II

# Loop group factorization method for the magnetic and thermostatic nonabelian
ray transforms

Shubham R. Jathar  Indian Institute of Science Education and Research (IISER)
Bhopal, India<EMAIL_ADDRESS>,  Manas Kar  Indian Institute of
Science Education and Research (IISER) Bhopal, India<EMAIL_ADDRESS>and
Jesse Railo  Department of Pure Mathematics and Mathematical Statistics,
University of Cambridge, Cambridge, UK & Computational Engineering, School of
Engineering Sciences, Lappeenranta-Lahti University of Technology LUT,
Lappeenranta, Finland<EMAIL_ADDRESS>
###### Abstract.

We study the injectivity of the matrix attenuated and nonabelian ray
transforms on compact surfaces with boundary for nontrapping
$\lambda$-geodesic flows and the general linear group of invertible complex
matrices. We generalize the loop group factorization argument of Paternain and
Salo to reduce to the setting of the unitary group when $\lambda$ has the
vertical Fourier degree at most $1$. This covers the magnetic and thermostatic
flows as special cases. Our article settles the general injectivity question
of the nonabelian ray transform for the simple magnetic geodesic flows in
combination with an earlier result by Ainsworth. We stress that the
injectivity question in the unitary case for simple Gaussian thermostats
remains open. Furthermore, we observe that the loop group argument does not
apply when $\lambda$ has higher Fourier modes.

Keywords. Attenuated ray transform, nonabelian ray transform, magnetic
geodesics, Gaussian thermostats.

Mathematics Subject Classification (2020): 44A12, 58C99, 37E35

##  1\. Introduction

Our main goal in this article is to study the nonabelian $X$-ray transform for
general connection $A$ and a matrix-valued field $\Phi$ for the
$\lambda$-geodesic flows. We define $G$ to be the Lie group of matrices with a
Lie algebra $\mathfrak{g}$ and denote by $\Omega^{1}(M,\mathfrak{g})$ the
space of smooth $\mathfrak{g}$-valued $1$-forms. We will assume that
$A\in\Omega^{1}(M,\mathfrak{g})$ and $\Phi\in C^{\infty}(M,\mathfrak{g})$.
Formally, the quantities $A$ and $\Phi$ are called a Yang-Mills potential and
Higgs field, respectively. For a given pair $(A,\Phi)$, suppose that the map
$U:[0,\tau(x,v)]\rightarrow G$ satisfies the matrix ordinary differential
equation (ODE)

$\dot{U}+\left[A_{\gamma_{x,v}(t)}\left(\dot{\gamma}_{x,v}(t)\right)+\Phi\left(\gamma_{x,v}(t)\right)\right]U=0,\quad
U(\tau(x,v))=\mathrm{Id}$ (1.1)

along the $\lambda$-geodesic $\gamma_{x,v}$ starting at the influx boundary
point $(x,v)\in\partial_{+}SM$. We define the scattering data (or the
nonabelian $\lambda$-ray transform) along $\lambda$-geodesic $\gamma_{x,v}$ by
requiring $C_{A,\Phi}^{\lambda}(\gamma_{x,v}):=U(0).$ The identity (1.1)
reduces to

$\log
U(0)=\int_{0}^{\tau(x,v)}\left[A_{\gamma_{x,v}(t)}\left(\dot{\gamma}_{x,v}(t)\right)+\Phi\left(\gamma_{x,v}(t)\right)\right]dt$

if $A$ and $\Phi$ are assumed to be scalar $\mathbb{C}$-valued functions and
this transformation can be regarded as the classical $X$-ray transform of
$A+\Phi$ along the twisted geodesic flow. Note that $C_{A,\Phi}^{\lambda}$
maps $C_{A,\Phi}^{\lambda}:\partial_{+}SM\rightarrow G$.

######  Question 1.

Can one recover the pair $(A,\Phi)$ from the knowledge of the scattering data
$C_{A,\Phi}^{\lambda}$ corresponding to the $\lambda$-geodesic flow?

The nonabelian ray transform has appeared in many different areas of physics,
geometry and applied mathematics. In the case when the Higgs fields $\Phi=0$,
the scattering data $C_{A}$ represents the parallel transport of the
connection $A$ and the injectivity for the nonabelian $X$-ray transform is
related to the recovery of a connection up to a gauge from $C_{A}$ along the
geodesics of the metric $g$. For the unitary connections $A$, the scattering
data $C_{A}$ can be recovered from the connection Laplacian $d_{A}^{*}d_{A}$,
where $d_{A}=d+A$, at a high energy limit, and also this data can be generated
from the corresponding wave equation, see for instance [JS07, OSSU20, Uhl04]
for further details. When the connection $A=0$ and $\Phi\in
C^{\infty}(M,\mathfrak{so}(3))$, the nonabelian ray transform appears in
polarimetric neutron tomography, see [DLS+20, HMK+18], and in the case of
general attenuation pairs $(A,\Phi)$, it has applications in solitons, see
[MZ81, War88]. Other applications include coherent quantum tomography [Ilm16].
See also [Nov19] for further applications.

There is a natural obstruction to solve the above question. Let us define the
gauge group $\mathcal{G}$ consisting of smooth $u:M\rightarrow G$ such that
$u|_{\partial M}=\text{Id}$ and we note that the gauge group $\mathcal{G}$
acts on pairs $(A,\Phi)$ by requiring $(A,\Phi)\cdot
u=\left(u^{-1}du+u^{-1}Au,u^{-1}\Phi u\right)$. Observe that for any
$u\in\mathcal{G}$, one has $C_{(A,\Phi)\cdot
u}^{\lambda}=C_{A,\Phi}^{\lambda}.$ Therefore, the main interest of this
article is to understand the injectivity question for the nonabelian twisted
X-ray transform $(A,\Phi)\mapsto C_{A,\Phi}^{\lambda}$ up to the natural
gauge. Throughout the article, we will denote the set of $n\times n$ matrices
by $\mathbb{C}^{n\times n}$ or $\mathfrak{gl}(n,\mathbb{C})$. We can now state
the following conjecture related to the uniqueness of the nonabelian ray
transform up to the natural gauge for general connection and Higgs fields for
$\lambda$-geodesic flows.

######  Conjecture 1.1.

Let $(M,g,\lambda)$, $\lambda\in C^{\infty}(SM)$, be a compact Riemannian
surface with strictly $\lambda$-convex boundary and nontrapping
$\lambda$-geodesic flow. Let there be two pairs $(A,\Phi)$ and $(B,\Psi)$ such
that $A,B\in\Omega^{1}(M,\mathfrak{gl}(n,\mathbb{C}))$ and $\Phi,\Psi\in
C^{\infty}(M,\mathfrak{gl}(n,\mathbb{C}))$. If

$C_{A,\Phi}^{\lambda}=C_{B,\Psi}^{\lambda},$

then there exists $u\in\mathcal{G}$ such that $(A,\Phi)\cdot u=(B,\Psi)$.

Conjecture 1.1 has been settled in [PS20] when $\lambda=0$ and with no
conjugate point assumption. Notice that the gauge is trivial with the priori
assumption $A,B=0$. The principle idea behind solving Conjecture 1.1 in the
case of $\lambda=0$ is to reduce the problem into the attenuated $X$-ray
transform for general connection and Higgs fields via a pseudolinearization
identity and then study this attenuated $X$-ray transform. In particular, they
proved Conjecture 1.2 for $\lambda=0$. To proceed further, we consider the
unique solution $u(t)$ for the vector-valued ordinary differential equation

$\dot{u}+\left[A_{\gamma_{x,v}(t)}\left(\dot{\gamma}_{x,v}(t)\right)+\Phi\left(\gamma_{x,v}(t)\right)\right]u=-f\left(\gamma_{x,v}(t),\dot{\gamma}_{x,v}(t)\right),\quad
u(\tau(x,v))=0,$

where ${\gamma}_{x,v}(t)$ is a $\lambda$-geodesic with $\gamma_{x,v}(0)=x$,
$\dot{\gamma}_{x,v}(0)=v$ and
$A\in\Omega^{1}(M,\mathfrak{gl}(n,\mathbb{C})),\Phi\in
C^{\infty}(M,\mathfrak{gl}(n,\mathbb{C}))$, $f\in
C^{\infty}\left(SM,\mathbb{C}^{n}\right)$ are given known quantities. The
attenuated twisted $X$-ray transform of $f$ can be defined by

$I_{A,\Phi}^{\lambda}(f)(x,v):=u(0)$

where $(x,v)\in\partial_{+}SM$. We now formulate another conjecture.

######  Conjecture 1.2.

Let $(M,g,\lambda)$, $\lambda\in C^{\infty}(SM)$, be a compact Riemannian
surface with strictly $\lambda$-convex boundary and nontrapping
$\lambda$-geodesic flow. Let $(A,\Phi)$ be an arbitrary attenuation pair with
$A\in\Omega^{1}(M,\mathfrak{gl}(n,\mathbb{C}))$ and $\Phi\in
C^{\infty}(M,\mathfrak{gl}(n,\mathbb{C}))$. Let $f=f_{0}+\alpha\in
C^{\infty}(SM,{\mathbb{C}}^{n})$ be a smooth function where $f_{0}\in
C^{\infty}(M,{\mathbb{C}}^{n})$ and $\alpha$ is a ${\mathbb{C}}^{n}$-valued
$1$-form. If $I_{A,\Phi}^{\lambda}(f)=0$, then $f_{0}=\Phi p$ and
$\alpha=dp+Ap$ for some $p\in C^{\infty}(M,{\mathbb{C}}^{n})$ with
$\left.p\right|_{\partial M}=0.$

In [PS20], Conjecture 1.2 has been proved in the case of $\lambda=0$ and
absence of conjugate points. They used a factorization theorem for Loop groups
to reduce the problem of attenuated $X$-ray transform for general connections
and Higgs fields to the problem of attenuated $X$-ray transform for unitary
connections and skew-Hermitian Higgs fields, which is precisely Conjecture 1.3
for $\lambda=0$. Conjectures 1.1 and 1.3 were proved in [PSU12] for unitary
connections $A$ and skew-Hermitian Higgs fields $\Phi$ for usual geodesic
flows. In [PS21], Conjecture 1.2 has been proved for $\lambda=0$ in the case
of compact Riemannian nontrapping negatively curved manifold with smooth
strictly convex boundary for general connections and Higgs fields, and they
used certain Carleman estimates related to the geodesic flow. This result has
been extended to the case of attenuated ray transform for Gaussian thermostat
on a compact Riemannian surface with negative thermostat curvature in [AR21].
In particular, they proved an injectivity result for thermostat ray transform
with a general connection and Higgs fields up to a gauge transformation under
curvature assumptions. We denote by $\mathfrak{u}(n)$ the Lie algebra of the
unitary group. We now state the following conjecture.

######  Conjecture 1.3.

Let $(M,g,\lambda)$, $\lambda\in C^{\infty}(SM)$, be a compact Riemannian
surface with strictly $\lambda$-convex boundary and nontrapping
$\lambda$-geodesic flow. Let $A:SM\rightarrow\mathfrak{u}(n)$ be a unitary
connection and $\Phi:M\rightarrow\mathfrak{u}(n)$ be a skew-Hermitian matrix
function. Let $f=f_{0}+\alpha\in C^{\infty}(SM,{\mathbb{C}}^{n})$ be a smooth
function where $f_{0}\in C^{\infty}(M,{\mathbb{C}}^{n})$ and $\alpha$ is a
${\mathbb{C}}^{n}$-valued $1$-form. If $I_{A,\Phi}^{\lambda}(f)=0$, then
$f_{0}=\Phi p$ and $\alpha=dp+Ap$ for some $p\in
C^{\infty}(M,{\mathbb{C}}^{n})$ with $\left.p\right|_{\partial M}=0.$

Conjecture 1.3 has been established in the case of $\lambda=0$ and no
conjugate point assumption, see [PSU12, Theorem 1.3]. In addition to that,
they also proved Conjecture 1.1 for Hermitian connections and skew-Hermitian
Higgs fields in a compact simple Riemannian surface in the case of usual
geodesics, see for instance [PSU12, Theorem 1.5]. The idea of the proof of
Conjecture 1.3 for $\lambda=0$ in [PSU12] is based on a proof similar to
Kodaira vanishing theorem in complex geometry to the transport problem
relevant for the corresponding nonabelian X-ray transform. In particular, they
used a scalar holomorphic integrating factor and applied a suitable Pestov
identity. A similar approach has been followed in [Ain13, Theorem 1.2] to
prove Conjecture 1.3 in the presence of magnetic geodesic flows. So far,
[Ain13, Theorem 1.2] is the best-known result when $\lambda$ is assumed to be
a function on $M$.

###  1.1. Main results

In this article, we first show that Conjecture 1.2 implies Conjecture 1.1 via
a pseudolinearization identity for nontrapping $\lambda$-geodesic flows. The
main contribution of this article is to find out under what conditions on
$\lambda$, Conjecture 1.3 implies Conjecture 1.2. We also state here
interesting conclusions on the generalization of the results in [Ain13] for
the simple magnetic flows on surfaces with boundary.

We now state our main theorems in this article and some of its corollaries
related to the nonabelian ray transform. To do that, let us first introduce
$\Omega_{k}=C^{\infty}(SM,{\mathbb{C}}^{n})\cap H_{k}$, where $H_{k}$ is the
eigenspace of $-iV$ corresponding to the eigenvalue $k$ and $V$ denotes the
vertical vector field. In more detail, we will discuss such function spaces
and related vector fields in Section 2. Our first theorem is an application of
pseudolinearization identity to prove the uniqueness of the nonabelian ray
transform up to the natural gauge for general $\lambda$ whenever one assumes
the injectivity result for the attenuated ray transform for general
connections and Higgs fields for $\lambda\in C^{\infty}(SM)$ up to a natural
obstruction.

######  Theorem 1.4.

Let $(M,g,\lambda)$, $\lambda\in C^{\infty}(SM)$, be a compact Riemannian
surface with strictly $\lambda$-convex boundary and nontrapping
$\lambda$-geodesic flow. If Conjecture 1.2 holds for $\lambda$, then
Conjecture 1.1 is true for $\lambda$.

The next theorem shows that Conjecture 1.3 implies Conjecture 1.2 whenever the
vertical Fourier degree of $\lambda$ is at most $1$.

######  Theorem 1.5.

Let $(M,g,\lambda)$, $\lambda\in C^{\infty}(SM)$, be a compact Riemannian
surface with strictly $\lambda$-convex boundary and nontrapping
$\lambda$-geodesic flow. Let
$\lambda\in\Omega_{-1}\oplus\Omega_{0}\oplus\Omega_{1}$ and assume that
Conjecture 1.3 holds for $\lambda$, then Conjecture 1.2 is true for $\lambda$.

Of course, trivially for any $\lambda\in C^{\infty}(SM)$ Conjecture 1.2
implies Conjecture 1.3. In the proof of Theorem 1.5, we mainly employ the loop
group factorization arguments to reduce the attenuated ray transform for
general connections and Higgs fields to the attenuated ray transform for the
unitary connections and skew-Hermitian Higgs fields when $\lambda$ is assumed
to be the sum of the function and $1$-form. However, in the case where
$\lambda\in\oplus_{k=-j}^{k=j}\Omega_{k}$, for $j\geq 2$, the argument based
on the loop group factorization no longer works.

######  Corollary 1.6.

Let $(M,g,\lambda)$, $\lambda\in C^{\infty}(SM)$, be a compact Riemannian
surface with strictly $\lambda$-convex boundary and nontrapping
$\lambda$-geodesic flow. Let
$\lambda\in\Omega_{-1}\oplus\Omega_{0}\oplus\Omega_{1}$ and assume that
Conjecture 1.3 holds for $\lambda$. Suppose we are given pairs $(A,\Phi)$ and
$(B,\Psi)$ with $A,B\in\Omega^{1}(M,\mathfrak{gl}(n,\mathbb{C}))$ and
$\Phi,\Psi\in C^{\infty}(M,\mathfrak{gl}(n,\mathbb{C}))$. If

$C_{A,\Phi}^{\lambda}=C_{B,\Psi}^{\lambda},$

then there is $u\in\mathcal{G}$ such that $(A,\Phi)\cdot u=(B,\Psi)$.

The above corollary is a direct consequence of Theorem 1.4 and Theorem 1.5.

In the case of magnetic geodesics, we can prove the following injectivity
result for the attenuated ray transform for general connections and Higgs
fields utilizing [Ain13, Theorem 1.2] and loop group factorization arguments
summarized in the earlier discussions and results. We say that $(M,g,\lambda)$
is a simple magnetic surface if $\lambda\in C^{\infty}(M)$, $M$ is compact
with smooth strictly magnetic convex boundary and the magnetic flow is
nontrapping without conjugate points.

######  Theorem 1.7 (Uniqueness for the attenuated magnetic ray transform).

Let $(M,g,\lambda)$ be a simple magnetic surface. Let $(A,\Phi)$ be an
arbitrary attenuation pair $(A,\Phi)$ with
$A\in\Omega^{1}(M,\mathfrak{gl}(n,\mathbb{C}))$ and $\Phi\in
C^{\infty}(M,\mathfrak{gl}(n,\mathbb{C}))$. Let $f=f_{0}+\alpha\in
C^{\infty}(SM,{\mathbb{C}}^{n})$ be a smooth function where $f_{0}\in
C^{\infty}(M,{\mathbb{C}}^{n})$ and $\alpha$ is a ${\mathbb{C}}^{n}$-valued
$1$-form. If $I_{A,\Phi}^{\lambda}f=0$, then $f_{0}=\Phi p$ and $\alpha=dp+Ap$
for some $p\in C^{\infty}(M,{\mathbb{C}}^{n})$ with $\left.p\right|_{\partial
M}=0.$

######  Corollary 1.8 (Uniqueness for the nonabelian magnetic ray transform).

Let $(M,g,\lambda)$ be a simple magnetic surface. Suppose we are given pairs
$(A,\Phi)$ and $(B,\Psi)$ with
$A,B\in\Omega^{1}(M,\mathfrak{gl}(n,\mathbb{C}))$ and $\Phi,\Psi\in
C^{\infty}(M,\mathfrak{gl}(n,\mathbb{C}))$. If

$C_{A,\Phi}^{\lambda}=C_{B,\Psi}^{\lambda},$

then there is $u\in\mathcal{G}$ such that $(A,\Phi)\cdot u=(B,\Psi)$.

The above corollary is a direct consequence of Corollary 1.6 and Theorem 1.7.

###  1.2. Further discussion on the earlier literature

There is an extensive literature to study the nonabelian ray transform in the
case of $\lambda=0$. For the usual geodesic case, Conjectures 1.1 and 1.2 are
proved in $\mathbb{R}^{3}$, see [Nov02], and in [PSUZ19] in the case of
compact manifolds with strictly convex boundary of dimensions three and
higher, and admitting a strictly convex function. This is based on an approach
for inverting the local geodesic ray transform [UV16]. For the pair $(A,\Phi)$
taking value in $\mathfrak{u}(n)$, Conjectures 1.1 and 1.2 are proved in
[PSU12] when $\lambda=0$ on a simple nontrapping Riemannian surface. In
[Nov02], Novikov proved the local uniqueness results in $\mathbb{R}^{n},n\geq
2$ under the assumption that the attenuation pairs $(A,\Phi)$ are not
necessarily compactly supported with certain decay conditions at infinity.
However, he constructed an example to show that the global uniqueness result
fails in general. Later, a similar to Conjecture 1.1, an injectivity question
for the nonabelian Radon transform has been considered by Eskin in [Esk04] for
compactly supported attenuated pairs $(A,\Phi)$ and the main idea of their
proof uses the existence of matrix holomorphic integrating factors in the
vertical variable. We also mention the work of Finch and Uhlmann [FU01] that
proves Conjecture 1.1 in the domains of $\mathbb{R}^{2}$ for the connections
having small curvature.

Under certain restrictions on the size of the attenuation pairs $(A,\Phi)$,
the nonabelian ray transform has been first considered in [Ver91, Ver00,
Sha00] in a manifold setting. In [Sha00], Sharafutdinov established Conjecture
1.1 for the case $\Phi=0$ and assuming that the connections are $C^{1}$ close
to another connection with small curvature. Conjecture 1.2 was proved in
[SU11] for $A=0$ and $n=1$ when $\lambda=0.$ Generic injectivity and stability
issues are discussed and some Fredholm alternatives are considered for
connections in [MP20, Zho17]. Conjecture 1.1 is proved under negative
curvature assumption for $\lambda=0$ in [PS21]. In [BP23], the authors
established the existence of matrix holomorphic integrating factors for any
simple surface where they provided a full characterization of the range of the
nonabelian X-ray transform. For the closed Riemannian surfaces, the nonabelian
ray transform problems are considered in [Pat13].

Compared to the usual geodesic case, not much is known in the nonabelian ray
transform literature for $\lambda\neq 0$. In [Ain13], Ainsworth proved
Conjectures 1.1 and 1.3 for the magnetic geodesics on simple surfaces with a
unitary connection and a skew-Hermitian matrix Higgs field. Recently,
Conjectures 1.1 and 1.2 have been established on a compact Riemannian surface
under the assumption of negative thermostat curvature in [AR21]. Finally, we
mention that many other integral geometry problems are studied extensively for
magnetic and thermostatic flows [DP07a, DP07b, DPSU07, Ain13, AA15, Ain15,
AZ17, AD18, AR21, MST23].

###  1.3. Organization of the article

In Section 2, we introduce some basic notations, definitions and properties of
twisted geodesic flows that will be required in the later sections. Section 3
deals with the attenuated and nonabelian ray transform for twisted geodesics.
The proofs of main results are given in Section 4. In Appendices A and B, we
consider some auxiliary results for nontrapping $\lambda$-geodesic flows and
the smoothness of the kernel of the (matrix) attenuated $\lambda$-ray
transforms, respectively.

### Acknowledgments

J.R. thanks Gabriel P. Paternain for many helpful discussions related to the
topics of this work. S.R.J. and J.R. would like to thank the Isaac Newton
Institute for Mathematical Sciences, Cambridge, UK, for support and
hospitality during _Rich and Nonlinear Tomography - a multidisciplinary
approach_ in 2023 where part of this work was done (supported by EPSRC Grant
Number EP/R014604/1). S.R.J. acknowledges the Prime Minister’s Research
Fellowship (PMRF) from the Government of India for his PhD work. M.K. was
supported by MATRICS grant (MTR/2019/001349) of SERB. J.R. was supported by
the Vilho, Yrjö and Kalle Väisälä Foundation of the Finnish Academy of Science
and Letters.

##  2\. Preliminaries

Throughout this article, we mainly follow [PSU23] for the used notations,
definitions and some relevant properties. Similar concepts appear earlier in
many works, for example, in [GK80, ST76]. Basics related to the twisted
geodesic flows can be recalled from [MP11].

Assume $(M,g)$ to be a compact Riemannian surface with smooth boundary and $g$
being a Riemannian metric. Let $SM=\\{(x,v)\in TM:\langle v,v\rangle_{g}=1\\}$
be the unit tangent bundle. We denote $\partial_{+}SM$ by the influx boundary
where the tangent vectors point inside the manifold and $\partial_{-}SM$ by
the outflux boundary where the tangent vectors point outside the manifold.
More precisely, these sets are defined by

$\partial_{\pm}SM:=\\{(x,v)\in SM:x\in\partial M,\ \pm\langle
v,\nu(x)\rangle_{g}\geq 0\\}$ (2.1)

where $\nu$ is the inward unit normal to $\partial M.$ For any $\lambda\in
C^{\infty}(SM)$, a curve $\gamma$ is called a $\lambda$-geodesic (or twisted
geodesic) if it satisfies the $\lambda$-geodesic equation

$D_{t}\dot{\gamma}=\lambda(\gamma,\dot{\gamma})i\dot{\gamma}$ (2.2)

where $i$ is the counterclockwise rotation by $90$ degrees [MP11, Definition
7.1]. In general, the term $\lambda(\gamma,\dot{\gamma})i\dot{\gamma}$
represents an external force that pushes a particle out from the usual
geodesic trajectory, corresponding to the case $\lambda=0$. We say that a
curve $\gamma$ is a magnetic geodesic if it satisfies

$D_{t}\dot{\gamma}=Y(\dot{\gamma}),$

where the Lorentz force $Y:TM\to TM$. The magnetic field $\Omega$, being a
closed $2$-form on $M$, is related to the Lorentz force via the unique bundle
map

$\Omega_{x}(\xi,\eta)=\left\langle
Y_{x}(\xi),\eta\right\rangle_{g},\quad\forall x\in M,\ \xi,\eta\in T_{x}M.$
(2.3)

The curve is said to be Gaussian thermostatic (or thermostatic geodesic) if
one has the identity

$D_{t}\dot{\gamma}=E(\gamma)-\frac{\langle
E(\gamma),\dot{\gamma}\rangle}{|\dot{\gamma}|^{2}}\dot{\gamma},$

where $E$ is a smooth vector field on $M$. In the case of magnetic geodesics,
$\lambda(x,v)$ only depends on $x$. In the case of Gaussian thermostats,
$\lambda(x,v)$ depends linearly on $v$.

By abuse of notation, we continue to write $\gamma_{x,v}$ for the twisted
geodesic starting at $x$ with the velocity $v$ where $(x,v)\in SM$. We define
the $\lambda$-geodesic flow $\phi_{t}:SM\rightarrow SM$ by

$\phi_{t}(x,v):=(\gamma_{x,v}(t),\dot{\gamma}_{x,v}(t)),$

where $\gamma_{x,v}$ is a $\lambda$-geodesic. In our main results, we assume
that the manifold $(M,g)$ has nontrapping $\lambda$-geodesic flow, which means
that the forward and backward exit times

$\tau^{\pm}(x,v):=\inf\\{\,t\geq 0\,:\,\phi_{\pm t}(x,v)\in\partial SM\,\\}$

of the $\lambda$-geodesics are finite for all $(x,v)\in\operatorname{int}SM$
and $\tau^{\pm}$ can be extended continuously to $\partial SM$. We also make a
strict $\lambda$-convexity assumption on the boundary of the manifold. The
boundary $\partial M$ is strictly $\lambda$-convex at a point $x\in\partial M$
if the following inequality holds:

$\sff_{x}(v,v)>-\left\langle\lambda(x,v)iv,\nu(x)\right\rangle\quad\text{ for
all }v\in S_{x}(\partial M),$

where $\sff$ is the second fundamental form of the boundary $\partial M.$ We
refer the reader to [JKR23] for a further discussion on strict
$\lambda$-convexity. If $(M,g,\lambda)$ is strictly $\lambda$-convex and
nontrapping without conjugate points, then we say that the $\lambda$-geodesic
flow is simple.

By [MP11, Lemma 7.4], the infinitesimal generator of $\phi_{t}$ can be written
as $X+\lambda V,$ where $X$ denote the usual geodesic vector field acting on
the unit circle bundle $SM$ and $V$ is the vertical vector field. Denote
$X_{\perp}:=[X,V]$ and that allows one to define $\\{X,X_{\perp},V\\}$ as a
global frame of $T(SM)$. The following structure equations hold:

$[X,V]=X_{\bot},\quad[X_{\bot},V]=-X,\quad[X,X_{\bot}]=-KV$ (2.4)

where $K$ is the Gaussian curvature of the Riemannian surface $(M,g)$, see
[PSU23, Lemma 3.5.5]. Since $\\{X,-X_{\bot},V\\}$ forms a positively oriented
orthonormal frame, it follows that the unit sphere bundle $SM$ possesses a
unique Riemmanian metric $G$, called the Sasaki metric and the volume form of
the metric satisfies $dV_{G}=d\Sigma^{3}$, see [PSU23, Lemma 3.5.11]. Here
$d\Sigma^{3}$ represents a measure on $SM$, defined by

$d\Sigma^{3}:=dV^{2}\wedge dS_{x}$ (2.5)

is usually called the Liouville form, where $dV^{2}$ is the volume form of
$(M,g)$ and $dS_{x}$ is the volume form of $(S_{x}M,g_{x})$ for any $x\in M$.
For any $u,v:SM\to{\mathbb{C}}^{n}$, we define the following $L^{2}$-inner
product

$(u,v)=\int_{SM}\langle u,v\rangle_{\mathbb{{\mathbb{C}}}^{n}}\,d\Sigma^{3}.$

We also define the spaces of eigenvectors of $V$ by requiring

$H_{k}:=\\{\,u\in
L^{2}(SM,{\mathbb{C}}^{n})\,;\,-iVu=ku\,\\},\quad\Omega_{k}:=\\{\,u\in
C^{\infty}(SM,{\mathbb{C}}^{n})\,;\,-iVu=ku\,\\}$ (2.6)

for any integer $k\in{\mathbb{Z}}$. Now we remark that the space
$L^{2}(SM,{\mathbb{C}}^{n})$ has a following orthogonal decomposition of
direct sum

$L^{2}(SM,{\mathbb{C}}^{n})=\bigoplus_{k\in\mathbb{Z}}H_{k}.$

For any $u\in L^{2}(SM)$, there is a unique $L^{2}$-orthogonal decomposition

$u=\sum_{k=-\infty}^{\infty}u_{k},\quad\lVert
u\rVert^{2}=\sum_{k=-\infty}^{\infty}\lVert u_{k}\rVert^{2},\quad u_{k}\in
H_{k}.$ (2.7)

If $u\in C^{\infty}(SM)$, then each $u_{k}\in C^{\infty}(SM)$ and the series
converges in the $C^{\infty}(SM)$ norm.

For any $I\subset\mathbb{Z}$, we denote by $\oplus_{k\in I}\Omega_{k}$ the set
of smooth functions $u$ satisfying $u_{k}=0$ whenever $k\notin I$. Note that
the vector fields $X$ and $V$ have the following mapping properties:

$X:\oplus_{k\geq 0}\Omega_{k}\to\oplus_{k\geq-1}\Omega_{k},\qquad
V:\Omega_{k}\to\Omega_{k}$

see for instance [PSU23].

If we assume that $\lambda\in C^{\infty}(SM,\mathbb{C})$ has finite degree
$m$. Then $\lambda$ can be written as

$\lambda=\sum_{k=-m}^{m}\lambda_{k}\
\in\Omega_{-m}\oplus\cdots\oplus\Omega_{m}.$

Notice that, if $u\in\Omega_{k}$ and $\lambda_{\pm m}\in\Omega_{\pm m}$, then
one could see that

$-iV(\lambda_{\pm m}Vu)=\lambda_{\pm m}(-iV(Vu))+(-iV(\lambda_{\pm
m}))Vu=(k\pm m)\lambda_{\pm m}Vu.$

Therefore $\lambda_{\pm m}V$ has the following mapping properties

$\lambda_{\pm m}V:\Omega_{k}\rightarrow\Omega_{k\pm m}.$

Note that all above smooth functions are taking values in complex matrices
$\mathfrak{gl}(n,{\mathbb{C}})$.

######  Definition 2.1.

A function $u:SM\to{\mathbb{C}}^{n}$ is said to be (fibrewise) holomorphic if
$u_{k}=0$ for all $k<0$. Similarly, $u$ is said to be (fibrewise)
antiholomorphic if $u_{k}=0$ for all $k>0$.

##  3\. Attenuated and nonabelian twisted ray transforms

Let $(M,g,\lambda),\lambda\in C^{\infty}(SM)$, be a compact Riemannian surface
with strictly $\lambda$-convex boundary and nontrapping $\lambda$-geodesic
flow. Assume $\mathcal{A}\in C^{\infty}\left(SM,\mathbb{C}^{n\times
n}\right)$. We introduce a closed manifold $(N,g)$ so that $(M,g)$ is
isometrically embedded in $(N,g)$ and we extend $\mathcal{A}$ and $\lambda$
smoothly to whole of $N$. On the closed manifold $(N,g)$, the matrix valued
function $\mathcal{A}$ defines a smooth cocycle $C$ over the
$\lambda$-geodesic flow $\phi_{t}$. We call $C:SN\times\mathbb{R}\to
GL(n,\mathbb{C})$ as a smooth cocycle if it satisfies the matrix ordinary
differential equation

$\frac{\mathrm{d}}{\mathrm{d}t}C(x,v,t)+\mathcal{A}\left(\phi_{t}(x,v)\right)C(x,v,t)=0,\quad
C(x,v,0)=\mathrm{Id}$ (3.1)

along the orbits of $\lambda$-geodesic flows. From the uniqueness of solution
of ODEs and $\phi_{t}$ is $\lambda$-geodesic flow, the function $C$ satisfies

$\displaystyle C(x,v,t+s)$ $\displaystyle=C(\phi_{t}(x,v),s)C(x,v,t)$ (3.2)

for all $(x,v)\in SN$ and $t,s\in\mathbb{R}$. The identity (3.2) explains the
name cocycle.

Let $(M,g,\lambda)$, with $\lambda\in C^{\infty}(SM)$, be a compact Riemannian
surface with strictly $\lambda$-convex boundary and nontrapping
$\lambda$-geodesic flow. From Lemma A.4, the existence of a nontrapping
$\lambda$-geodesic flow is an open condition. Similarly, from the definition
of strictly $\lambda$-convexity, it is also an open condition. Hence, there
exists a slightly larger manifold $(M_{0},g,\lambda_{0})$ engulfing $M$, such
that $\lambda_{0}$ is a smooth extension of $\lambda$ on $M_{0}$, and $M_{0}$
has strictly $\lambda_{0}$-convex boundary and a nontrapping
$\lambda_{0}$-geodesic flow.

######  Definition 3.1.

Let $(M,g,\lambda)$, $\lambda\in C^{\infty}(SM)$, be a compact Riemannian
surface with strictly $\lambda$-convex boundary and nontrapping
$\lambda$-geodesic flow. Suppose there exists a larger compact manifold
$(M_{0},g_{0},\lambda_{0})$ containing $M$ such that:

  * •

$\lambda_{0}\in C^{\infty}(SM_{0})$ is a smooth extension of $\lambda$ to
$M_{0}$,

  * •

the Riemannian metric $g_{0}$ is a smooth extension of $g$ to $M_{0}$

  * •

$M_{0}$ has strictly $\lambda_{0}$-convex boundary,

  * •

$M_{0}$ has nontrapping $\lambda_{0}$-geodesic flow.

Then $(M_{0},g_{0},\lambda_{0})$ is called as a _smooth strictly convex,
nontrapping, extension_ of $(M,g,\lambda)$.

Next lemma is a generalisation of [PSU23, Lemma 5.3.2] for the cocycle $C$
corresponding to the $\lambda$-geodesic flows $\phi_{t}$.

######  Lemma 3.2.

Let $(M,g,\lambda)$, $\lambda\in C^{\infty}(SM)$, be a compact Riemannian
surface with strictly $\lambda$-convex boundary and nontrapping
$\lambda$-geodesic flow. Let $(M_{0},g,\lambda_{0})$ be a smooth strictly
convex, nontrapping, extension as defined in Definition 3.1. and $\tau_{0}$
the forward travel time in $M_{0}$. Define a function $R:SM\to
GL(n,\mathbb{C})$ such that

$R(x,v):=\left[C\left(x,v,\tau_{0}(x,v)\right)\right]^{-1}.$

Then $R$ is a smooth function and satisfies

$\displaystyle(X+\lambda V)R+\mathcal{A}R=0,\qquad(X+\lambda
V)R^{-1}-R^{-1}\mathcal{A}=0.$ (3.3)

###### Proof.

Recall that $\tau_{0}|_{SM}$ is a smooth function on $SM$ (cf. [JKR23, Lemma
4.16]), and the cocycle is also a smooth function. Therefore, the function $R$
is smooth on $SM$. Replacing $\tau_{0}(\phi_{t}(x,v))=\tau_{0}(x,v)-t$ in
(3.2), we can write

$\displaystyle
C\left(x,v,t+\tau_{0}-t\right)=C\left(\varphi_{t}(x,v),\tau_{0}(x,v)-t\right)C(x,v,t),$

for all $t\in[0,\tau_{0}(x,v)]$ and simplifying further yields

$\displaystyle[C\left(x,v,\tau_{0}(x,v)\right)][C(x,v,t)]^{-1}=[C\left(\varphi_{t}(x,v),\tau_{0}({x,v})-t\right)].$

By the definition of $R$, we obtain

$\displaystyle R(\phi_{t}(x,v))$
$\displaystyle=[C(\phi_{t}(x,v),\tau_{0}({\phi_{t}(x,v)}))]^{-1}=[C(\phi_{t}(x,v),\tau_{0}({x,v})-t)]^{-1}$
$\displaystyle=C(x,v,t)[C\left(x,v,\tau_{0}({x,v})\right)]^{-1}=C(x,v,t)R(x,v).$

Differentiating the above identity with respect $t$ and evaluating at $t=0$,
we have

$\displaystyle(X+\lambda V)R(x,v)$
$\displaystyle=\left.\frac{d}{dt}R(\phi_{t}(x,v))\right|_{t=0}=\left.\frac{d}{dt}C(x,v,t)\right|_{t=0}R(x,v)$
$\displaystyle=-\left.\mathcal{A}\left(\phi_{t}(x,v)\right)C(x,v,t)\right|_{t=0}R(x,v)$
$\displaystyle=-\mathcal{A}(x,v)R(x,v).$

To prove the other identity, we first compute

$\displaystyle 0$
$\displaystyle=\frac{d}{dt}\left(R(\phi_{t}(x,v))(R(\phi_{t}(x,v))^{-1})\right)$
$\displaystyle=\left(\frac{d}{dt}R(\phi_{t}(x,v))\right)(R(\phi_{t}(x,v))^{-1}+R(\phi_{t}(x,v))\left(\frac{d}{dt}(R(\phi_{t}(x,v))^{-1}\right).$

Then we have

$\displaystyle(X+\lambda V)R^{-1}(x,v)$
$\displaystyle=(\left.\frac{d}{dt}\left(R\left(\phi_{t}(x,v)\right)^{-1})\right|_{t=0}\right.$
$\displaystyle=-(R(x,v))^{-1}(X+\lambda V)R(x,v)(R(x,v))^{-1}$
$\displaystyle=(R(x,v))^{-1}\mathcal{A}(x,v),$

which establishes the formula. ∎

Remark that the solution of equations (3.3) in the above lemma are usually
called the matrix integrating factor.

######  Definition 3.3 (Nonabelian $\lambda$-ray transform).

Let $(M,g,\lambda)$, $\lambda\in C^{\infty}(SM)$, be a compact Riemannian
surface with strictly $\lambda$-convex boundary and nontrapping
$\lambda$-geodesic flow. The $\lambda$-scattering data of $\mathcal{A}$ is the
map

$C_{{\mathcal{A}}}^{\lambda}:\partial_{+}SM\to GL(n,{\mathbb{C}})$

defined by $C_{{\mathcal{A}}}^{\lambda}:=U_{+}|_{\partial_{+}SM}$ where
$U_{+}(x,v):=[C(x,v,\tau(x,v))]^{-1}$ satisfies

$\displaystyle(X+\lambda V)U_{+}+{\mathcal{A}}U_{+}=0,\qquad
U_{+}|_{\partial_{-}SM}=\operatorname{Id}.$

We call $C_{{\mathcal{A}}}^{\lambda}$ the nonabelian $\lambda$-ray transform
of ${\mathcal{A}}$.

It is immediate to see that $C_{\mathcal{A}}^{\lambda}\in
C^{\infty}\left(\partial_{+}SM,\mathbb{C}^{n\times n}\right)$.

######  Definition 3.4 (Attenuated $\lambda$-ray transform).

Let $(M,g,\lambda)$, $\lambda\in C^{\infty}(SM)$, be a compact Riemannian
surface with strictly $\lambda$-convex boundary and nontrapping
$\lambda$-geodesic flow. The attenuated $\lambda$-ray transform of $f\in
C^{\infty}(SM,{\mathbb{C}}^{n})$ is defined by

$I_{{\mathcal{A}}}^{\lambda}f:=u^{f}|_{\partial_{+}SM}$

where $u^{f}:SM\to{\mathbb{C}}^{n}$ is a solution to the following transport
equation

$(X+\lambda V)u^{f}+{\mathcal{A}}u^{f}=-f\text{ in }SM,\qquad
u^{f}|_{\partial_{-}SM}=0.$ (3.4)

Next lemma is a generalisation of [PS20, Lemma 3.4] to the case of
$\lambda$-geodesic flow. The lemma allows one to write down the solution of
(3.4) as an integral representation using a matrix integrating factor.

######  Lemma 3.5.

Let $(M,g,\lambda)$, $\lambda\in C^{\infty}(SM)$, be a compact Riemannian
surface with strictly $\lambda$-convex boundary and nontrapping
$\lambda$-geodesic flow. If the function $R:SM\rightarrow GL(n,\mathbb{C})$
satisfies $(X+\lambda V)R+\mathcal{A}R=0$, then

$u^{f}(x,v)=R(x,v)\int_{0}^{\tau(x,v)}\left(R^{-1}f\right)\left(\varphi_{t}(x,v)\right)dt\
\text{ for }(x,v)\in SM.$

###### Proof.

Using $(X+\lambda V)R^{-1}=R^{-1}\mathcal{A}$, see Lemma 3.2, and $(X+\lambda
V)u^{f}+\mathcal{A}u^{f}=-f$, we obtain

$\displaystyle(X+\lambda V)(R^{-1}u^{f})$ $\displaystyle=(X+\lambda
V)(R^{-1})u^{f}+R^{-1}(X+\lambda V)u^{f}$
$\displaystyle=R^{-1}\mathcal{A}u^{f}-R^{-1}({\mathcal{A}}u^{f}+f)=-R^{-1}f.$
(3.5)

Since $R^{-1}u^{f}|_{\partial_{-}SM}=0$, one immediately has

$\displaystyle
u^{f}(x,v)=R(x,v)\int_{0}^{\tau(x,v)}\left(R^{-1}f\right)\left(\varphi_{t}(x,v)\right)dt\
\text{ for }(x,v)\in SM$ (3.6)

and hence the lemma follows. ∎

We state a regularity result for the transport equation for $\lambda\in
C^{\infty}(SM)$. In the case where $\lambda=0$, this is proved in [PSU12,
Proposition 5.2], and when $\lambda\in C^{\infty}(M)$, it is proved in [Ain13,
Proposition 4.3]. For the thermostatic case, see [AR21, Proposition 4.3]. The
proof is similar in nature and, for the sake of completeness, is given in
Appendix B.

######  Lemma 3.6.

Let $(M,g,\lambda)$, $\lambda\in C^{\infty}(SM)$, be a compact Riemannian
surface with strictly $\lambda$-convex boundary and nontrapping
$\lambda$-geodesic flow. If $I_{\mathcal{A}}^{\lambda}(f)=0$ for $f\in
C^{\infty}(SM;\mathbb{C}^{n})$, then $u^{f}:SM\to\mathbb{C}^{n}$ is a smooth
function.

###  3.1. Pseudolinearization identity

Let $\mathcal{A},\mathcal{B}\in C^{\infty}(SM,\mathbb{C}^{n\times n})$ be two
smooth matrix-valued functions on the unit tangent bundle $SM$. Our aim in
this subsection is to establish a relation between $C^{\lambda}_{\mathcal{A}}$
and $C^{\lambda}_{\mathcal{B}}$ using a specific attenuated $\lambda$-ray
transform. To this end, we define a map
$E({\mathcal{A}},{\mathcal{B}}):SM\to\operatorname{End}(\mathbb{C}^{n\times
n})$ by setting

$E({\mathcal{A}},{\mathcal{B}})U:={\mathcal{A}}U-U{\mathcal{B}},$

where $E({\mathcal{A}},{\mathcal{B}})$ denotes the linear endomorphisms of
$\mathbb{C}^{n\times n}.$ Next proposition is a generalisation of [PS20,
Proposition 3.5] to the case of the attenuated $\lambda$-ray transform.

######  Lemma 3.7.

Let $(M,g,\lambda)$, $\lambda\in C^{\infty}(SM)$, be a compact Riemannian
surface with strictly $\lambda$-convex boundary and nontrapping
$\lambda$-geodesic flow. Given any two matrix attenuation functions
${\mathcal{A}},{\mathcal{B}}\in C^{\infty}(SM,{\mathbb{C}}^{n\times n})$, we
have

$C_{{\mathcal{A}}}^{\lambda}\left[C^{\lambda}_{{\mathcal{B}}}\right]^{-1}=\operatorname{Id}+I_{E({\mathcal{A}},{\mathcal{B}})}^{\lambda}({\mathcal{A}}-{\mathcal{B}})$
(3.7)

where $I_{E({\mathcal{A}},{\mathcal{B}})}^{\lambda}$ denotes the attenuated
$\lambda$-ray transform with matrix attenuation
$E({\mathcal{A}},{\mathcal{B}})$.

###### Proof.

Similar to the proof of [PS20, Proposition 3.5], we consider

$\displaystyle\left\\{\begin{array}[]{l}(X+\lambda
V)U_{{\mathcal{A}}}+{\mathcal{A}}U_{{\mathcal{A}}}=0\\\
\left.U_{{\mathcal{A}}}\right|_{\partial_{-}SM}=\mathrm{Id}\end{array}\right.$

and

$\displaystyle\left\\{\begin{array}[]{l}(X+\lambda
V)U_{{\mathcal{B}}}+{\mathcal{B}}U_{{\mathcal{B}}}=0\\\
\left.U_{{\mathcal{B}}}\right|_{\partial_{-}SM}=\mathrm{Id}.\end{array}\right.$

Using Lemma 3.2, we have

$\displaystyle(X+\lambda V)(U_{\mathcal{A}}U_{{\mathcal{B}}}^{-1})$
$\displaystyle=[(X+\lambda
V)(U_{\mathcal{A}})]U_{{\mathcal{B}}}^{-1}+U_{\mathcal{A}}[(X+\lambda
V)(U_{{\mathcal{B}}}^{-1})]$ (3.8)
$\displaystyle=-{\mathcal{A}}U_{\mathcal{A}}U_{{\mathcal{B}}}^{-1}+U_{\mathcal{A}}U_{{\mathcal{B}}}^{-1}{\mathcal{B}}.$
(3.9)

Simplifying further, we obtain

$(X+\lambda
V)[(U_{\mathcal{A}}U_{{\mathcal{B}}}^{-1})]+{\mathcal{A}}U_{\mathcal{A}}U_{{\mathcal{B}}}^{-1}-U_{\mathcal{A}}U_{{\mathcal{B}}}^{-1}{\mathcal{B}}=0.$

This gives

$(X+\lambda
V)\left[U_{\mathcal{A}}U_{{\mathcal{B}}}^{-1}-\operatorname{Id}\right]+{\mathcal{A}}(U_{\mathcal{A}}U_{{\mathcal{B}}}^{-1}-\operatorname{Id})-(U_{\mathcal{A}}U_{{\mathcal{B}}}^{-1}-\operatorname{Id}){\mathcal{B}}=-{\mathcal{A}}+{\mathcal{B}}$

and hence if we define
$W:=U_{\mathcal{A}}U_{{\mathcal{B}}}^{-1}-\operatorname{Id}$, then $W$
satisfies

$\displaystyle\begin{split}\left\\{\begin{array}[]{l}(X+\lambda
V)W+E({\mathcal{A}},{\mathcal{B}})W=-({\mathcal{A}}-{\mathcal{B}})\\\
\left.W\right|_{\partial_{-}SM}=0.\end{array}\right.\end{split}$ (3.10)

Using Definition 3.4, we can write

$I_{E({\mathcal{A}},{\mathcal{B}})}^{\lambda}({\mathcal{A}}-{\mathcal{B}})=W|_{\partial_{+}SM}=U_{\mathcal{A}}U_{{\mathcal{B}}}^{-1}-\operatorname{Id}|_{\partial_{+}SM}.$
(3.11)

Since $C^{\lambda}_{\mathcal{A}}=U_{\mathcal{A}}|_{\partial_{+}SM}$ and
$C^{\lambda}_{\mathcal{B}}=U_{\mathcal{B}}|_{\partial_{+}SM}$, see Definition
3.3, the above identity becomes

$I_{E({\mathcal{A}},{\mathcal{B}})}^{\lambda}({\mathcal{A}}-{\mathcal{B}})=C^{\lambda}_{\mathcal{A}}\left[C^{\lambda}_{\mathcal{B}}\right]^{-1}-\operatorname{Id}$
(3.12)

and hence the lemma follows. ∎

The identity (3.7) is often called the pseudolinearization identity. The
following proposition generalizes [PS20, Proposition 3.7] to the case of
$\lambda$-geodesics.

######  Proposition 3.8.

Let $(M,g,\lambda)$, $\lambda\in C^{\infty}(SM)$, be a compact Riemannian
surface with strictly $\lambda$-convex boundary and nontrapping
$\lambda$-geodesic flow. Given any two matrix attenuation functions
$\mathcal{A},\mathcal{B}\in C^{\infty}\left(SM,\mathbb{C}^{n\times n}\right)$,
we have $C^{\lambda}_{\mathcal{A}}=C^{\lambda}_{\mathcal{B}}$ if and only if
there exists a smooth $U:SM\rightarrow GL(n,\mathbb{C})$ with
$\left.U\right|_{\partial SM}=\mathrm{Id}$ such that

$\mathcal{B}=U^{-1}(X+\lambda V)U+U^{-1}\mathcal{A}U.$

###### Proof.

Let $U$ be a smooth function satisfying the given conditions and define
$Z=UU_{\mathcal{B}}$. Now,

$\displaystyle(X+\lambda V)Z+\mathcal{A}Z$ $\displaystyle=\left((X+\lambda
V)U\right)U_{\mathcal{B}}+U\left((X+\lambda
V)U_{\mathcal{B}}\right)+{\mathcal{A}}UU_{{\mathcal{B}}}$
$\displaystyle=(U{\mathcal{B}}-{\mathcal{A}}U)U_{{\mathcal{B}}}-U{\mathcal{B}}U_{{\mathcal{B}}}+{\mathcal{A}}UU_{{\mathcal{B}}}=0.$

That is $Z$ satisfies both $(X+\lambda V)Z+$ $\mathcal{A}Z=0$ and
$\left.Z\right|_{\partial_{-}SM}=\mathrm{Id}$. It follows that
$Z=U_{\mathcal{A}}$ and consequently $C_{\mathcal{A}}=C_{\mathcal{B}}$.

Conversely, assume that the nonabelian $\lambda$-ray transforms agree for
$\mathcal{A}$ and $\mathcal{B}$. In the proof of Proposition 3.7, the function
$W=U_{\mathcal{A}}U_{\mathcal{B}}^{-1}-\mathrm{Id}$ has a zero boundary value.
By Lemma 3.6, $W$ is smooth. Therefore,
$U:=U_{\mathcal{A}}U^{-1}_{\mathcal{B}}=W+\operatorname{Id}$ is smooth and the
function $U$ satisfies $U|_{\partial_{-}SM}=\mathrm{Id}$. Using (3.8), we
obtain

$\displaystyle U^{-1}(X+\lambda V)U+U^{-1}\mathcal{A}U$
$\displaystyle=U_{{\mathcal{B}}}U_{{\mathcal{A}}}^{-1}(X+\lambda
V)(U_{\mathcal{A}}U_{{\mathcal{B}}}^{-1})+U_{{\mathcal{B}}}U_{{\mathcal{A}}}^{-1}{\mathcal{A}}U_{\mathcal{A}}U_{{\mathcal{B}}}^{-1}$
$\displaystyle=U_{{\mathcal{B}}}U_{{\mathcal{A}}}^{-1}(-{\mathcal{A}}U_{\mathcal{A}}U_{{\mathcal{B}}}^{-1}+U_{\mathcal{A}}U_{{\mathcal{B}}}^{-1}{\mathcal{B}})+U_{{\mathcal{B}}}U_{{\mathcal{A}}}^{-1}{\mathcal{A}}U_{\mathcal{A}}U_{{\mathcal{B}}}^{-1}$
$\displaystyle={\mathcal{B}}.$

That is $U$ satisfies the required equation. ∎

##  4\. Proofs of Main results

In [PS20, Lemma 5.1], it is proved for any skew-Hermitian ${\mathcal{B}}\in
C^{\infty}(SM,\mathfrak{u}(n))$ with
${\mathcal{B}}\in\bigoplus_{k\geq-1}\Omega_{k}$, it holds that
${\mathcal{B}}\in\Omega_{-1}\oplus\Omega_{0}\oplus\Omega_{1}$. Similarly to
their argument, one can also prove that if ${\mathcal{B}}\in
C^{\infty}(SM,\mathfrak{u}(n))$ and ${\mathcal{B}}\in\bigoplus_{k\leq
1}\Omega_{k}$, then
${\mathcal{B}}\in\Omega_{-1}\oplus\Omega_{0}\oplus\Omega_{1}$.

######  Lemma 4.1.

Let $M$ be a Riemannian surface with or without boundary. Let
${\mathcal{B}}\in C^{\infty}(SM,\mathfrak{gl}(n))$. If ${\mathcal{B}}$ is
skew-Hermitian, i.e., ${\mathcal{B}}\in C^{\infty}(SM,\mathfrak{u}(n))$, and
${\mathcal{B}}\in\oplus_{k\geq-m}\Omega_{k}$, then
${\mathcal{B}}\in\oplus_{-m\leq k\leq m}\Omega_{k}$.

###### Proof.

In the Fourier modes, one can write
${\mathcal{B}}=\sum_{k\geq-m}{\mathcal{B}}_{k}$. Note that

$\displaystyle{\mathcal{B}}^{*}=\left(\sum_{k\geq-m}{\mathcal{B}}_{k}\right)^{*}=\sum_{k\geq-m}{\mathcal{B}}_{k}^{*}$

with ${\mathcal{B}}_{k}^{*}\in\Omega_{-k}$. This implies
${\mathcal{B}}^{*}\in\oplus_{k\leq m}\Omega_{k}$. From skew-Hermitian
property, ${\mathcal{B}}^{*}=-{\mathcal{B}}=-\sum_{k\geq-m}{\mathcal{B}}_{k}$.
This implies ${\mathcal{B}}\in\oplus_{-m\leq k\leq m}\Omega_{k}$. ∎

######  Remark 4.2.

Similarly, if ${\mathcal{B}}$ is skew-Hermitian and
${\mathcal{B}}\in\oplus_{k\leq m}\Omega_{k}$, then
${\mathcal{B}}\in\oplus_{-m\leq k\leq m}\Omega_{k}$.

In [PS20, Theorem 4.2], it is proved that, given any smooth map
$R:SM\rightarrow GL(n,\mathbb{C})$, one can always find smooth maps
$U:SM\rightarrow U(n)$ and $F:SM\rightarrow GL(n,\mathbb{C})$ where the map
$F$ is fiberwise holomorphic with a fiberwise holomorphic inverse, and $R=FU$.
Additionally, one can decompose $R=\widetilde{F}\widetilde{U}$, where
$\widetilde{U}:SM\to U(n)$ is smooth and $\widetilde{F}:SM\rightarrow
GL(n,\mathbb{C})$ is smooth, fiberwise antiholomorphic, and has a fiberwise
antiholomorphic inverse. The following lemma generalizes [PS20, Lemma 5.2].
For the sake of completeness, we include the proof here.

######  Lemma 4.3.

Let $(M,g,\lambda)$, $\lambda\in C^{\infty}(SM)$, be a compact Riemannian
surface with strictly $\lambda$-convex boundary and nontrapping
$\lambda$-geodesic flow. Let ${\mathcal{A}}\in
C^{\infty}(SM,\mathfrak{gl}({\mathbb{C}}))$. Let $R:SM\to GL(n,{\mathbb{C}})$
be a smooth function solving $(X+\lambda V+{\mathcal{A}})R=0$ with a loop
group factorisation $R=FU$, where $F$ is fiberwise holomorphic with a
fiberwise holomorphic inverse and $U$ is unitary. Then

${\mathcal{B}}=F^{-1}(X+\lambda V+{\mathcal{A}})F$ (4.1)

is skew-Hermitian. In addition, if
${\mathcal{A}},\lambda\in\bigoplus_{k\geq-m}\Omega_{k}$ for $m\geq 1$, then
${\mathcal{B}}\in\bigoplus_{-m\leq k\leq m}\Omega_{k}$.

###### Proof.

Using our assumptions on $R$, we obtain

$\displaystyle 0=(X+\lambda V+{\mathcal{A}})R=((X+\lambda V)F)U+F(X+\lambda
V)U+{\mathcal{A}}FU.$ (4.2)

By multiplying $F^{-1}$ we have

$\displaystyle 0=F^{-1}((X+\lambda V)F)U+(X+\lambda
V))U+F^{-1}{\mathcal{A}}FU.$

This implies

$\displaystyle(X+\lambda V+{\mathcal{B}})U=0$ (4.3)

where ${\mathcal{B}}=F^{-1}(X+\lambda V+{\mathcal{A}})F$. By rearranging
(4.3), we get

${\mathcal{B}}=-((X+\lambda V)U)U^{-1}.$ (4.4)

As $U$ is unitary, we have

$\displaystyle\left(((X+\lambda V)U)U^{-1}\right)^{*}$
$\displaystyle=(U^{*})^{*}(X+\lambda V)U^{*}$ $\displaystyle=U(X+\lambda
V)U^{-1}$ $\displaystyle=-((X+\lambda V)U)U^{-1}.$

This proves that ${\mathcal{B}}$ is skew-Hermitian.

Now let us assume that ${\mathcal{A}},\lambda$ is in
$\bigoplus_{k\geq-m}\Omega_{k}$ for $m\geq 1$. Under this assumption, we have
the following mapping property:

$X+\lambda V:\oplus_{k\geq 0}\Omega_{k}\rightarrow\oplus_{k\geq-m}\Omega_{k}.$

Since $F$ and $F^{-1}$ are holomorphic, i.e., $F,F^{-1}\in\oplus_{k\geq
0}\Omega_{k}$, we have $F^{-1}(X+\lambda V)F\in\oplus_{k\geq-m}\Omega_{k}$.
Similarly, since ${\mathcal{A}}\in\bigoplus_{k\geq-m}\Omega_{k}$, we have
$F^{-1}{\mathcal{A}}F\in\bigoplus_{k\geq-m}\Omega_{k}$. This implies
${\mathcal{B}}\in\bigoplus_{k\geq-m}\Omega_{k}$. By Lemma 4.1, we have
$\mathcal{B}\in\bigoplus_{-m\leq k\leq m}\Omega_{k}$. ∎

In similar spirit, one can prove the following lemma using antiholmorphic and
unitary loop group decomposition.

######  Lemma 4.4.

Let $(M,g,\lambda)$, $\lambda\in C^{\infty}(SM)$, be a compact Riemannian
surface with strictly $\lambda$-convex boundary and nontrapping
$\lambda$-geodesic flow. Let ${\mathcal{A}}\in
C^{\infty}(SM,\mathfrak{gl}({\mathbb{C}}))$. Let $R:SM\to GL(n,{\mathbb{C}})$
be a smooth function solving $(X+\lambda V+{\mathcal{A}})R=0$ with a loop
group factorisation $R=\widetilde{F}\widetilde{U}$, where $\widetilde{F}$ is
fiberwise antiholomorphic with a fiberwise antiholomorphic inverse and
$\widetilde{U}$ is unitary. Then

${\mathcal{B}}=\widetilde{F}^{-1}(X+\lambda V+{\mathcal{A}})\widetilde{F}$
(4.5)

is skew-Hermitian. In addition, if ${\mathcal{A}},\lambda\in\bigoplus_{k\leq
m}\Omega_{k}$ for $m\geq 1$, then ${\mathcal{B}}\in\bigoplus_{-m\leq k\leq
m}\Omega_{k}$.

The following Theorem generalises [PS20, Theorem 5.5] to the case of
$\lambda$-geodesic flow for
$\lambda\in\Omega_{-1}\oplus\Omega_{0}\oplus\Omega_{1}$. Recall that in the
case of $\lambda\in\Omega_{-1}\oplus\Omega_{0}\oplus\Omega_{1}$, the vector
field $X+\lambda V$ has the following mapping property:

$X+\lambda V:\oplus_{k\geq 0}\Omega_{k}\to\oplus_{k\geq-1}\Omega_{k}.$

######  Theorem 4.5.

Let $(M,g,\lambda)$, $\lambda\in C^{\infty}(SM)$, be a compact Riemanian
surface with strictly $\lambda$-convex boundary and nontrapping
$\lambda$-geodesic flow. Let ${\mathcal{A}}\in
C^{\infty}(SM,\mathfrak{gl}(n,{\mathbb{C}}))$ and assume that
${\mathcal{A}},\lambda\in\Omega_{-1}\oplus\Omega_{0}\oplus\Omega_{1}$.
Additionally, assume that Conjecture 1.3 is true. Let $u\in
C^{\infty}(SM,\mathbb{C}^{n})$ satisfy $u|_{\partial SM}=0$ and

$(X+\lambda V)u+{\mathcal{A}}u=-f\in\oplus_{k\geq-1}\Omega_{k}.$

Then $u$ is holomorphic.

###### Proof.

From Lemma 3.2, there exists a smooth function $R$ such that $(X+\lambda
V)R+\mathcal{A}R=0$ and $(X+\lambda V)R^{-1}-R^{-1}\mathcal{A}=0$. As in Lemma
4.3, $R$ can be written as $R=FU$, where $F$ is fiberwise holomorphic with a
fiberwise holomorphic inverse, and $U$ is unitary. Now using ${\mathcal{B}}$
as in Lemma 4.3, we have

$\displaystyle(X+\lambda V)$ $\displaystyle(F^{-1}u)+{\mathcal{B}}(F^{-1}u)$
$\displaystyle=-F^{-1}((X+\lambda V)F)F^{-1}u+F^{-1}(X+\lambda
V)u+F^{-1}((X+\lambda V+\mathcal{A})F)F^{-1}u$ $\displaystyle=F^{-1}(X+\lambda
V+{\mathcal{A}})u=-F^{-1}f.$

and $F^{-1}u|_{\partial SM}=0$. Since $f\in\oplus_{k\geq-1}\Omega_{k}$ and
$F^{-1}$ is holomorphic, it follows that
$F^{-1}f\in\oplus_{k\geq-1}\Omega_{k}$. Similar to the proof of [PS20, Theorem
5.5], we consider $q:=\sum_{-\infty}^{-1}(F^{-1}u)_{k}$. From

$\displaystyle(X+\lambda
V)(F^{-1}u)+{\mathcal{B}}(F^{-1}u)=-F^{-1}f\in\oplus_{k\geq-1}\Omega_{k},$

we obtain

$\displaystyle(X+\lambda V)q+{\mathcal{B}}q\in\Omega_{-1}\oplus\Omega_{0}.$

Since $q|_{\partial SM}=0$ and ${\mathcal{B}}$ is skew-Hermitian, it follows
from Conjecture 1.3 that

$(X+\lambda V+{\mathcal{B}})p=(X+{\mathcal{B}})p=(X+\lambda V+{\mathcal{B}})q$

for some $p\in C^{\infty}(M)$ with $p|_{\partial M}=0$. Therefore $r:=p-q$
solves the equation $(X+\lambda V+{\mathcal{B}})r=0$ with the boundary
condition $r|_{\partial SM}=0$. Since the solution to this problem is unique
due to the uniqueness of solutions for a linear system of first order ODEs
(using the fact that $\lambda$-geodesic flow is nontrapping), we find that
$r=0$. Therefore $q=p\in\Omega_{0}$, which gives that $q=0$.

This implies that $F^{-1}u$ is holomorphic and hence $u=F(F^{-1}u)$ is also
holomorphic. ∎

###### Proof of Theorem 1.4.

From Proposition 3.8, we infer that
$C_{A,\Phi}^{\lambda}=C_{B,\Psi}^{\lambda}$ implies the existence of a smooth
function $U:SM\to GL(n,\mathbb{C})$ such that $U|_{\partial SM}=\mathrm{Id}$
and

${\mathcal{B}}=U^{-1}(X+\lambda V)U+U^{-1}{\mathcal{A}}U,$ (4.6)

where ${\mathcal{B}}(x,v)=B_{x}(v)+\Psi(x)$ and
${\mathcal{A}}(x,v)=A_{x}(v)+\Phi(x)$. We can rephrase this information in
terms of an attenuated $\lambda$-ray transform. If we define
$W=U-\mathrm{Id}$, then $W|_{\partial SM}=0$ and

$(X+\lambda V)W+{\mathcal{A}}W-W{\mathcal{B}}=-({\mathcal{A}}-{\mathcal{B}}).$

That is, $W$ satisfies (3.10). Consequently, $W$ is associated with the
attenuated $\lambda$-ray transform
$I_{E({\mathcal{A}},{\mathcal{B}})}^{\lambda}({\mathcal{A}}-{\mathcal{B}})$,
and if $C_{{\mathcal{A}}}^{\lambda}=C_{{\mathcal{B}}}^{\lambda}$, then this
transform vanishes by Lemma 3.7. Note that
${\mathcal{A}}-{\mathcal{B}}\in\Omega_{-1}\oplus\Omega_{0}\oplus\Omega_{1}$.

We introduce a new connection $\hat{A}$ on the trivial bundle
$M\times\mathbb{C}^{n\times n}$ and a new Higgs field $\hat{\Phi}$ as follows:
for any matrix $H\in\mathbb{C}^{n\times n}$, we define $\hat{A}(H):=AH-HB$ and
$\hat{\Phi}(H):=\Phi H-H\Psi$. Observe that
$E({\mathcal{A}},{\mathcal{B}})=\hat{A}+\hat{\Phi}$, which is an arbitrary
attenuation pair as assumed in Conjecture 1.2. Applying Conjecture 1.2, we
deduce the existence of a function $\widetilde{W}\in C^{\infty}(M)$ satisfying
$\widetilde{W}|_{\partial M}=0$ and

$\displaystyle(X+\lambda V+\hat{A}+\hat{\Phi})\widetilde{W}$
$\displaystyle=(X+\hat{A}+\hat{\Phi})\widetilde{W}=(X+\lambda
V+\hat{A}+\hat{\Phi})W.$

Consequently, the function $r:=\widetilde{W}-W$ solves the equation
$(X+\lambda V+\hat{A}+\hat{\Phi})r=0$ with $r|_{\partial SM}=0$. Following a
similar argument as used in the proof of Theorem 4.5, we conclude that $r=0$.
Therefore, $\widetilde{W}=W\in\Omega_{0}$, indicating that $W$ smooth function
on $M$. Thus, $U=W+\mathrm{Id}$, which depends only on $x$, and setting
$u(x)=U_{0}(x)=U(x)$, equation (4.6) can be reformulated as
$B=u^{-1}du+u^{-1}Au$ and $\Psi=u^{-1}\Phi u$, by examining solely the
components of degree $0$ and $\pm 1$. This proves the existence of
$u\in\mathcal{G}$ such that $(A,\Phi)\cdot{u}=(B,\Psi)$. ∎

###### Proof of Theorem 1.5.

Consider an arbitrary attenuation pair $(A,\Phi)$, where

$A\in\Omega^{1}(M,\mathfrak{gl}(n,\mathbb{C})),\qquad\Phi\in
C^{\infty}(M,\mathfrak{gl}(n,\mathbb{C})).$ (4.7)

Define ${\mathcal{A}}(x,v)=A(x,v)+\Phi(x)$. If
$I_{{\mathcal{A}}}^{\lambda}f=0$, then by Lemma 3.6, we have $u\in
C^{\infty}(SM)$, $u|_{\partial SM}=0$, and

$(X+\lambda
V)u+{\mathcal{A}}u=-f\in\Omega_{-1}\oplus\Omega_{0}\oplus\Omega_{1}.$ (4.8)

Given that ${\mathcal{A}},\lambda\in\oplus_{-1\leq k\leq 1}\Omega_{k}$,
Theorem 4.5 implies $u$ is holomorphic. The complex conjugate takes
$w_{k}\in\Omega_{k}$ to $\overline{w_{k}}\in\Omega_{-k}$. Taking the complex
conjugate of equation (4.8), and considering that $\overline{{\mathcal{A}}}$,
$\overline{\lambda}$ and $\overline{f}$ belong to $\oplus_{-1\leq k\leq
1}\Omega_{k}$, we deduce that $\overline{u}$ is also holomorphic (using
Theorem 4.5). This leads to the conclusion that $u=u_{0}$ is solely dependent
on $x$. Setting $p=-u$, we then obtain $f=(X+{\mathcal{A}})p$. ∎

From [Ain13, Theorem 1.2] and Theorem 1.5, we can easily deduce Theorem 1.7.

##  Appendix A Some basic properties of the nontrapping $\lambda$-geodesic
flows

Our main goal in this appendix is to show that the existence of a nontrapping
$\lambda$-geodesic flow is an open condition. For that, we construct a smooth
function $f$ on $SM$ satisfying the property that $(X+\lambda V)f>0$. For the
case of $\lambda=0$, this is shown in [PSU23, Proposition 3.3.1]. The proof
for the general case follows similarly. For the sake of completeness, we
present it in this appendix. Before that, we restate one technical lemma from
[PSU23].

######  Lemma A.1 ([PSU23, Lemma 3.2.10]).

Let $h(t,y)$ be a smooth function near $\left(0,y_{0}\right)$ in
$\mathbb{R}\times\mathbb{R}^{N}$. If

$h\left(0,y_{0}\right)=0,\quad\partial_{t}h\left(0,y_{0}\right)=0,\quad\partial_{t}^{2}h\left(0,y_{0}\right)<0,$

then one has

$h(t,y)=0\text{ near }\left(0,y_{0}\right)\text{ when }h(0,y)\geq
0\quad\Longleftrightarrow\quad t=Q(\pm\sqrt{a(y)},y)$

where $Q$ is a smooth function near $\left(0,y_{0}\right)$ in
$\mathbb{R}\times\mathbb{R}^{N}$, a is a smooth function near $y_{0}$ in
$\mathbb{R}^{N}$, and $a(y)\geq 0$ when $h(0,y)\geq 0$. Moreover,
$Q(\sqrt{a(y)},y)\geq Q(-\sqrt{a(y)},y)$ when $h(0,y)\geq 0$.

The following lemma generalizes [PSU23, Lemma 3.2.9] in the context of the
forward and backward travel times of $\lambda$-geodesic flow.

######  Lemma A.2.

Let $(M,g,\lambda)$, $\lambda\in C^{\infty}(SM)$, be a compact Riemmanian
surface. Let $(x_{0},v_{0})\in\partial_{0}SM$ and suppose that $\partial M$ is
strictly $\lambda$-convex near $x_{0}$. Assume that $M$ is isometrically
embedded in a closed manifold $N$ and $\lambda_{0}$ is smooth extension of
$\lambda$ to $N$. Then, near $(x_{0},v_{0})$ in $SM$, one has

$\displaystyle\begin{split}\tau^{+}(x,v)&=Q(\sqrt{a(x,v)},x,v),\\\
-\tau^{-}(x,v)&=Q(-\sqrt{a(x,v)},x,v),\end{split}$ (A.1)

where $Q$ is a smooth function near $(0,x_{0},v_{0})$ in $\mathbb{R}\times
SN$, and $a$ is a smooth function near $(x_{0},v_{0})$ in $SN$ with $a\geq 0$
in $SM$.

###### Proof.

Following the approach of [PSU23, Lemma 3.2.9], we define $h:\mathbb{R}\times
SN\to\mathbb{R}$, $h(t,x,v):=\rho\left(\gamma_{x,v}(t)\right)$, where $\rho$
is a boundary defining function satisfying, $\rho|_{\operatorname{int}M}>0$,
$\rho|_{\partial M}=0$ and $\rho|_{N\setminus M}<0$. According to [JKR23,
Lemma A.12] and [JKR23, Lemma 3.15], for any $(x_{0},v_{0})\in\partial_{0}SM$,
we have the following:

$\displaystyle
h\left(\pm\tau^{\pm}\left(x_{0},v_{0}\right),x_{0},v_{0}\right)$
$\displaystyle=h\left(0,x_{0},v_{0}\right)=0,$
$\displaystyle\partial_{t}h\left(\pm\tau^{\pm}\left(x_{0},v_{0}\right),x_{0},v_{0}\right)$
$\displaystyle=\partial_{t}h\left(0,x_{0},v_{0}\right)=0,$
$\displaystyle\partial_{t}^{2}h\left(\pm\tau^{\pm}\left(x_{0},v_{0}\right),x_{0},v_{0}\right)$
$\displaystyle=\partial_{t}^{2}h\left(0,x_{0},v_{0}\right)<0.$

Also, $h(\pm\tau^{\pm}(x,v),x,v)=0$ for any $(x,v)\in SM$ near
$(x_{0},v_{0})$. Note that $\tau^{+}(x,v)\geq 0\geq-\tau^{-}(x,v)$. Applying
Lemma A.1, we express $\tau^{+}(x,v)$ and $-\tau^{-}(x,v)$ as in equation
(A.1), confirming that $Q$ and $a$ have the stated properties. ∎

######  Lemma A.3.

Let $(M,g,\lambda),\lambda\in C^{\infty}(SM)$, be a compact Riemanian surface
with strictly $\lambda$-convex boundary and nontrapping $\lambda$-geodesic
flow. Define $\tilde{\tau}:SM\to\mathbb{R}$ by

$\tilde{\tau}(x,v):=\tau^{+}(x,v)-\tau^{-}(x,v)$ (A.2)

Then $\tilde{\tau}\in C^{\infty}(SM)$.

###### Proof.

Assume that $(M,g)$ is isometrically embedded in a closed manifold $(N,g)$ of
the same dimension and $\lambda_{0}$ is a smooth extension of $\lambda$ in
$N$. From [JKR23, Lemma 3.15], it suffices to show smoothness of
$\widetilde{\tau}$ in the neighbourhood of $\partial_{0}SM$.

From Lemma A.2, we have

$\displaystyle\widetilde{\tau}(x,v)=Q(\sqrt{a(x,v)},x,v)+Q(-\sqrt{a(x,v)},x,v)$

where $Q$ is a smooth function near $(0,x_{0},v_{0})$ in $\mathbb{R}\times SN$
and $a$ is a smooth function near $(x_{0},v_{0})$ in $SN$, and $a\geq 0$ in
$SM$ Now, as $Q(r,x,v)+Q(-r,x,v)$ is an even function in $r$, using [Whi43,
Theorem 1], there exists a smooth function $H$ near $(0,x_{0},v_{0})$ such
that $Q(r,x,v)+Q(-r,x,v)=H(r^{2},x,v)$. This implies
$\widetilde{\tau}(x,v)=H(a(x,v),x,v)$, which proves the smoothness of
$\widetilde{\tau}$. ∎

Finally, we prove that having nontrapping $\lambda$-geodesic flow is an open
condition.

######  Lemma A.4.

Let $(M,g,\lambda),\lambda\in C^{\infty}(SM)$, be a compact Riemanian surface
with strictly $\lambda$-convex boundary. The $\lambda$-geodesic flow is
nontrapping if and only if there exists a function $f\in C^{\infty}(SM)$ such
that the inequality $(X+\lambda V)f>0$ holds.

###### Proof.

First, assume that the $\lambda$-geodesic flow is nontrapping on $M$. By Lemma
A.3, the function $\widetilde{\tau}$ is smooth on $SM$. Consider the function
$f=-\widetilde{\tau}$. Then, we have

$\displaystyle(X+\lambda V)f(x,v)$
$\displaystyle=\left.-\frac{d}{dt}\left(\tau^{+}(\phi_{t}(x,v))-\tau^{-}(\phi_{t}(x,v))\right)\right|_{t=0}$
$\displaystyle=-\left.\frac{d}{dt}\left(\tau^{+}(x,v)-t-(\tau^{-}(x,v)+t)\right)\right|_{t=0}=2>0.$

Conversely, assume there exists $f\in C^{\infty}(SM)$ such that $(X+\lambda
V)f\geq c>0$ for some constant $c$. Suppose, for the sake of contradiction,
that there exists a trapped $\lambda$-geodesic $\gamma$. Applying the
fundamental theorem of calculus, we obtain
$f\left(\phi_{t}(x,v)\right)-f(x,v)\geq ct$ for all $t>0$. This implies that
$f\circ(\gamma,\dot{\gamma})$ is unbounded, which contradicts the compactness
of $SM$. Hence, the assumption of the existence of a trapped
$\lambda$-geodesic is wrong, completing the proof. ∎

##  Appendix B Regularity of the kernel of the attenuated $\lambda$-ray
transform

Given any function $w\in C^{\infty}(\partial_{+}SM,\mathbb{C}^{n})$, consider
the unique solution $w^{\sharp}:SM\to\mathbb{C}^{n}$ to the transport
equation:

$\displaystyle\begin{cases}(X+\lambda V)(w^{\sharp})=0,\\\
w^{\sharp}|_{\partial_{+}(SM)}=w.\end{cases}$

Similar to the concept of scattering relation along geodesics, as defined in
[PSU23, Definition 3.3.4], we define the $\lambda$-scattering relation
$\alpha$ as a map which associates the initial point and direction of a
$\lambda$-geodesic flow with its endpoint and direction. More precisely, let
$(M,g,\lambda)$, with $\lambda\in C^{\infty}(SM)$, be a compact Riemannian
surface with a strictly $\lambda$-convex boundary and a nontrapping
$\lambda$-geodesic flow. Then, the $\lambda$-scattering relation
$\alpha:\partial SM\to\partial SM$ is given by
$\alpha(x,v):=\phi_{\tilde{\tau}}(x,v)$, where $\tilde{\tau}$ is defined as in
Lemma A.3.

Similar to the approach in [PU05], we introduce the operator
$E:C^{\infty}(\partial_{+}SM)\to C^{\infty}(\partial SM)$ defined as

$\displaystyle Ew(x,v)=\begin{cases}w(x,v)&\text{if
}(x,v)\in\partial_{+}SM,\\\ (w\circ\alpha)(x,v)&\text{if
}(x,v)\in\partial_{-}SM.\end{cases}$

Additionally, we define the space

$\displaystyle C_{\alpha,\lambda}^{\infty}(\partial_{+}SM):=\\{w\in
C^{\infty}(\partial_{+}SM):Ew\in C^{\infty}(\partial SM)\\}.$

The following lemma generalizes [PU05, Lemma 1.1] or [PSU23, Theorem 5.1.1] to
the context of $\lambda$-geodesic flow.

######  Lemma B.1.

Let $(M,g,\lambda)$, $\lambda\in C^{\infty}(SM)$, be a compact Riemannian
surface with strictly $\lambda$-convex boundary and nontrapping
$\lambda$-geodesic flow. Then

$\displaystyle C_{\alpha,\lambda}^{\infty}(\partial_{+}SM):=\\{w\in
C^{\infty}(\partial_{+}SM):w^{\sharp}\in C^{\infty}(SM)\\}.$

###### Proof.

If $w^{\sharp}\in C^{\infty}(SM)$, then obviously
$Ew=\left.w^{\sharp}\right|_{\partial_{+}SM}\in C^{\infty}(\partial SM)$.
Conversely, assume that $Ew\in C^{\infty}(\partial SM)$. It suffices to show
that $w^{\sharp}\in C^{\infty}(SM)$.

Assume that $(M,g)$ is isometrically embedded in a closed manifold $(N,g)$ of
the same dimension, as stated in [PSU23, Lemma 3.1.8]. Extend $\lambda$
smoothly to the whole $sN$. Consider a smooth extension $\tilde{w}$ of $Ew$
into $SN$. Define $\mathfrak{F}(x,v,t)=\frac{1}{2}\tilde{w}(\phi_{t}(x,v))$,
then

$\displaystyle w^{\sharp}(x,v)$
$\displaystyle=\frac{1}{2}\left[\tilde{w}(\phi_{\tau^{+}(x,v)}(x,v))+\tilde{w}(\phi_{-\tau^{-}(x,v)}(x,v))\right]$
$\displaystyle=\mathfrak{F}(x,v,\tau^{+}(x,v))+\mathfrak{F}(x,v,-\tau^{-}(x,v)).$

As $\tau^{\pm}$ is smooth on $SM\setminus\partial_{0}SM$, $w^{\sharp}$ is
smooth on $SM\setminus\partial_{0}SM$. Fixing any point
$(x_{0},v_{0})\in\partial_{0}SM$, and from Lemma A.2, we have

$\displaystyle w^{\sharp}(x,v)$
$\displaystyle=\mathfrak{F}(x,v,Q(\sqrt{a(x,v)},x,v))+\mathfrak{F}(x,v,Q(-\sqrt{a(x,v)},x,v))$

near $(x_{0},v_{0})$ in $SM$, where $Q$ is a smooth function near
$(0,x_{0},v_{0})\in\mathbb{R}\times SN$. Set
$\mathcal{G}(r,x,v):=\mathfrak{F}(x,v,Q(r,x,v))$. This implies

$\displaystyle w^{\sharp}(x,v)$
$\displaystyle=\mathcal{G}(\sqrt{a(x,v)},x,v)+\mathcal{G}(-\sqrt{a(x,v)},x,v)$

near $(x_{0},v_{0})$ in $SM$, where $\mathcal{G}$ is a smooth function near
$(0,x_{0},v_{0})\in\mathbb{R}\times SN$. Now, as
$\mathcal{G}(r,x,v)+\mathcal{G}(-r,x,v)$ is even function in $r$, by using
[Whi43, Theorem 1], there exists smooth function $H$ near $(0,x_{0},v_{0})$
such that $\mathcal{G}(r,x,v)+\mathcal{G}(-r,x,v)=H(r^{2},x,v)$. This indeed
shows that

$\displaystyle w^{\sharp}(x,v)=H(a(x,v),x,v)$

near $(x_{0},v_{0})$ in $SM$, which finally proves that $w^{\sharp}(x,v)$ is
smooth near the point $(x_{0},v_{0})$ in $SM$. As $(x_{0},v_{0})$ is any
arbitrary point in $\partial_{0}SM$, we have $w^{\sharp}\in C^{\infty}(SM)$. ∎

###### Proof of Lemma 3.6.

From (3.6) and $I_{{\mathcal{A}}}^{\lambda}(f)=0$, we have

$\displaystyle(X+\lambda V)(R^{-1}u^{f})=-R^{-1}f\quad\text{ in }SM\quad
R^{-1}u^{f}|_{\partial SM}=0.$

Similar to the proof of [PSU23, Theorem 5.3.6], by enlarging $M$ to $M_{0}$
(as defined in Definition 3.1) with the travel time $\tau_{0}$, and also
extending $R^{-1}f$ smoothly to $M_{0}$, consider

$\displaystyle h(x,v)=\int_{0}^{\tau_{0}(x,v)}R^{-1}f(\phi_{t}(x,v))\,dt$

for $(x,v)\in SM$. Note that $\tau_{0}|_{SM}$ is smooth, which implies $h\in
C^{\infty}(SM)$. Also, $(h-R^{-1}u^{f})|_{\partial SM}=h|_{\partial SM}\in
C^{\infty}(\partial SM,\mathbb{C}^{n})$. As $(X+\lambda V)(h-R^{-1}u^{f})=0$
and $(h-R^{-1}u^{f})|_{\partial SM}\in C^{\infty}(\partial
SM,\mathbb{C}^{n})$, then by Lemma B.1, $h-R^{-1}u^{f}$ is smooth in $SM$. The
smoothness of $h$ in $SM$ implies that $R^{-1}u^{f}$ is smooth, which in turn
proves that $u^{f}$ is smooth. ∎

## References

  * [AA15] Gareth Ainsworth and Yernat M. Assylbekov.  On the range of the attenuated magnetic ray transform for connections and Higgs fields.  Inverse Probl. Imaging, 9(2):317–335, 2015.
  * [AD18] Yernat M. Assylbekov and Nurlan S. Dairbekov.  The X-ray transform on a general family of curves on Finsler surfaces.  J. Geom. Anal., 28(2):1428–1455, 2018.
  * [Ain13] Gareth Ainsworth.  The attenuated magnetic ray transform on surfaces.  Inverse Probl. Imaging, 7(1):27–46, 2013.
  * [Ain15] Gareth Ainsworth.  The magnetic ray transform on Anosov surfaces.  Discrete Contin. Dyn. Syst., 35(5):1801–1816, 2015.
  * [AR21] Yernat M. Assylbekov and Franklin T. Rea.  The attenuated ray transforms on gaussian thermostats with negative curvature, 2021.  arXiv:2102.04571.
  * [AZ17] Yernat M. Assylbekov and Hanming Zhou.  Invariant distributions and tensor tomography for Gaussian thermostats.  Comm. Anal. Geom., 25(5):895–926, 2017.
  * [BP23] Jan Bohr and Gabriel P. Paternain.  The transport Oka-Grauert principle for simple surfaces.  J. Éc. polytech. Math., 10:727–769, 2023.
  * [DLS+20] Naeem M. Desai, William R. B. Lionheart, Morten Sales, Markus Strobl, and Sø ren Schmidt.  Polarimetric neutron tomography of magnetic fields: uniqueness of solution and reconstruction.  Inverse Problems, 36(4):045001, 17, 2020.
  * [DP07a] Nurlan S. Dairbekov and Gabriel P. Paternain.  Entropy production in Gaussian thermostats.  Comm. Math. Phys., 269(2):533–543, 2007.
  * [DP07b] Nurlan S. Dairbekov and Gabriel P. Paternain.  Entropy production in thermostats. II.  J. Stat. Phys., 127(5):887–914, 2007.
  * [DPSU07] Nurlan S. Dairbekov, Gabriel P. Paternain, Plamen Stefanov, and Gunther Uhlmann.  The boundary rigidity problem in the presence of a magnetic field.  Adv. Math., 216(2):535–609, 2007.
  * [Esk04] G. Eskin.  On non-abelian Radon transform.  Russ. J. Math. Phys., 11(4):391–408, 2004.
  * [FU01] David Finch and Gunther Uhlmann.  The x-ray transform for a non-abelian connection in two dimensions.  volume 17, pages 695–701. 2001.  Special issue to celebrate Pierre Sabatier’s 65th birthday (Montpellier, 2000).
  * [GK80] V. Guillemin and D. Kazhdan.  Some inverse spectral results for negatively curved $2$-manifolds.  Topology, 19(3):301–312, 1980.
  * [HMK+18] A Hilger, I Manke, N Kardjilov, M Osenberg, H Markötter, and J Banhart.  Tensorial neutron tomography of three-dimensional magnetic vector fields in bulk materials.  Nature communications, 9(1):4023, 2018.
  * [Ilm16] Joonas Ilmavirta.  Coherent quantum tomography.  SIAM J. Math. Anal., 48(5):3039–3064, 2016.
  * [JKR23] Shubham R. Jathar, Manas Kar, and Jesse Railo.  Broken ray transform for twisted geodesics on surfaces with a reflecting obstacle, 2023.  arXiv:2306.17604.
  * [JS07] Dmitry Jakobson and Alexander Strohmaier.  High energy limits of Laplace-type and Dirac-type eigenfunctions and frame flows.  Comm. Math. Phys., 270(3):813–833, 2007.
  * [MP11] Will J. Merry and Gabriel P. Paternain.  Inverse Problems in Geometry and Dynamics.  2011\.  Lecture notes, University of Cambridge, March 2011.
  * [MP20] François Monard and Gabriel P. Paternain.  The geodesic X-ray transform with a $GL(n,\mathbb{C})$-connection.  J. Geom. Anal., 30(3):2515–2557, 2020.
  * [MST23] Marco Mazzucchelli, Mikko Salo, and Leo Tzou.  A general support theorem for analytic double fibration transforms, 2023.
  * [MZ81] S. V. Manakov and V. E. Zakharov.  Three-dimensional model of relativistic-invariant field theory, integrable by the inverse scattering transform.  Lett. Math. Phys., 5(3):247–253, 1981.
  * [Nov02] R. G. Novikov.  On determination of a gauge field on $\mathbb{R}^{d}$ from its non-abelian Radon transform along oriented straight lines.  J. Inst. Math. Jussieu, 1(4):559–629, 2002.
  * [Nov19] R. G. Novikov.  Non-Abelian Radon transform and its applications.  In The Radon transform—the first 100 years and beyond, volume 22 of Radon Ser. Comput. Appl. Math., pages 115–127. Walter de Gruyter, Berlin, [2019] ©2019.
  * [OSSU20] Lauri Oksanen, Mikko Salo, Plamen Stefanov, and Gunther Uhlmann.  Inverse problems for real principal type operators, 2020.
  * [Pat13] Gabriel P. Paternain.  Inverse problems for connections.  In Inverse problems and applications: inside out. II, volume 60 of Math. Sci. Res. Inst. Publ., pages 369–409. Cambridge Univ. Press, Cambridge, 2013.
  * [PS20] Gabriel P Paternain and Mikko Salo.  The non-abelian x-ray transform on surfaces.  2020\.  arXiv=2006.02257,J. Diff. Geom. (to appear).
  * [PS21] Gabriel P. Paternain and Mikko Salo.  Carleman estimates for geodesic x-ray transforms, 2021.  arXiv=1805.02163, Ann. Sci. Éc. Norm. Supér. (to appear).
  * [PSU12] Gabriel P. Paternain, Mikko Salo, and Gunther Uhlmann.  The attenuated ray transform for connections and Higgs fields.  Geom. Funct. Anal., 22(5):1460–1489, 2012.
  * [PSU23] Gabriel P. Paternain, Mikko Salo, and Gunther Uhlmann.  Geometric inverse problems—with emphasis on two dimensions, volume 204 of Cambridge Studies in Advanced Mathematics.  Cambridge University Press, Cambridge, 2023.  With a foreword by András Vasy.
  * [PSUZ19] Gabriel P. Paternain, Mikko Salo, Gunther Uhlmann, and Hanming Zhou.  The geodesic X-ray transform with matrix weights.  Amer. J. Math., 141(6):1707–1750, 2019.
  * [PU05] Leonid Pestov and Gunther Uhlmann.  Two dimensional compact simple Riemannian manifolds are boundary distance rigid.  Ann. of Math. (2), 161(2):1093–1110, 2005.
  * [Sha00] V. A. Sharafutdinov.  On the inverse problem of determining a connection on a vector bundle.  J. Inverse Ill-Posed Probl., 8(1):51–88, 2000.
  * [ST76] I. M. Singer and J. A. Thorpe.  Lecture notes on elementary topology and geometry.  Springer-Verlag, New York-Heidelberg, 1976.  Reprint of the 1967 edition.
  * [SU11] Mikko Salo and Gunther Uhlmann.  The attenuated ray transform on simple surfaces.  J. Differential Geom., 88(1):161–187, 2011.
  * [Uhl04] Gunther Uhlmann.  The Cauchy data and the scattering relation.  In Geometric methods in inverse problems and PDE control, volume 137 of IMA Vol. Math. Appl., pages 263–287. Springer, New York, 2004.
  * [UV16] Gunther Uhlmann and András Vasy.  The inverse problem for the local geodesic ray transform.  Invent. Math., 205(1):83–120, 2016.
  * [Ver91] L. B. Vertgeĭm.  Integral geometry with a matrix weight and a nonlinear problem of the reconstruction of matrices.  Dokl. Akad. Nauk SSSR, 319(3):531–534, 1991.
  * [Ver00] L. B. Vertgeĭm.  Weighted integral geometry of matrices.  Sibirsk. Mat. Zh., 41(6):1325–1337, ii, 2000.
  * [War88] R. S. Ward.  Soliton solutions in an integrable chiral model in $2+1$ dimensions.  J. Math. Phys., 29(2):386–389, 1988.
  * [Whi43] Hassler Whitney.  Differentiable even functions.  Duke Math. J., 10:159–160, 1943.
  * [Zho17] Hanming Zhou.  Generic injectivity and stability of inverse problems for connections.  Comm. Partial Differential Equations, 42(5):780–801, 2017.