume dM y = |b 2 (x, v, t)| dt dS(v), the Schwarz kernels of W A and W A,⊥ are of the form k(x,v(y),d(x,y)) b 2 (x,v(y),d(x,y)) with k ∈ {w A , w A,⊥ }, and they are bounded near the diagonal and away from the diagonal. Since M × M has finite volume, these kernels belong to L 2 (M × M ), and thus the operators W A , W A,⊥ : L 2 (M, C n ) → L 2 (M, C n ) are compact.

Analytic Fredholm approach -Proof of Theorem 3
We start with some preliminary estimates for ordinary differential equations. Let us define, for A ∈ C 1 (M, (Λ 1 ) n×n ), In addition, for a function B : D → C n×n , we define B F,∞ := sup (x,v,t)∈D B(x, v, t) , making (C 0 (D, C n×n ), · F,∞ ) into a Banach space. Moreover, since the Frobenius norm is submultiplicative, so is · F,∞ . We also consider the Banach space (C 1 (D, C n×n ), · F,∞,1 ) with the norm also submultiplicative. Now for a problem of the form a priori estimates yield an estimate of the form Moreover, for Y ∈ {X, X ⊥ , V }, one may derive ODE's for Y U of the form which upon using (36) implies an estimate of the form where C ′ is independent of U , A or F . Similarly, a problem of the form is equivalent to a problem for W = U − I n : for which (37) applies. Combining this with the triangle inequality, and using that A(ϕ t ) F,∞,1 ≤ A C 1 (M,(Λ 1 ) n×n ) , we arrive at: Finally, let us note that estimates (36), (37) and (38) also hold if the connection is rightmultiplied in the ODEs considered instead of left-multiplied. With these estimates in mind, we are ready to prove Theorem 3.
Proof of Theorem 3. We prove the statement for W A λ only, as the proof from W A λ ,⊥ is similar.
Recall that w A , the kernel of W A up to exponential map, is given by with E A as defined in (11). Estimates on W A λ boil down to studying how λ → w A λ behaves in the C 0 (D, C n×n ) topology, which in turn requires to look at how λ → E A λ behaves in the C 1 (D, C n×n ) topology. Denote E λ = E A λ for short. Fix λ 0 ∈ C and consider λ close to λ 0 , write by assumption Since lim λ→λ 0 B λ (ϕ t )E λ + A ′ λ 0 (ϕ t )(E λ − E λ 0 ) C 1 (D,C n×n ) = 0, then estimate (37) implies that lim λ→λ 0 F λ C 1 (D,C n×n ) = 0 thus λ → E λ is an analytic C 1 (D, C n×n ) function. Similarly, we obtain Thus, upon defining where the estimate on v λ easily follows from the estimates on G λ and B λ . In addition, let us analyze the behavior of w ′ A λ 0 near t = 0 since the ratio w ′ A λ 0 /b 2 will be continuous, hence bounded, elsewhere. Looking at the ODE satisfied by E ′ λ 0 , we have, near t = 0, the expansions With the additional vanishing of V (b 1 /b 2 ) as t → 0, this is enough to establish that lim t→0 w ′ which, by virtue of estimate (35), implies that the operator W ′ Reasoning similary on V λ , we also obtain that Theorem 3 is proved.

Further error estimates -proof of Theorem 4
We now refine the previous result by estimating the operator norms of W A and W A,⊥ explicitly. In particular, we now define two functions of interest which appeared in the proof of Lemma 16: in terms of which the kernels w A and w A,⊥ are written as Using the fact that we now establish the following Lemma 17. The functions K ℓ for ℓ = 1, 2 satisfy the following ODEs on D: Proof. We only treat K 2 , as the case of K 1 is similar. We compute directly, keeping the variables (x, v) implicit and writingĖ ≡ dE dt : Note that b 2 is a scalar function, so we can commute it. We also have used thatḃ 2 = −c 2 . The proof is complete.
The next result, whose proof we relegate to the Appendix, is an explicit bound on the quantities V b 1 b 2 and V 1 b 2 . Such quantities first appeared in [26] and arise as the kernels, up to exponential map, of the error operators W and W * of the geodesic ray transform without connection. In what follows, we recall that for (M, g) a simple surface, we may define C 1 (M, g) := min D Lemma 18. Let (M, g) a simple surface with constants C 1 , C 2 as in (7). Then the functions satisfy the following estimates: We now prove the main result of this section, Theorem 4.
Proof of Theorem 4. In order to apply estimate (35) to W A and W A,⊥ , we now bound the functions w A and w A,⊥ using expressions in (41). The equation satisfied by so that we may obtain the estimate The same estimate holds for E −1 . Integrating (42) using E as integrating factor, we deduce the following integral representations (we keep (x, v) implicit) where in the last equality, we have used (52) (proved in the appendix). We now bound the Frobenius norm of the left hand sides, using submultiplicativity of · F : Similarly, using that |b 2 (ϕ s , t − s)| ≤ C 2 (t − s), we can arrive at the exact same bound for b 2 K 1 − b 1 K 2 (t). Given the form of w A and w A,⊥ in (41), and the fact that bounds on V b 1 b 2 and V 1 b 2 are the same and bounds on K 2 and b 2 K 1 − b 1 K 2 are the same, this will yield the same bound on w A or w A,⊥ . Therefore, let us focus on w A,⊥ : using the previous bound together with Lemma 18 Using (a + b) 2 ≤ 2(a 2 + b 2 ) to bound w A,⊥ (x, v, t) 2 and using (35), we arrive at Therefore, (8) holds with valid both for W A and W A,⊥ as explained above. Theorem 4 is proved.

Injectivity equivalences and implications
The purpose of this section is twofold. It first clarifies the relation between the transform I A restricted to one-forms, and the transform I A,⊥ . Second, it serves as preparation for the range characterization results stated in the next section.

On the range decomposition of I A
We now prove that the range of I A acting on 1-forms (i.e. acting on Ω −1 ⊕Ω 1 ) decomposes into (i) the range of I A,⊥ defined on H 1 0 and (ii) the ranges of I A restricted to ker ±1 µ * ± := Ω ±1 ∩ ker µ * ± (or their L 2 versions), and that the sum vanishes if and only if all three components vanish. The first thing to observe is the following lemma which follows right away form the ellipticity of µ ± . Lemma 19. Let (M, g) be a Riemannian surface with boundary and A a C 1 connection. The following decompositions hold, orthogonal for the L 2 (SM, C n ) inner product (hence unique): (i) For every f ∈ Ω 1 , there exists v ∈ Ω 0 with v| ∂M = 0 and g 1 ∈ ker 1 µ * + such that f = µ + v + g 1 .
(ii) For every f ∈ Ω −1 , there exists v ∈ Ω 0 with v| ∂M = 0 and g −1 ∈ ker −1 µ * − such that Lemma 19 implies that any one-form ω = ω 1 + ω −1 decomposes uniquely as follows: write ω 1 = µ + a + g 1 and ω −1 = µ − b + g −1 , with a, b ∈ H 1 0 (M ) and g ±1 ∈ ker ±1 µ * ± . Upon defining g p := (a + b)/2 and g s := (i(a − b)/2), the sum can be rewritten as The transport equation can then be rewritten as where the functions u and u + g p agree on ∂SM so that We will say that I A acting on 1-forms is solenoidal injective if whenever I A (ω) = 0, there is smooth p : M → C n with p| ∂M = 0 such that ω = d A p = dp + Ap. Lemma 19 implies the following: Lemma 20. For any C 1 connection A, I A is solenoidal injective on one-forms if and only if, for any f ∈ C 1 0 (M ), g 1 ∈ ker 1 µ * + and g −1 ∈ ker −1 µ * − , Proof. ( =⇒ ) Suppose I A solenoidal injective and assume that I A g −1 + I A,⊥ f + I A g 1 = 0. Then from solenoidal injectivity, this means that there exists a function h defined on M vanishing on ∂M such that rewritten differently this means that (µ + +µ − )h = g 1 −i(µ + −µ − )f +g −1 , which upon projecting onto Fourier modes 1 and −1, implies

Uniqueness of such decompositions implies
Let ω be such that I A = 0. Using Lemma 19, we can write ω = (X + A)g p + (X ⊥ − A V )g s + g −1 + g 1 , with g p , g s functions on M vanishing at ∂M and g ±1 ∈ ker ±1 µ * ± . Then which by assumption implies g s = g −1 = g 1 = 0, thus ω = (X + A)g p . Hence I A is solenoidal injective on one-forms.
Given a C n -valued 1-form ω = ω −1 + ω 1 there is an alternative decomposition to (43) which uses slightly different boundary conditions. For this one considers the elliptic operator where Λ 1 (M ) is the set of all C n -valued 1-forms, given by Note that H −A * is the ortho-complement to the range of D (compare this with [23, Lemma 6.1]).
Observe also that we can express (44) as where h ±1 ∈ ker ±1 µ * ± , but the difference with (43) is that now we do not require f to vanish at the boundary and instead we have j * h = 0. We will return to this alternative decomposition after proving Theorem 6.

Injectivity for scalar perturbations of connections
Given a connection A on a simple surface (M, g), we first start by giving a characterization of the injectivity for I A,0 . Recall that a function f defined on SM is so-called (fiberwise) holomorphic Proposition 21 (Characterization of injectivity of I A,0 ). Let A be a GL(n, C)-connection. Then I A,0 is injective if and only if the following is true: for any f, u ∈ C ∞ (SM, C n ) satisfying (X + A)u = −f with u| ∂SM = 0, (i) If f is holomorphic and even, then u is holomorphic, odd.
(ii) If f is antiholomorphic and even, then u is antiholomorphic and odd.
Proof. ( =⇒ ) Suppose I A,0 injective. We only prove (i), as (ii) is similar. Let u, f as in the statement with f holomorphic. Then (Id − iH)f = f 0 . Moreover, projecting the transport equation onto odd harmonics, we obtain (X + A)u + = 0 with boundary condition u + | ∂SM = 0, hence u + = 0, thus u is odd. We then compute which upon integrating along geodesics implies that In particular, (X + A)(Id − iH)u = 0 with (Id − iH)u| ∂SM = 0, hence (Id − iH)u = 0, which means that u is holomorphic, hence the proof. ( ⇐= ) Suppose (i), (ii) are satisfied. Let f be a smooth function such that I A,0 f = 0, then there exists u : SM → C n with u| ∂SM = 0 and such that (X + A)u = −f . f is even, both holomorphic and antiholomorphic, thus by (i) and (ii), u is odd, both holomorphic and antiholomorphic, thus u = 0, hence f = 0. Proposition 21 is proved.
The next result relies on the key concept of holomorphic integrating factor for scalar connections, which we now recall. Given a one-form ω, there exists v : SM → C holomorphic, even solution of Xv = −ω. This is based on injectivity of the unattenuated transform I 0 , cf. [21,Theorem 4.1]. The construction goes as follows. First one may write ω = Xf + X ⊥ g for g vanishing at ∂M . Then we are left looking for v such that X(v + f ) = −X ⊥ g. One can construct u = (Id + iH)h ψ with h ψ even such that By surjectivity of I * 0 , one can find h such that I * 0 h = 2π(h ψ ) 0 = −2πig and for such an h, the function v = −f + (Id + iH)h ψ is a holomorphic, even solution of Xv = −ω. As a result, the functions e v and e −v are non-vanishing holomorphic, even, solutions of Xe ±v ± ωe ±v = 0. Using the same h, we can then construct w = −f − (Id − iH)h ψ , anti-holomorphic solution of Xw = −ω giving rise to anti-holomorphic integrating factors e ±w solutions of Xe ±w ±ωe ±w = 0.
With the use of such integrating factors, we are then able to establish the following.
Proposition 22. For any GL(n, C)-connection A, if I A,0 is injective, then for any smooth one-form ω, so is I A+ωIn,0 .
Proof. Suppose I A,0 injective and let ω be a one-form. We use the characterization from Proposition 21 to show that I A+ωIn,0 is injective by satisfying (i), (ii). Let u, f be such that (X +A+ω)u = −f with u| ∂SM = 0. If f is holomorphic even, then u is odd since (X +A+ω)u + = 0 with zero boundary condition. Let e v a holomorphic, even, integrating factor for ω, then we can recast (X + A + ω)u = −f as (X + A)(e −v u) = −e −v f , where e −v f is holomorphic, even and e −v u vanishes at ∂SM . Then since A satisfies (i), this implies that e −v u is holomorphic, odd, and hence u = e v (e −v u) is holomorphic, odd. The proof of (ii) is similar.
Such a result allows to derive injectivity results for several restrictions of I A to other subspaces of C ∞ (SM ), as they amount to studying transforms with connections which are translated from one another by a scalar one-form. Here and below, we denote I A,k the transform I A restricted to Ω k . Proposition 23. Suppose I A,0 injective, then the following conclusions hold.
(i) For any k ∈ Z, the transform I A,k is injective.
(ii) I A is solenoidal injective over one-forms. In particular, I A,⊥ is injective.
Remark 24. In particular, both statements imply that I A,k | ker k µ * + and I A,−k | ker −k µ * − are both injective for every k = 0, 1, 2 . . . . However this can be proved to always hold, see Proposition 27 below.
Proof. Suppose I A,0 injective. Proof of (i). Let f ∈ Ω k such that I A,k f = 0. Write f = q kf for q a non-vanishing section of Ω 1 andf : M → C n . Then if u is the unique solution to so that (q −k u)| ∂ + SM = I A+kq −1 XqIn,0f . In particular, this implies that Since I A+kq −1 XqIn,0 is injective by virtue of Proposition 22, thenf = 0, hence f = 0.
Proof of (ii). Suppose I A (ω 1 + ω −1 ) = 0, then there exists u such that (X + A)u = −ω −1 − ω 1 with u| ∂SM = 0. In particular, u is even since u − is a first integral of X + A vanishing at ∂SM . If q ∈ Ω 1 is non-vanishing, the equation (X + A)u = −ω −1 − ω 1 can be rewritten as If e v is a holomorphic, even, solution of Xe v − q −1 Xqe v = 0, then this equation can be rewritten as (X + A)(e −v qu) = −e −v q(ω −1 + ω 1 ), (e −v qu)| ∂SM = 0, and since the right hand side is holomorphic and even, then by injectivity of I A,0 , e −v qu is holomorphic and odd. Then u = q −1 e v (e −v qu) has harmonic content no less than −1 and since u is even, u −1 = 0 as well, so u is holomorphic. Similarly using an antiholomorphic integrating factor, one may show that u is antiholomorphic, so we conclude that u = u 0 with u 0 | ∂M = (u| ∂SM ) 0 = 0, and the relation (X + A)u 0 = −ω 1 − ω −1 implies that I A is solenoidal injective over one-forms. The proof is complete.
Finally, the next two propositions aim at showing that I A,⊥ injective implies that I A,0 injective.
Proposition 25 (Characterization of injectivity of I A,⊥ ). Let A be a smooth GL(n) connection.
is injective if and only the following is true: for any f, u ∈ C ∞ (SM ) satisfying (X + A)u = −f with u| ∂SM = 0, f odd and u even, (i) If f k = 0 for all k < −1 and f −1 ⊥ ker −1 µ * − , then u is holomorphic.
Proof. ( =⇒ ) Suppose I A,⊥ injective. We only prove (i) as (ii) is similar. Let u, f as in the statement with f k = 0 for all k < −1 and f −1 ⊥ ker −1 µ * − . In particular, from Lemma 19, we can Upon integrating along geodesics, we get which by injectivity of I A,⊥ implies u 0 + v 0 = 0. Then the transport equation above becomes Let e w a holomorphic, even function such that Xe w + ωe w = 0, then the equation above can be rewritten as where qf e −w is odd and que −w is even. Moreover, qf e −w is holomorphic, thus satisfies the requirement for (i), hence que −w is holomorphic, hence u = u −1 + u 1 + u 3 . . . . Using a similar argument with (ii), we can then cancel all u k 's for k ≥ 2. Thus u = u −1 + u 1 . Projecting the equation (X + A)u = −f onto Ω 2 and Ω −2 gives µ + u 1 = µ − u −1 = 0, and since u 1 | ∂SM = u −1 | ∂SM = 0, this implies u 1 = u −1 = 0, hence f = 0.
We conclude by proving the following result which has independent interest.
Proposition 27. Suppose there is u ∈ Ω k such that I A,k (u) = 0. Then u has vanishing jet at ∂M . In particular I A,k is injective when restricted to Ker µ ± .
Proof. The main observation is that N = I * A,k I A,k is an elliptic classical ΨDO of order −1 in the interior of any simple manifold engulfing M , see [23,Section 5] and references therein. Hence consider a slightly larger simple manifold M 1 containing M and extend u by zero to M 1 (A is extended in any smooth way). Thus N u = 0 in the interior of M 1 and by elliptic regularity we deduce that u is smooth in M 1 . Since u vanishes outside M , this clearly imply that u has zero jet at the boundary of M .
Suppose in addition µ − (u) = 0. If we write u = he ikθ then using (13) we see that∂(he kλ ) + Azhe kλ = 0. Using the existence of F : M → GL(n, C) such that∂F + AzF = 0 as in Lemma 7 we see that∂(F −1 he kλ ) = 0. Since h vanishes on ∂M , this is enough to conclude that u = 0. A similar argument applies to elements in the kernel of µ + (or their adjoints).

Range characterization
We start with a standard surjectivity result.
The proof of this result is now well-understood and we omit it. It follows from injectivity of I A,0 and the fact that I * A,0 I A,0 is an elliptic classical ΨDO of order −1 in the interior of any simple manifold engulfing M , see [23, Section 5] and references therein. From the expression I * A,0 h = 2π(h −A * ,ψ ) 0 , upon setting u = 2π h −A * ,ψ ∈ C ∞ (SM, C n ), Theorem 28 is equivalent to stating that for every f ∈ C ∞ (M, C n ), there exists u ∈ C ∞ (SM, C n ) satisfying (X − A * )u = 0 and u 0 = f . The next result is less standard and it is based on the solvability result given by Lemma 7 and follows the strategy of the proof of [23,Theorem 5.5].
Consider the purely imaginary 1-form where q ∈ Ω 1 is nowhere vanishing (e.g. in global isothermal coordinates q = e iθ ). Observe that if u : SM → C n is any smooth function then where m ∈ Z. First we show the following result which is interesting in its own right: Proof. Since I A,0 is injective, by Proposition 22, I A−maIn,0 is injective (with a defined in (45)), thus by Theorem 28, there is u ∈ C ∞ (SM, C n ) such that 0 = (X − A * + māI n )u = (X − A * − maI n )u and u 0 = q −m f . If we let w := q m u, then clearly w m = f and by (46) we also have (X − A * )w = 0.
As before, consider the operators µ ± = η A ± = η ± + A ±1 . Clearly, We need the following solvability result which is a direct consequence of Lemma 7.
We are now in good shape to complete the proof of Theorem 29. Given f ∈ C ∞ (M, C n ), we consider the functions w ±1 ∈ Ω ±1 given by Lemma 31. By Lemma 30 we can find odd functions p, q ∈ C ∞ (SM, C n ) solving the transport equation (X − A * )p = (X − A * )q = 0 and with p −1 = w −1 and q 1 = w 1 . Then the smooth function satisfies (X −A * )w = 0 thanks to equation (47). Upon defining h = w| ∂ + SM so that w = h ψ,−A * , we then obtain that h satsfies Finally, with the surjectivity Theorems 28 and 29, we are now ready to prove Theorem 6. As explained in the Introduction, define P A : S ∞ A (∂ + (SM ), C n ) → C ∞ (M, C n ), as follows P A := B A,− HQ A,+ .
The operator P A is a boundary operator which only depends on the scattering relation and the scattering data C A . Upon splitting the Hilbert transform H into its projections onto even and odd harmonics (call them H + and H − ), we obtain the splitting P A = P A,+ + P A,− , where we have defined P A,± := B A,− H ± Q A,+ .
Proof of Theorem 6. For w defined on ∂ + (SM ), recall that Q A,+ w = w ψ,A | ∂SM and that B A,− (u| ∂(SM ) ) = I A (−(X + A)u). Using these considerations and the commutator formulas, we are able to derive Similarly for P A,− , Since it is assumed that I −A * ,0 is injective, by virtue of Theorems 28 and 29, the operators I * −A * ,0 , I * −A * ,⊥ : S ∞ A (∂ + (SM ), C n ) → C ∞ (M, C n ) are surjective. Combining this surjectivity with the two factorizations above, claims (i) and (ii) follow.
Remark 32. Examining the proof above, we then see that In addition, on the direct sum S ∞ A (∂ + (SM ), C n ) = V A,+ ⊕ V A,− , since I * −A * ,0 vanishes on V A,− and I * −A * ,⊥ vanishes on V A,+ , one realizes that P A,± coincides with the restriction P A | V A,± . This is also true since, following previous observations, if h ∈ V A,+ , then Q A,+ h is even and if h ∈ V A,− , then Q A,+ h is odd, which justifies the corresponding splitting of the Hilbert transform into odd and even parts in the previous definitions.
6.1 Comparison with the range characterization in [23] We conclude this section by making a comparison between Theorem 6 and the range characterization of I A acting on 1-forms in [23,Theorem 1.3] when A is skew-hermitian. The first thing to observe is that due to our sign conventions P A,+ is precisely −P + in [23], so the main difference is the presence of I A (H A ), where H A was introduced in Subsection 5.1. The reason why H A does not appear in Theorem 6 is that we are only considering the range of I A,⊥ . In fact for a general GL(n, C)-connection A we have: Lemma 33. Assume I A is solenoidal injective on 1-forms. Then range I A = range I A,⊥ ⊕ I A (H −A * ).
Proof. The fact that the range splits follows directly from the decomposition (44) and the definitions. The sum is direct because of the following observation: if h ∈ H −A * is such that I A (h) ∈ range I A,⊥ then h = 0. Indeed, in this case there is f ∈ C ∞ (M, C n ) such that I A (⋆d A f + h) = 0. Since I A is solenoidal injective, we have that there is p ∈ C ∞ (M, C n ) with p| ∂M = 0 such that d A p = ⋆d A f + h. This implies right away that h = 0.
We conclude with an example showing that H A could be non-trivial. We note that H A transforms isomorphically under gauge equivalences and it is trivial for A = 0 (hence it is zero for any flat connection). For the example, suppose M is the unit disk with the standard metric. Consider the following map F : ∂M = S 1 → SU (2) given by Since SU (2) is simply connected F can be extended to a smooth map F : M → SU (2). Define the GL(2, C)-connection A := −(∂F )F −1 dz = Azdz. Thus ∂F + AzF = 0, .
We claim that there is a non-zero 1-form h such that d A h = d A ⋆ h = 0 and j * h = 0. Indeed, let h := hzdz + h z dz where hz(x, y) := 1 0 , h z (x, y) := F (x, y) 1 0 .
It is now natural to ask: is there a way to characterize the finite dimensional subspace I A (H −A * ) in terms of boundary data?
where b and b solve the following problems: where we have −V (K • ϕ t ) = b 2 (x, v, t)κ ⊥ (ϕ t (x, v)) [ 0 0 1 0 ] with κ ⊥ = X ⊥ κ. Using Φ itself as an integrating factor of the latter equation, we can arrive at the integral formula (keeping (x, v) implicit) (s), and picking particular entries of V Φ, we deduce the expressions In particular, we get Notice further the following cocycle property: Φ(x, v, t) = Φ(ϕ s (x, v), t − s)Φ(x, v, s), t ≥ s, true since, as functions of t both sides satisfy the ODE d dt U + K(ϕ t )U = 0 with matching condition at t = s. This equality can be recasted as Φ(ϕ s (x, v), t − s) = Φ(x, v, t)Φ −1 (x, v, s).
In particular, looking at the (1,2) entry in this matrix equality, we obtain the relation b 2 (ϕ s , t − s) = b 2 (t)b 1 (s) − b 1 (t)b 2 (s). (52) In particular, when (M, g) is simple with constants C 1 , C 2 as in (7), we can deduce the following estimate We then obtain the exact same estimate for V (b 1 /b 2 ): The lemma is proved.
As a corollary of Lemma 18, we can revisit the case without connection here. Indeed, the function w(x, v, t) = −V b 1 b 2 (t) is the kernel of the error operator in the case without connection, so we record the operator estimate here. We recall that where we have defined W(x, y) := 1 2π w(x,Exp −1 x (y)) b 2 (x,Exp −1 x (y)) . Then we write As a result, we obtain the bound: Vol (M ) 2π 1 2 .

B Proof of Lemma 15
Proof of Lemma 15. Let us define the function We will show that G vanishes identically by proving that G| ∂ − (SM ) = 0 and XG = 0 on SM .
In short, we obtain the matrix equation out of which XG = 0 is just Liouville's formula.