The attenuated ray transforms on Gaussian thermostats with negative curvature
aa r X i v : . [ m a t h . DG ] F e b THE ATTENUATED RAY TRANSFORMS ON GAUSSIANTHERMOSTATS WITH NEGATIVE CURVATURE
YERNAT M. ASSYLBEKOV AND FRANKLIN T. REA
Abstract.
In the present paper we consider a Gaussian thermostat on a com-pact Riemannian surface with negative thermostat curvature. In the case ofsurfaces withboundary, we show that the thermostat ray transform with atten-uation given by a general connection and Higgs fieldis injective, modulo thenatural obstruction, for tensors. We also prove that the connection and Higgsfield can be determined, up to a gauge transformation, from the knowledge ofthe parallel transport between boundary points along all possible thermostatgeodesics. In the case of closed surfaces, we obtain similar results for the raytransform with some additional conditions on the connection.Under the samecondition, we study connections and Higgs fields whose parallel transport alongperiodic thermostat geodesics coincides with the ones for the flat connection. Introduction and statement of results
Gaussian Thermostats.
Let (
M, g ) be a compact oriented Riemannian man-ifold and let E be a smooth vector field on M which we refer to as the externalfield . We say that a parameterized curve γ ( t ) on M is a thermostat geodesic if itsatisfies the equation D t ˙ γ = E ( γ ) − h E ( γ ) , ˙ γ i| ˙ γ | ˙ γ, (1)where D t is the covariant derivative along γ . This differential equation defines aflow φ t = ( γ ( t ) , ˙ γ ( t )) on the unit sphere bundle SM which is called a Gaussianthermostat . We will also refer to φ t as the thermostat flow . The flow φ t reduces tothe geodesic flow when E = 0. As in the case of geodesic flows, thermostat flowsare reversible in the sense that the flip ( x, v ) ( x, − v ) conjugates φ t with φ − t . Wedenote the Gaussian thermostat by ( M, g, E ) and the generating vector field of φ t by G E .Gaussian thermostats were first introduced in [21] by Hoover who studied aclass of dynamical systems in mechanics with thermostatic term satisfying Gauss’least Constraint Principle for nonholonomic constraints. It is known that suchconstraints do not have variational principles in general. Interesting models in non-equilibrium statistical mechanics are provided by Gaussian thermostats [14, 13, 43].In this model, temperature fluctuations prevented by maintaining a constant kineticenergy. More precisely, the external field E in (1) contributes to the kinetic energy.But the second term in (1) is the “thermostat”, which damps the component of E parallel to the velocity of the trajectory, and, therefore, retains the kinetic energy to be constant. For this reason, Gaussian thermostats are also known as isokineticdynamics .Gaussian thermostats also arise in Weyl geometry. More precisely, when thegeodesics of a Weyl manifold are reparameterized with respect to the arc-length,they turn out to be identical with thermostat geodesics; see [42]. Weyl geometryhas been actively studied by many authors in mathematics [5, 12, 17, 29] withapplications to static Yang-Mills-Higgs theory and twistor theory [18, 19, 24]. Seealso [44] for the applications of the space-time version of Weyl geometry in thetheories of gravity, quantum mechanics, elementary particle physics, and cosmology.More recently, some inverse problems for Gaussian thermostats were addressedin [3, 4, 6, 7]. Dynamical and geometrical properties were studied in [2, 29, 30, 33,46, 47].In the present paper we study several geometric inverse problems for Gaussianthermostats when M is a surface. In this setting, the equation (1) can be rewrittenas D t ˙ γ = λ ( γ, ˙ γ ) i ˙ γ, λ ( x, v ) := h E ( x ) , iv i . (2) Definition 1.1.
For a given Gaussian thermostat (
M, g, E ), the thermostat curva-ture is the quantity K E := K − div g E , where K is the Gaussian curvature of thesurface ( M, g ).We note that K E is precisely the sectional curvature of the Weyl connection intwo dimensions [47]. We will mainly focus on Gaussian thermostats with negativethermostat curvature.1.2. Connections and Higgs fields.
We define a connection on the trivial bundle M × C n as a matrix-valued smooth map A : T M → gl ( n, C ) which is linear in v ∈ T x M for a fixed x ∈ M , and define a Higgs field as a matrix-valued smoothmap Φ : M → gl ( n, C ). The connection A induces a covariant derivative which actson sections of M × C n by d A := d + A . Pairs of connections and Higgs fields ( A, Φ)are very important in the Yang-Mills-Higgs theories; see [9, 20, 27, 28]. It is naturalto consider connections and Higgs fields for Gaussian thermostats, since the latter,as mentioned before, is also related to Yang-Mills-Higgs theories via Weyl geometry.1.3.
Main results on surfaces with boundary.
Let (
M, g ) be a smooth compactand oriented Riemannian surface with boundary ∂M and let E be an external field.We consider the set of inward and outward unit vectors on the boundary of M defined as ∂ ± SM = { ( x, v ) ∈ SM : x ∈ ∂M, ±h v, ν ( x ) i ≥ } , where ν is the inward unit normal to ∂M . The thermostat geodesics entering M can be parametrized by ∂ + SM . We assume that ( M, g, E ) is non-trapping , i.e. forany ( x, v ) ∈ SM the first non-negative time τ ( x, v ) when the thermostat geodesic γ x,v , with x = γ x,v (0), v = ˙ γ x,v (0), exits M is finite.For the rest of the paper, if ∂M is non-empty, we will work with strictly convex Gaussian thermostats, i.e. the ones satisfyingΛ( x, v ) > h E ( x ) , ν ( x ) i g ( x ) for all ( x, v ) ∈ S ( ∂M ) , HE ATTENUATED THERMOSTAT RAY TRANSFORMS 3 where Λ is the second fundamental form of ∂M ; see Appendix A.1 for explanationand more details.For a pair of connection and Higgs field ( A, Φ) on the trivial bundle M × C n ,we have a scattering data defined as follows. Consider the unique solution U A, Φ : SM → GL ( n, C ) for the transport equation( G E + A + Φ) U A, Φ = 0 in SM, U A, Φ | ∂ + SM = Id . (3) Definition 1.2.
The scattering data of a pair of connection and Higgs field ( A, Φ)is the map C A, Φ : ∂ − SM → GL ( n, C ) defined as C A, Φ ( x, v ) := U A, Φ ( x, v ) , ( x, v ) ∈ ∂ − SM.
We are interested in the inverse problem of determining ( A, Φ) from the knowl-edge of C A, Φ . However, there is a natural gauge invariance for the scattering data C A, Φ . Indeed, suppose that Q ∈ C ∞ ( M ; GL ( n, C )) such that Q | ∂M = Id and U A, Φ satisfies (3). Then, using the fact d ( Q − ) Q = − Q − dQ , it is not difficult to seethat Q − U A, Φ satisfies (cid:0) G E + Q − ( d + A ) Q + Q − Φ Q (cid:1) ( Q − U A, Φ ) = 0 in SM, Q − U A, Φ | ∂ + SM = Id . Thus, we have C Q − ( d + A ) Q,Q − Φ Q = ( Q − U A, Φ ) | ∂ − SM = U A, Φ | ∂ − SM = C A, Φ . Therefore, we can only hope to recover ( A, Φ) from C A, Φ up to such an obstruction.Our first main result says that it is possible when the thermostat curvature K E isnegative. Theorem 1.3.
Let ( M, g, E ) be a strictly convex and non-trapping Gaussian ther-mostat on a compact oriented surface with boundary such that K E < . Let also A, B : T M → gl ( n, C ) be two connections and Φ , Ψ : M → gl ( n, C ) be two Higgsfields. If C A, Φ = C B, Ψ , then B = Q − ( d + A ) Q and Ψ = Q − Φ Q for some Q ∈ C ∞ ( M ; GL ( n, C )) such that Q | ∂M = Id . Similar result was earlier established in [38] for geodesic flows in arbitrary di-mensions. For earlier related results in the case of geodesic flows, see [10, 11, 15,32, 39, 41, 45, 48]. In the presence of a magnetic field some positive result is givenin [1].The proof is based on the reduction of the problem to an integral geometryproblem. This motivates us to study attenuated ray transforms along thermostatgeodesics which is the central object of the paper.Given f ∈ C ∞ ( SM ; C n ), consider the following transport equation for u : SM → C n ( G E + A + Φ) u = − f in SM, u (cid:12)(cid:12) ∂ − SM = 0 . Here A and Φ act on functions on SM by matrix multiplication. This equation hasa unique solution u f , since on any fixed thermostat geodesic the transport equationis a linear system of ordinary differential equations with zero initial condition. YERNAT M. ASSYLBEKOV AND FRANKLIN T. REA
Definition 1.4.
The attenuated thermostat ray transform I A, Φ f of f ∈ C ∞ ( SM ; C n ), with attenuation given by a connection A : T M → gl ( n, C ) anda Higgs field Φ : M → gl ( n, C ), is defined as I A, Φ f := u f (cid:12)(cid:12) ∂ + SM . It is clear that a general f ∈ C ∞ ( SM ; C n ) cannot be determined by its atten-uated thermostat ray transform, since f depends on more variables than I A, Φ f .Moreover, one can easily see that the functions of the following type are always inthe kernel of I A, Φ ( G E + A + Φ) p, p ∈ C ∞ ( SM ; C n ) such that p | ∂ ( SM ) = 0 . However, one often encounters I A, Φ acting on functions on SM arising from sym-metric tensor fields on M . We will further consider this particular case.Let f i ··· i m ( x ) dx i ⊗ · · · ⊗ dx i m be a smooth C n -valued symmetric m -tensor fieldon M . Then f induces the corresponding function f on SM , defined by f ( x, v ) := f i ··· i m ( x ) v i · · · v i m , ( x, v ) ∈ SM.
We denote by C ∞ ( S mM ; C n ) the space of smooth C n -valued symmetric (covariant) m -tensor fields on M . We will also use the notations C ∞ (Λ M ; C n ) and C ∞ ( M ; C n )for C ∞ ( S M ; C n ) and C ∞ ( S M ; C n ), respectively. In what follows, we will identify f ∈ C ∞ ( S mM ; C n ) with its corresponding induced function f ∈ C ∞ ( SM ; C n ).For a given [ f, h ] ∈ C ∞ ( S mM ; C n ) × C ∞ ( S m − M ; C n ), m ≥
1, we define I A, Φ [ f, h ] := I A, Φ f + I A, Φ h. This operator also has non-trivial kernel since I A, Φ [ G E p + Ap, Φ p ] = 0 for all p ∈ C ∞ ( S m − M ; C n ) such that p | ∂M = 0. In the present paper, we are interested inthe question whether these are the only kernels in I A, Φ . Our second main resultgives an affirmative answer provided that the thermostat curvature is negative. Theorem 1.5.
Let ( M, g, E ) be a strictly convex and non-trapping Gaussian ther-mostat on a compact oriented surface with boundary such that K E < . Let also A : T M → gl ( n, C ) be a connection and Φ : M → gl ( n, C ) be a Higgs field. As-sume that [ f, h ] ∈ C ∞ ( S mM ; C n ) × C ∞ ( S m − M ; C n ) , m ≥ . If I A, Φ [ f, h ] = 0 , then f = G E p + Ap and h = Φ p for some p ∈ C ∞ ( S m − M ; C n ) with p | ∂M = 0 . The above question was extensively studied in [15, 38, 39, 41, 48] for geodesicflows and also for magnetic flows in [1]. In fact, a result similar to ours was proven in[38] for geodesic flows in any dimension. For Gaussian thermostats on surfaces, suchresults were established in [4] in the absence of connections and Higgs fields. When m = 1, these results hold on Finsler surfaces without any curvature constraints [3].1.4. Main results on closed surfaces.
Next, we consider similar problems dis-cussed in Section 1.3 but on closed oriented Riemannian surfaces. Let (
M, g ) be aclosed Riemannian surface and let E be an external field. Consider the canonicalprojection π : SM → M . Since there is no boundary, we need to work with closedthermostat geodesics. Clearly, there should exist sufficiently many closed thermo-stat geodesics for the question to be sound. This is so, for example, in the case when HE ATTENUATED THERMOSTAT RAY TRANSFORMS 5 the thermostat flow is Anosov, (see [47, Theorem 5.2]), the Gaussian thermostat(
M, g, E ) with K E < A, Φ), thereis a naturally induced U ( n )-cocycle over the thermostat flow φ t on SM associatedwith the Gaussian thermostat ( M, g, E ). The cocyle corresponding to ( A, Φ) is theunique solution C A, Φ : SM × R → U ( n ) of (cid:2) ∂ t + A ( φ t ( x, v )) + Φ( π ◦ φ t ( x, v )) (cid:3) C A, Φ ( x, v ; t ) = 0 , C A, Φ ( x, v ; 0) = Id . Definition 1.6.
The pair ( A, Φ) is said to be transparent if C A, Φ ( x, v ; T ) = Id forevery time T such that φ T ( x, v ) = ( x, v ).Suppose that ( A, Φ) is transparent and Q ∈ C ∞ ( M ; U ( n )). Then it is not difficultto show that C ( x, v ; t ) := Q ( φ t ( x, v )) − C A, Φ ( x, v ; t ) Q ( x, v ) satisfies h ∂ t + ( Q − ( d + A ) Q )( φ t ( x, v )) + ( Q − Φ Q )( π ◦ φ t ( x, v )) i C ( x, v ; t ) = 0and C ( x, v ; 0) = Id. Moreover, one can see that C ( x, v ; T ) = Id for all T satisfying φ T ( x, v ) = ( x, v ). Therefore, we can claim that C Q − ( d + A ) Q,Q − Φ Q ( x, v ; t ) = Q ( φ t ( x, v )) − C A, Φ ( x, v ; t ) Q ( x, v ) . This says that the set of transparent pairs is invariant under the gauge change:( A, Φ) ( Q − ( d + A ) Q, Q − Φ Q ) , Q ∈ C ∞ ( M ; U ( n )) . We are interested in the problem of classifying all transparent pairs up to gaugeequivalence.To state our result, we need to introduce a few more notions. Since A is aunitary connection, its curvature F A , defined as F A = dA + A ∧ A , is a smooth u ( n )-valued 2-form on M . Hence, ⋆F A ∈ C ∞ ( M ; u ( n )), where ⋆ is the Hodge staroperator on ( M, g ). Therefore, there exist Lipschitz continuous λ min , λ max : M → R such that λ min ( x ) and λ max ( x ) are smallest and largest, respectively, eigenvaluesof i ⋆ F A ( x ) ∈ u ( n ) at every x ∈ M . As usual, by χ ( M ) we denote the Eulercharacteristic of M . Our third main result is as follows. Theorem 1.7.
Let ( M, g, E ) be a Gaussian thermostat on a closed oriented surfacesuch that K E < . Let also A : T M → u ( n ) be a unitary connection and Φ : M → gl ( n, C ) be a skew-Hermitian Higgs field. Suppose that either πkχ ( M ) < Z M λ min d Vol g , Z M λ max d Vol g < − πkχ ( M ) (4) or k > k i ⋆ F A k L ∞ ( M ) κ (5) is satisfied for all k ≥ , where κ > is a constant such that K E ≤ − κ . If C A, Φ ( x, v ; T ) = Id for all ( x, v ) ∈ SM and T ∈ R such that φ T ( x, v ) = ( x, v ) , thenthe pair ( A, Φ) is gauge equivalent to the trivial pair, i.e. Q − ( d + A ) Q = 0 and Φ = 0 for some Q ∈ C ∞ ( M ; U ( n )) . YERNAT M. ASSYLBEKOV AND FRANKLIN T. REA
For geodesic flows on higher dimensional manifolds, this result was obtained in[15, 38]. For earlier related results in the case of geodesic flows, see [34, 35, 36, 37].As in the case of surfaces with boundary, this problem is reduced to of theproblem to an integral geometry problem. Our final main result is an analog ofTheorem 1.5 for closed surfaces.
Theorem 1.8.
Let ( M, g, E ) be a Gaussian thermostat on a closed oriented surfacesuch that K E < . Suppose A : T M → u ( n ) is a unitary connection for which either (4) or (5) holds for all k ≥ . Let also Φ : M → gl ( n, C ) be a Higgs field. Assumethat [ f, h ] ∈ C ∞ ( S mM ; C n ) × C ∞ ( S m − M ; C n ) , m ≥ . If u ∈ C ∞ ( SM ; C n ) satisfies ( G E + A + Φ) u = f + h , then u ∈ C ∞ ( S m − M ; C n ) . Hence, f = G E u + Au and h = Φ u . This result was obtained in [15, 38] for geodesic flows on higher dimensionalmanifolds. In the absence of connections and Higgs fields, various versions of thisresult were established in [3, 4, 6, 7, 23] for Gaussian thermostats.1.5.
Structure of the paper.
The paper is organized as follows. In Section 2we collect certain preliminary facts about geometry and Fourier analysis on SM and Gaussian thermostats. Then we derive a Carleman type estimate for Gaussianthermostats with negative curvature in Section 3. Next, in Section 4 we provecertain regularity results for solutions to the transport equation on non-trapping andstrictly convex Gaussian thermostats. Section 5 contains the proofs of Theorem 1.5and Theorem 1.8. In Section 6 we prove certain technical results which we usedin Section 5. Finally, the proofs of Theorem 1.3 and Theorem 1.7 are presented inSection 7. Acknowledgements.
YA would like to thank Total E & P Research & TechnologyUSA and the members of the Geo-Mathematical Imaging Group at Rice Universityfor financial support. 2.
Preliminaries
Geometry on SM . Since M is assumed to be oriented there is a circle actionon the fibres of SM with infinitesimal generator V called the vertical vector field .Let X denote the generator of the geodesic flow of g . We complete X, V to a globalframe of T ( SM ) by defining the vector field X ⊥ := [ V, X ], where [ · , · ] is the Liebracket for vector fields. Using this frame we can define a Riemannian metric on SM by declaring { X, X ⊥ , V } to be an orthonormal basis and the volume form ofthis metric will be denoted by d Σ . The vector fields X, X ⊥ , V satisfy the followingstructural equations X = [ V, X ⊥ ] , X ⊥ = [ X, V ] , [ X, X ⊥ ] = − KV, (6)where K is the Gaussian curvature of the surface. HE ATTENUATED THERMOSTAT RAY TRANSFORMS 7
Fourier analysis on SM . For any two functions u, v : SM → C n define the L inner product as ( u, v ) := Z SM h u, v i C n d Σ with the corresponding norm denoted by k · k . The space L ( SM ; C n ) decomposesorthogonally as a direct sum L ( SM ; C n ) = L k ∈ Z H k ( SM ; C n ) where H k ( SM ; C n )is the eigenspace of V corresponding to the eigenvalue ik . A function u ∈ L ( SM ; C n )has a Fourier series expansion u = ∞ X k = −∞ u k , u k ∈ H k ( SM ; C n ) . Then k u k = P ∞ k = −∞ k u k k . We denote the subspaceΩ k ( SM ; C n ) := H k ( SM ; C n ) ∩ C ∞ ( SM ; C n ) . We say that u ∈ C ∞ ( SM ; C n ) is of degree m if u k = 0 for all | k | ≥ m + 1. If f ∈ C ∞ ( S mM ; C n ), then the corresponding f ∈ C ∞ ( SM ; C n ) is of degree m ; see [40,Section 2] for more details. Moreover, if m is odd/even then f k = 0 for all even/odd k ∈ Z . In particular, for any α ∈ C ∞ (Λ M ; C n ) its induced function α on SM canbe written as α = α − + α with α ± ∈ Ω ± ( SM ; C n ).2.3. Generating vector field of a Gaussian thermostat.
Let (
M, g, E ) be aGaussian thermostat with dim( M ) = 2. Then according to [6], the generatingvector field G E of the thermostat flow φ t can be written as G E = X + λV. For the adjoints of X ⊥ , V and G E , w.r.t. ( · , · ), we have X ∗⊥ = − X ⊥ , V ∗ = − V, G ∗ E = − ( G E + V ( λ ));see [4, 6].2.4. Guillemin-Kazhdan type operators.
Consider the following first order dif-ferential operators introduced by Guillemin and Kazhdan [16] η − = 12 ( X − iX ⊥ ) , η + = 12 ( X + iX ⊥ ) . (7)Then X can be expressed as X = η − + η + . It was shown that η ± : Ω k ( SM ; C n ) → Ω k ± ( SM ; C n ), k ∈ Z . Moreover, these operators are elliptic and satisfy the follow-ing commutation relation [ η + , η − ] = i KV. (8)It is also important to note that η ∗− = − η + and η ∗ + = − η − ; see [16].For the Gaussian thermostat ( M, g, E ), we consider the modified versions of η + and η − defined as µ − := η − + λ − V, µ + := η + + λ + V, where λ = λ − + λ + with λ ± ∈ Ω ± ( SM ; C ). Then it is not difficult to see that G E = µ − + µ + . YERNAT M. ASSYLBEKOV AND FRANKLIN T. REA
Using V ∗ = − V and λ − = λ + , one can see that( µ − ) ∗ = − µ + − iλ + , ( µ + ) ∗ = − µ − + iλ − . (9)We also have the following analog of (8). Lemma 2.1. If u ∈ C ∞ ( SM ; C n ) , then [ µ + , µ − ] u = i K E V u − iλ − µ + u − iλ + µ − u. Proof.
It is enough to give the proof in the case u ∈ Ω k ( SM ; C n ). Using (8) andthe fact that V u = iku ,[ µ + , µ − ] u = [ η + , η − ] u + [ η + , λ − V ] u + [ λ + V, η − ] u + [ λ + V, λ − V ] u = i KV u + η + ( ikλ − u ) − i ( k + 1) λ − η + u + i ( k − λ + η − u − η − ( ikλ + u ) − k ( k − λ + λ − u + k ( k − λ + λ − u = i KV u + ik ( η + λ − − η − λ + ) u − iλ − η + u − iλ + η − u + 2 kλ + λ − u = i KV u + ( η + λ − − η − λ + ) V u − iλ − µ + u − iλ + µ − u. Now, let θ be a 1-form on M dual to E . Then λ = V θ , and hence λ − = − iθ − and λ + = iθ . Using these, a straightforward calculation shows that η + λ − − η − λ + = − i Xθ − X ⊥ V θ ) . However, Xθ − X ⊥ V θ = div g E ; see the proof of [33, Lemma 5.2]. Thus, we have( η + λ − − η − λ + ) = − i g E, which completes the proof. (cid:3) Lemma 2.2. If A : T M → gl ( n, C ) is a connection, then [ µ + + A + , µ − + A − ] u = i K E V u + i ⋆ F A u − iλ − ( µ + + A + ) u − iλ + ( µ − + A − ) u for u ∈ Ω k ( SM ; C n ) .Proof. Using Lemma 2.1,[ µ + + A + , µ − + A − ] u = [ µ + , µ − ] u + [ A + , µ − ] u + [ µ + , A − ] u + [ A + , A − ] u = i K E V u − iλ − ( µ + + A + ) u − iλ + ( µ − + A − ) u + ( η + A − − η − A + ) u + [ A + , A − ] u. A straightforward calculation shows that η + A − − η − A + = i X ⊥ A + XV A ) , [ A + , A − ] = − i V A, A ] . Hence, by (6) in [39], we have η + A − − η − A + + [ A + , A − ] = i ⋆ F A , Using this, we get the desired identity. (cid:3)
HE ATTENUATED THERMOSTAT RAY TRANSFORMS 9 Carleman estimate for Gaussian thermostats with negativecurvature
In the present section, we prove a Carleman estimate for Gaussian thermostatswith negative curvature, which will be used in Section 5 to prove our main results.
Theorem 3.1.
Let ( M, g, E ) be a Gaussian thermostat on a compact oriented sur-face and m ≥ be an integer. Assume that K E ≤ − κ for some κ > constant.Then for any s > , we have ∞ X k = m k s +1 (cid:16) k u k k + k u − k k (cid:17) ≤ κs ∞ X k = m +1 k s +1 (cid:16) k ( G E u ) k k + k ( G E u ) − k k (cid:17) , for all u ∈ C ∞ ( SM ; C n ) , with u | ∂ ( SM ) = 0 in the case ∂M = ∅ . To prove Theorem 3.1, we follow the arguments in [38, Section 6]. Our startingpoint will be the following analog of Guillemin-Kazhdan energy identity [16].
Proposition 3.2.
Let ( M, g, E ) be a Gaussian thermostat on a compact orientedsurface. Then for any u ∈ Ω k ( SM ; C n ) , k ∈ Z , with u | ∂ ( SM ) = 0 in the case ∂M = ∅ , we have k µ + u k = k µ − u k − k K E u, u ) . Remark 3.3.
In fact, this identity was derived earlier in [4] (see the the proofof Lemma 6.5 therein) via the Pestov identity. Here we give an alternative proofwithout the involvement of the Pestov identity. It also follows from more generalProposition 6.2 in Section 6.
Proof.
Using (9) and then Lemma 2.1, k µ + u k = ( µ + u, µ + u ) = ( µ ∗ + µ + u, u ) = ( − µ − µ + u, u ) + ( iλ − µ + u, u )= − ( µ + µ − u, u ) + i K E V u, u ) − ( iλ + µ − u, u )= ( µ ∗− µ − u, u ) + i K E V u, u ) = k µ − u k − k K E u, u ) . The proof is thus complete. (cid:3)
The proposition above now yields the following result.
Proof of Theorem 3.1.
Using the hypothesis K E ≤ − κ , for k ≥ m integer, we show k µ − u k k + k µ + u − k k + kκ (cid:16) k u k k + k u − k k (cid:17) ≤ k µ − u k k − k K E u k , u k ) + k µ + u − k k + ( − k )2 ( K E u − k , u − k )= k µ + u k k + k µ − u − k k , where at the last step we used Proposition 3.2 to u k and u − k . Multiplying thisestimate by k s for some s >
0, we get k s (cid:16) k µ − u k k + k µ + u − k k (cid:17) + κ k s +1 (cid:16) k u k k + k u − k k (cid:17) ≤ k s (cid:16) k µ + u k k + k µ − u − k k (cid:17) ≤ k s (cid:16) ε k (cid:17)(cid:16) k ( G E u ) k +1 k + k ( G E u ) − k − k (cid:17) + k s (1 + ε k ) (cid:16) k µ − u k +2 k + k µ + u − k − k (cid:17) , where { ε k } ∞ k = m is a sequence of positive numbers to be chosen later. Summing theseestimates over k from m to N , we obtain N X k = m k s (cid:16) k µ − u k k + k µ + u − k k (cid:17) + N X k = m κ k s +1 (cid:16) k u k k + k u − k k (cid:17) ≤ N +1 X k = m +1 ( k − s (cid:16) ε k − (cid:17)(cid:16) k ( G E u ) k k + k ( G E u ) − k k (cid:17) + N +2 X k = m +2 ( k − s (1 + ε k − ) (cid:16) k µ − u k k + k µ + u − k k (cid:17) . Note here we are only considering the first N terms of the series, with the intent ofachieving convergence later. Now, we make an appropriate choice of ε k , i.e. ε k − = k s ( k − s − > , k ≥ m + 2 , which yields ( k − s (1 + ε k − ) = k s , for all k ≥ m + 2 , and ( k − s (cid:16) ε k − (cid:17) = ( k − s ( k + 1) s − ( k − s ≤ k s +1 s , where in the last step we used [38, Lemma 6.8]. Inserting these into the aboveestimate, we get m +1 X k = m k s (cid:16) k µ − u k k + k µ + u − k k (cid:17) + N X k = m κ k s +1 (cid:16) k u k k + k u − k k (cid:17) ≤ N +1 X k = m +1 k s +1 s (cid:16) k ( G E u ) k k + k ( G E u ) − k k (cid:17) + N +2 X k = N +1 k s (cid:16) k µ − u k k + k µ + u − k k (cid:17) . Now we take the limit as N → ∞ . Since the weights k s +1 grow at most polyno-mially for each fixed s >
0, the last term goes to zero. Therefore, in particular, we
HE ATTENUATED THERMOSTAT RAY TRANSFORMS 11 have ∞ X k = m k s +1 (cid:16) k u k k + k u − k k (cid:17) ≤ κs ∞ X k = m +1 k s +1 (cid:16) k ( G E u ) k k + k ( G E u ) − k k (cid:17) as desired. This completes the proof. (cid:3) Regularity results for the transport equations
Let (
M, g, E ) be a non-trapping and strictly convex Gaussian thermostat on acompact manifold with boundary. Suppose that A : T M → gl ( n, C ) is a connectionand Φ : M → gl ( n, C ) is a Higgs field on the trivial bundle M × C n .4.1. Scattering relation.
Recall that τ ( x, v ), for ( x, v ) ∈ SM , is the first non-negative time when the thermostat geodesic γ x,v exits M . Since ( M, g, E ) is non-trapping and strictly convex, τ is continuous on SM and smooth on SM \ S ( ∂M ). Lemma 4.1.
For a non-trapping and strictly convex Gaussian thermostat ( M, g, E ) on a compact manifold with boundary, the restricted function τ | ∂ + SM is smooth on ∂ + SM .Proof. Consider a smooth ρ : M → [0 , ∞ ) such that ρ − (0) = ∂M and |∇ ρ | g ≡ ∂M . Clearly, ∇ ρ = ν . Write h ( x, v ; t ) = ρ ( γ x,v ( t )) for ( x, v ) ∈ ∂ + SM . Then h ( x, v ; 0) = 0 , ∂ t h ( x, v ; 0) = h ν ( x ) , v i g ( x ) ,∂ t h ( x, v ; 0) = Hess x ρ ( v, v ) + h ν ( x ) , E ( x ) i g ( x ) . Therefore, we have h ( x, v ; t ) = h ν ( x ) , v i g ( x ) t + 12 (Hess x ρ ( v, v ) + h ν ( x ) , E ( x ) i g ( x ) ) t + R ( x, v ; t ) t for a smooth function R . Since h ( x, v ; τ ( x, v )) = 0, one can see that T = τ ( x, v )solves the equation h ν ( x ) , v i g ( x ) + 12 (Hess x ρ ( v, v ) + h ν ( x ) , E ( x ) i g ( x ) ) T + R ( x, v ; T ) T = 0 . Let us denote the left side of the above equation by F ( x, v ; T ). Then for ( x, v ) ∈ S ( ∂M ), 2 ∂ T F ( x, v ; 0) = ( − Λ( x, v ) + h ν ( x ) , E ( x ) i g ( x ) ) . Here we used the fact that Hess x ρ ( v, v ) = − Λ( x, v ) for ( x, v ) ∈ S ( ∂M ). By strictconvexity, we have ∂ T F ( x, v ; 0) < x, v ) ∈ S ( ∂M ). Therefore, the ImplicitFunction Theorem implies the smoothness of τ in a neighborhood of S ( ∂M ). Thiscompletes the proof, since τ is smooth on SM \ S ( ∂M ). (cid:3) The scattering relation is the map S : ∂ + SM → ∂ − SM defined as S ( x, v ) = (cid:0) γ x,v ( τ ( x, v )) , ˙ γ x,v ( τ ( x, v )) (cid:1) , ( x, v ) ∈ ∂ + SM.
In other words, S maps the point and direction of entrance of a thermostat geodesicto the point and direction of the exit. By Lemma 4.1, the scattering relation S is adiffeomorphism between ∂ + SM and ∂ − SM . Functions constant along the thermostat flow.
For w ∈ C ∞ ( ∂ + SM ; C n ),the problem G E w ψ = 0 in SM, w ψ (cid:12)(cid:12) ∂ + SM = w has the unique solution w ψ : SM → C n . In other words, w ψ is the function that isconstant along the orbits of the thermostat flow and equals w on ∂ + SM .Define the space C ∞ α ( ∂ + SM ; C n ) := { w ∈ C ∞ ( ∂ + SM ; C n ) : w ψ ∈ C ∞ ( SM ; C n ) } . Consider the operator A : C ( ∂ + SM ; C n ) → C ( ∂ ( SM ); C n ) given by A w ( x, v ) := ( w ( x, v ) , ( x, v ) ∈ ∂ + SM, ( w ◦ S − )( x, v ) , ( x, v ) ∈ ∂ − SM.
Clearly, w ψ | ∂ ( SM ) = A w . According to Theorem A.2, the space C ∞ α ( ∂ + SM ; C n )can be characterized as follows: C ∞ α ( ∂ + SM ; C n ) := { w ∈ C ∞ ( ∂ + SM ; C n ) : A w ∈ C ∞ ( ∂ ( SM ); C n ) } . (10)4.3. Homogeneous equations.
For a given w ∈ C ∞ ( ∂ + SM ; C n ), consider theunique solution w ♯ : SM → C n for the transport equation( G E + A + Φ) w ♯ = 0 in SM, w ♯ (cid:12)(cid:12) ∂ + SM = w. Define the space S ∞ A, Φ ( ∂ + SM ; C n ) := { w ∈ C ∞ ( ∂ + SM ; C n ) : w ♯ ∈ C ∞ ( SM ; C n ) } . One can see that w ♯ ( x, v ) = U A, Φ ( x, v ) w ψ ( x, v ) , ( x, v ) ∈ SM, where U A, Φ introduced in Section 1.3. Introduce the operator Q : C ( ∂ + SM ; C n ) → C ( ∂ ( SM ); C n )defined, using the scattering relation S and the scattering data C A, Φ , as Q w ( x, v ) = ( w ( x, v ) , ( x, v ) ∈ ∂ + SM,C A, Φ ( x, v )( w ◦ S − )( x, v ) , ( x, v ) ∈ ∂ − SM.
Then note that w ♯ | ∂ ( SM ) = Q w . Now, we characterize the space S ∞ A, Φ ( ∂ + SM ; C n )in terms of the operator Q . Lemma 4.2.
We have the following characterization S ∞ A, Φ ( ∂ + SM ; C n ) := { w ∈ C ∞ ( ∂ + SM ; C n ) : Q w ∈ C ∞ ( ∂ ( SM ); C n ) } . Proof.
Consider a closed manifold N containing M . We smoothly extend the metric g and the vector field E to N and denote extensions by the same notations. Next,embed M into the interior of a compact manifold M ⊂ N with boundary. Choose M to be sufficiently close to M so that ( M , g, E ) is also non-trapping and strictlyconvex. We also extend A and Φ smoothly to N .Consider the unique solution R : SM → GL ( n, C ) to the transport equation( G E + A + Φ) R = 0 in SM , R (cid:12)(cid:12) ∂ + SM = Id . HE ATTENUATED THERMOSTAT RAY TRANSFORMS 13
Then the restriction of R to SM , still denoted by R , is in C ∞ ( SM ; GL ( n, C )) andsolves ( G E + A + Φ) R = 0 in SM . Set r := R − | ∂ + SM . Then we can express w ♯ and Q w as w ♯ = R ( rw ) ψ and Q w ( x, v ) = ( R ( x, v ) r ( x, v ) w ( x, v ) , if ( x, v ) ∈ ∂ + SM,R ( x, v )(( rw ) ◦ S − )( x, v ) , if ( x, v ) ∈ ∂ − SM.
From this it is obvious that Q w ∈ C ∞ ( ∂ ( SM ); C n ) if w ♯ ∈ C ∞ ( SM ; C n ). Nowsuppose Q w ∈ C ∞ ( ∂ ( SM ); C n ). Since R ∈ C ∞ ( SM ; GL ( n, C )) and A ( rw ) = R − | ∂ ( SM ) Q w , this, together with (10), implies that ( rw ) ψ ∈ C ∞ ( SM ; C n ). Usingthe fact R ∈ C ∞ ( SM ; GL ( n, C )) once again, we can claim w ♯ ∈ C ∞ ( SM ; C n ),finishing the proof. (cid:3) Non-homogeneous equations.
Given f ∈ C ∞ ( SM ; C n ), consider the uniquesolution u f : SM → C n for the transport equation( G E + A + Φ) u = − f in SM, u (cid:12)(cid:12) ∂ − SM = 0 . Note that U − A, Φ solves G E U − A, Φ − U − A, Φ ( A + Φ) = 0. Therefore, G E ( U − A, Φ u f ) = − U − A, Φ f . Integrating along γ x,v : [0 , τ ( x, v )] → M , for ( x, v ) ∈ ∂ + SM , we obtainthe following integral expression u f ( x, v ) = Z τ ( x,v )0 U − A, Φ ( γ x,v ( t ) , ˙ γ x,v ( t )) f ( γ x,v ( t ) , ˙ γ x,v ( t )) dt, ( x, v ) ∈ ∂ + SM.
Since τ is a non-smooth function on SM in general, the function u f also may fail tobe smooth on SM . But we can and shall show that u f ∈ C ∞ ( SM ; C n ) if I A, Φ f = 0. Proposition 4.3.
Let ( M, g, E ) be a non-trapping and strictly convex Gaussianthermostat on a compact manifold with boundary. Suppose that A : T M → gl ( n, C ) is a connection and Φ : M → gl ( n, C ) is a Higgs field on the trivial bundle M × C n .If I A, Φ f = 0 for f ∈ C ∞ ( SM ; C n ) , then u f ∈ C ∞ ( SM ; C n ) .Proof. Let N and M be as in the proof of Lemma 4.2. We consider the sameextensions of g , E , A and Φ to N . We also extend f smoothly to SN preservingthe former notation for extension.Consider the unique solution a : SM → C n for the transport equation( G E + A + Φ) a = − f in SM , a (cid:12)(cid:12) ∂ − SM = 0 . Then a | SM , which we keep denoting by a , is in C ∞ ( SM ; C n ) and solves ( G E + A +Φ) a = − f in SM . Then w := u f − a satisfies ( G E + A + Φ) w = 0 in SM . Since u f | ∂ − SM = 0 and u f | ∂ + SM = I A, Φ f = 0, we have Q ( w | ∂ + SM ) = w | ∂ ( SM ) = − a | ∂ ( SM ) which is in C ∞ ( ∂ ( SM ); C n ). Then w ∈ C ∞ ( SM ; C n ) by Lemma 4.2, and hence, u f ∈ C ∞ ( SM ; C n ). The proof is complete. (cid:3) Injectivity results for the linear problems
In the present section, we prove injectivity results for the linear problems statedin the introduction, namely Theorem 1.5 and Theorem 1.8. To that end, we firstneed the following theorem.
Theorem 5.1.
Let ( M, g, E ) be a Gaussian thermostat on a compact orientedsurface such that K E < . Let also A : T M → gl ( n, C ) be a connection and Φ : M → gl ( n, C ) be a Higgs field. Assume that f ∈ C ∞ ( SM ; C n ) has finite degree.If u ∈ C ∞ ( SM ; C n ) , with u | ∂ ( SM ) = 0 in the case ∂M = ∅ , satisfies ( G E + A + Φ) u = f in SM, then u also has finite degree.Proof. Suppose that f k = 0 for all | k | ≥ m ′ for some m ′ ≥ m ≥ m ′ .Then, writing A = A − + A + with A ± ∈ Ω ± ( SM ; C n ),( G E u ) k = − A − u k +1 − A + u k − − Φ u k for all | k | ≥ m. Therefore, there is
R > k ( G E u ) k k ≤ R (cid:16) k u k +1 k + k u k − k + k u k k (cid:17) for all | k | ≥ m. Using these in the estimate from Theorem 3.1, we obtain ∞ X k = m k s +1 (cid:16) k u k k + k u − k k (cid:17) ≤ Rκs ∞ X k = m +1 k s +1 (cid:16) k u k +1 k + k u k − k + k u k k + k u − k +1 k + k u − k − k + k u − k k (cid:17) ≤ Cκs ∞ X k = m ( k + 1) s +1 (cid:16) k u k k + k u − k k (cid:17) . Let us take s > m ≥ s + 1. Then( k + 1) s +1 = (cid:16) k (cid:17) s +1 k s +1 ≤ (cid:16) k (cid:17) k k s +1 ≤ ek s +1 for all k ≥ m. Hence, for m ≥ s + 1, we have ∞ X k = m k s +1 (cid:16) k u k k + k u − k k (cid:17) ≤ eCκs ∞ X k = m k s +1 (cid:16) k u k k + k u − k k (cid:17) Now, we fix s > eC/κ and m = max(2 s + 1 , m ′ ). Then (cid:16) − eCκs (cid:17) ∞ X k = m k s +1 (cid:16) k u k k + k u − k k (cid:17) ≤ . Since (1 − eC/κs ) >
0, this allows us to conclude u k = 0 for all | k | ≥ m . (cid:3) HE ATTENUATED THERMOSTAT RAY TRANSFORMS 15
Surfaces with boundary.
The following result is a restatement of Theo-rem 1.5.
Theorem 5.2.
Let ( M, g, E ) be a strictly convex and non-trapping Gaussian ther-mostat on a compact oriented surface with boundary such that K E < . Let also A : T M → gl ( n, C ) be a connection and Φ : M → gl ( n, C ) be a Higgs field. For m ≥ integer, assume that f ∈ C ∞ ( SM ; C n ) with f k = 0 for all | k | ≥ m + 1 . If I A, Φ f = 0 , then f = ( G E + A + Φ) u for some u ∈ C ∞ ( SM ; C n ) with u | ∂ ( SM ) = 0 such that u k = 0 for all | k | ≥ m .Proof. By the results of Section 4, there is u ∈ C ∞ ( SM ; C n ) with u | ∂ ( SM ) = 0 suchthat ( G E + A + Φ) u = f in SM. (11)Then u is of finite degree by Theorem 5.1. Hence, there is l > u k = 0for all | k | ≥ l . Now, our goal is to show that l ≤ m . Suppose this is not the case,i.e. l > m . Then from (11), we get ( µ + + A + ) u l − = 0 and ( µ − + A − ) u − l +1 = 0.Since u − l +1 | ∂ ( SM ) = u l − | ∂ ( SM ) = 0, Theorem 6.1 of the next section implies that u − l +1 = u l − = 0. Continuing this process, we show that u k = 0 for all | k | ≥ m .This concludes the proof. (cid:3) Closed surfaces.
Similarly, we restate Theorem 1.8 as follows.
Theorem 5.3.
Let ( M, g, E ) be a Gaussian thermostat on a closed oriented surfacesuch that K E ≤ − κ for some constant κ > . Suppose A : T M → u ( n ) is aunitary connection for which either (4) or (5) is satisfied for all k ≥ . Let also Φ : M → gl ( n, C ) be a Higgs field. For m ≥ integer, assume that f ∈ C ∞ ( SM ; C n ) with f k = 0 for all | k | ≥ m + 1 . If u ∈ C ∞ ( SM ; C n ) satisfies ( G E + A + Φ) u = f ,then u k = 0 for all | k | ≥ m .Proof. Theorem 5.1 allows us to claim that u is of finite degree. Then, parallel toarguments in Theorem 5.3, the proof is reduced to the injectivity of µ ± + A ± onΩ ± k ( SM ; C n ) for all ± k ≥
1. For unitary A , Theorem 6.4 and Corollary 6.3 ensureinjectivity of these operators under the hypotheses (4) and (5), respectively. Theproof is complete. (cid:3) Injectivity of µ ± + A ± operators In the present section we prove injectivity results for µ ± + A ± operators whichare one of the crucial components in the proofs of Theorem 1.5 and Theorem 1.8.6.1. Surfaces with boundary.
This case can be reduced to a similar result forgeodesic flows obtained in [15].
Theorem 6.1.
Let ( M, g, E ) be a Gaussian thermostat on a compact oriented sur-face with boundary and let A : T M → gl ( n, C ) be a connection. If u ∈ Ω ± k ( SM ; C n ) , k ≥ , satisfy ( µ ± + A ± ) u = 0 and u | ∂ ( SM ) = 0 , then u = 0 . Proof.
Let θ be a 1-form on M dual to E . Then λ + = iθ and λ − = − iθ − .Therefore, the operators µ + + A + and µ − + A − restricted to Ω k ( SM ; C n ) andΩ − k ( SM ; C n ), respectively, have the forms µ + + A + = η + + ( A + − kθ ) and µ − + A − = η − + ( A − − kθ − ) . Now, fix k ≥ A k := A − kθ as a connection. Then, according to[15, Theorem 5.2], any u ∈ Ω ± m ( SM ; C n ), m ≥
1, such that ( η ± + A k ± ) u = 0 and u | ∂ ( SM ) = 0 must vanish identically. In the case m = k , this, in particular, yieldsthe desired injectivity result for ( µ ± + A ± ). (cid:3) Closed surfaces.
This case is more complicated. We closely follow the argu-ments of [15, Section 8]. We remind the reader that on closed surfaces we alwayswork with unitary connections. First, we need the following weighted analog ofProposition 3.2 for µ ± + A ± operators. Proposition 6.2.
Let ( M, g, E ) be a Gaussian thermostat on a compact orientedsurface and let A : T M → u ( n ) be a unitary connection. Suppose that ϕ ∈ C ∞ ( M ; R ) . Then for any u ∈ Ω k ( SM ; C n ) , k ∈ Z , with u | ∂ ( SM ) = 0 in thecase ∂M = ∅ , we have k e − ϕ ( µ + + A + )( e ϕ u ) k = k e ϕ ( µ − + A − )( e − ϕ u ) k − k K E u, u ) −
12 ((∆ g ϕ ) u, u ) + i ⋆F A u, u ) . Proof.
Define the operators P ϕ = e − ϕ ◦ ( µ + + A + ) ◦ e ϕ = ( µ + + A + ) + ( η + ϕ ) ,Q ϕ = e ϕ ◦ ( µ − + A − ) ◦ e − ϕ = ( µ − + A − ) − ( η − ϕ ) . It suffices to prove P ∗ ϕ P ϕ u − Q ∗ ϕ Q ϕ u = − k K E u −
12 (∆ g ϕ ) u + i ⋆ F A u, u ∈ Ω k ( SM ; C n ) . (12)Using (9) and the fact that A is unitary, it is easy to check that P ∗ ϕ = − ( µ − + A − ) + iλ − + ( η − ϕ ) , Q ∗ ϕ = − ( µ + + A + ) − iλ + − ( η + ϕ ) . Then for u ∈ Ω k ( SM ; C n ), P ∗ ϕ P ϕ u = ( − ( µ − + A − ) + iλ − + ( η − ϕ ))(( µ + + A + ) + ( η + ϕ )) u = − ( µ − + A − )( µ + + A + ) u − ( η − η + ϕ ) u − iλ − ( η + ϕ ) u − ( η + ϕ )( µ − + A − ) u + iλ − ( µ + + A + ) u + iλ − ( η + ϕ ) u + ( η − ϕ )( µ + + A + ) u + ( η − ϕ )( η + ϕ ) u = − ( µ − + A − )( µ + + A + ) u − ( η − η + ϕ ) u − ( η + ϕ )( µ − + A − ) u + iλ − ( µ + + A + ) u + ( η − ϕ )( µ + + A + ) u + ( η − ϕ )( η + ϕ ) u HE ATTENUATED THERMOSTAT RAY TRANSFORMS 17 and Q ∗ ϕ Q ϕ u = ( − ( µ + + A + ) − iλ + − ( η + ϕ ))(( µ − + A − ) − ( η − ϕ )) u = − ( µ + + A + )( µ − + A − ) u + ( η + η − ϕ ) u − iλ + ( η − ϕ ) u + ( η − ϕ )( µ + + A + ) u − iλ + ( µ − + A − ) u + iλ + ( η − ϕ ) u − ( η + ϕ )( µ − + A − ) u + ( η + ϕ )( η − ϕ ) u = − ( µ + + A + )( µ − + A − ) u + ( η + η − ϕ ) u + ( η − ϕ )( µ + + A + ) u − iλ + ( µ − + A − ) u − ( η + ϕ )( µ − + A − ) u + ( η + ϕ )( η − ϕ ) u. Therefore, P ∗ ϕ P ϕ u − Q ∗ ϕ Q ϕ u = [ µ + + A + , µ − + A − ] u − ( η − η + ϕ + η + η − ϕ ) u + iλ − ( µ + + A + ) u + iλ + ( µ − + A − ) u. This gives (12) by applying Lemma 2.2 and the fact η − η + ϕ + η + η − ϕ = ∆ g ϕ . (cid:3) As an immediate consequence of Proposition 6.2, we obtain our first result oninjectivity of µ ± + A ± on closed surfaces. Corollary 6.3.
Let ( M, g, E ) be a Gaussian thermostat on a closed oriented surfacewith K E ≤ − κ for some constant κ > and let A : T M → u ( n ) be a unitaryconnection. Suppose that for all k ≥ , k > k i ⋆ F A k L ∞ ( M ) κ . Then µ ± + A ± is injective on Ω ± k ( SM ; C n ) for all k ≥ . Now, we are ready to state our second result on injectivity of µ ± + A ± on closedsurfaces which is an analog of [15, Theorem 8.1]. Theorem 6.4.
Let ( M, g, E ) be a Gaussian thermostat on a closed oriented surfaceand let A : T M → u ( n ) be a unitary connection. Denote by λ min and λ max thesmallest and largest, respectively, eigenvalues of i ⋆ F A . (a) If k ∈ Z and Z M λ min d Vol g > πkχ ( M ) , then any u ∈ Ω k ( SM ; C n ) satisfying ( µ + + A + ) u = 0 vanishes identically. (b) If k ∈ Z and Z M λ max d Vol g < − πkχ ( M ) , then any u ∈ Ω − k ( SM ; C n ) satisfying ( µ − + A − ) u = 0 vanishes identically.Proof. We prove (a). Suppose that there is ϕ ∈ C ∞ ( M ; R ) and some constant C > − kK E − (∆ g ϕ ) + i ⋆ F A ≥ C Id on M (13)as positive definite endomorphisms. Then Proposition 6.2 below yields C k w k ≤ k e − ϕ ( µ + + A + )( e ϕ w ) k , w ∈ Ω k ( SM ; C n ) . If ( µ + + A + ) u = 0, taking w = e − ϕ u in this estimate will give u = 0 as desired. To find ϕ satisfying (13), we need f ∈ C ∞ ( M ; R ) such that f + λ > Z M f d Vol g = − πkχ ( M ) . (14)Such f , by the Gauss-Bonnet theorem, satisfies Z M [ kK E + f ] d Vol g = Z M [ kK + f ] d Vol g = 0 . Therefore, there exists ϕ ∈ C ∞ ( M ; R ) solving the equation − ∆ g ϕ = kK E + f in M . Using this, one can see that − kK E − (∆ g ϕ ) + i ⋆ F A = f + i ⋆ F A ≥ f + λ ≥ C > f satisfying (14). To that end, define ε := 1Vol g ( M ) h Z M λ d Vol g − πkχ ( M ) i > . Since λ ∈ C ( M ; R ) there is h ∈ C ∞ ( M ; R ) such that k h − λ k ≤ ε/
4. Define f := − h + ε , where ε is the constant determined by Z M f d Vol g = − πkχ ( M ) . Then f + λ = ε − [ h − λ ] ≥ ε − ε/
4. However, ε = 1Vol g ( M ) Z M [ f + h ] d Vol g = 1Vol g ( M ) h Z M h d Vol g − πkχ ( M ) i = ε + 1Vol g ( M ) Z M [ h − λ ] d Vol g ≥ ε − ε ε . Hence, we finally get f + λ ≥ ε/
2. The proof of (b) is analogous. (cid:3) Injectivity results for the nonlinear problems
In the present section we prove injectivity results for the nonlinear problems,namely Theorem 1.3 and Theorem 1.7.7.1.
Surfaces with boundary.
We use a pseudolinearization to reduce the non-linear problem to Theorem 1.5. This argument was used in earlier works [15, 38,39, 41, 48].
Proof of Theorem 1.3.
Let U A, Φ : SM → GL ( n, C ), U B, Ψ : SM → GL ( n, C ) be theunique solutions for the problems( G E + A + Φ) U A, Φ = 0 in SM, U A, Φ | ∂ + SM = Id , ( G E + B + Ψ) U B, Ψ = 0 in SM, U B, Ψ | ∂ + SM = Id , respectively. Then C A, Φ = U A, Φ | ∂ − SM and C B, Ψ = U B, Ψ | ∂ − SM . It is not difficultto show that Q := U A, Φ U − B, Ψ : SM → GL ( n, C ) satisfies G E Q + AQ − QB + Φ Q − Q Ψ = 0 in
SM, Q | ∂ ( SM ) = Id . HE ATTENUATED THERMOSTAT RAY TRANSFORMS 19
Set U := Q − Id. Then U : SM → C n × n solves G E U + AU − U B + Φ U − U Ψ = − ( A − B + Φ − Ψ) in
SM, U | ∂ ( SM ) = 0 . Let A : T M → gl ( n × n, C ) be a connection on the trivial bundle M × C n × n definedas A U := AU − U B , and let Φ : M → gl ( n × n, C ) be a Higgs field defined as Φ U := Φ U − U Ψ. Then we can rewrite the above transport equation as G E U + A U + Φ U = − ( A − B + Φ − Ψ) in
SM, U | ∂ ( SM ) = 0 . In other words, I A , Φ [ A − B, Φ − Ψ] = 0. Then by Theorem 1.5, there is p ∈ C ∞ ( M ; C n × n ) with p | ∂M = 0 such that A − B = d A p = dp + Ap − pB, Φ − Ψ = Φ p = Φ p − p Ψ . Uniqueness for solutions of transport equations imply that U = − p . Setting back Q := U + Id, we finish the proof. (cid:3) Closed surfaces.
Let (
M, g, E ) be a Gaussian thermostat on a closed orientedsurface M with K E <
0. Let ( A, Φ) be a pair of a unitary connection and a skew-Hermitian Higgs field and let C A, Φ : SM × R → U ( n ) be the corresponding cocycle. Definition 7.1.
The pair ( A, Φ) will be called cohomologically trivial if there is u ∈ C ∞ ( SM ; U ( n )) such that C A, Φ ( x, v ; t ) = u ( φ t ( x, v )) u ( x, v ) − , In this case, u is referred to as a trivializing function .Clearly, cohomologically trivial pairs are transparent. The opposite does notneed to be true in general. However, in our setting, transparent pairs are alwayscohomologically trivial. Indeed, as mentioned before, the assumption K E < φ t is Anosov and transitive. Then cohomologicallytriviality of transparent pairs follows from the Livsic theorem for U ( n )-cocycles[25, 26] combined with regularity results in [31].Now we are in position to prove Theorem 1.7. Proof of Theorem 1.7.
According to the above discussion, there is a trivializingfunction u ∈ C ∞ ( SM ; U ( n )). It is easy to see that u satisfies( G E + A + Φ) u = 0 in SM. (15)By Theorem 1.8, we deduce that u ∈ C ∞ ( M ; U ( n )). Then (15) becomes du + Au + Φ u = 0, which can be separated as du + Au = 0 and Φ u = 0. Setting Q := u ∈ C ∞ ( M ; U ( n )), this gives Q − ( d + A ) Q = 0 and Φ = 0. (cid:3) Appendix A. Gaussian thermostats on manifolds with boundary
Throughout this section, (
M, g, E ) is a non-trapping Gaussian thermostat on acompact manifold with boundary.
A.1.
Convexity.
Consider a manifold M whose interior contains M . We extendthe metric g and the external field E to M smoothly. We use the same notationsfor extensions. We say that ( M, g, E ) is convex at x ∈ ∂M if x has a neighborhood U ⊂ M such that all thermostat geodesics in U , passing through x and tangent to ∂M at x , lie in M \ M int . Furthermore, ( M, g, E ) is said to be strictly convex at x if these thermostat geodesics do not intersect M except at x . Lemma A.1.
The following holds if ( M, g, E ) is convex at x ∈ ∂M Λ( x, v ) ≥ h E ( x ) , ν ( x ) i g ( x ) for all v ∈ S x ( ∂M ) . (16) Moreover, if this inequality is strict then ( M, g, E ) is strictly convex at x .Proof. For sufficiently small U , take ρ ∈ C ∞ ( U ; R ) such that |∇ ρ | g ≡ ∂M ∩ U = ρ − (0). Then the convexity of ( M, g, E ) at x implies that ρ ( γ x,v ( t )) ≤ v ∈ S x ( ∂M ) and small enough t ∈ R . Since ρ ( γ x,v ( t )) (which is considered as afunction of t ) attains its maximum value at t = 0, dρ ( γ x,v ( t )) dt (cid:12)(cid:12)(cid:12) t =0 = 0 and d ρ ( γ x,v ( t )) dt (cid:12)(cid:12)(cid:12) t =0 ≤ . As in the proof of Lemma 4.1, it is straightforward to check that dρ ( γ x,v ( t )) dt (cid:12)(cid:12)(cid:12) t =0 = h ν ( x ) , v i g ( x ) ,d ρ ( γ x,v ( t )) dt (cid:12)(cid:12)(cid:12) t =0 = Hess x ρ ( v, v ) + h ν ( x ) , E ( x ) i g ( x ) . Since x ∈ ∂M and v ∈ S x ( ∂M ), we always have h ν ( x ) , v i g ( x ) = 0 and Hess x ρ ( v, v ) = − Λ( x, v ). Hence, we get (16).Now, suppose that the inequality (16) is strict. Then there is δ > v ∈ S x M , d ρ ( γ x,v ( t )) dt (cid:12)(cid:12)(cid:12) t =0 ≤ − δ. Therefore, there is a sufficiently small ε > ρ ( γ x,v ( t )) ≤ − δt / t ∈ ( − ε, ε ). This implies the strict convexity at x . (cid:3) A.2.
Scattering relation and folds.
In this section we prove the following result.
Theorem A.2.
Let ( M, g, E ) be a non-trapping and strictly convex Gaussian ther-mostat on a compact manifold with boundary. We have the following characteriza-tion: C ∞ α ( ∂ + SM ; C n ) := { w ∈ C ∞ ( ∂ + SM ; C n ) : A w ∈ C ∞ ( ∂ ( SM ); C n ) } . For the proof, we use the notion of a Whitney fold.
Definition A.3.
Let M and N be smooth manifolds having the same dimensionand let f : M → N be a smooth map. We say that f is a Whitney fold , with fold L ⊂ M , at m ∈ L if { x ∈ M : d x f is singular } is a smooth hypersurface near m and ker( d m f ) is transverse to T m L . HE ATTENUATED THERMOSTAT RAY TRANSFORMS 21
Consider a closed manifold N containing M . Extend the metric g and the vectorfield E smoothly to N , preserving the former notations for extensions. Next, embed M into the interior of a compact manifold M ⊂ N with boundary. Choose M to be sufficiently close to M so that ( M , g, E ) is also non-trapping and strictlyconvex.For ( x, v ) ∈ SM , we denote by ℓ ( x, v ) the first non-negative time such that thethermostat geodesic γ x,v exits M . Lemma A.4.
The map
Ψ : ∂ ( SM ) → ∂ − SM defined as Ψ( x, v ) := (cid:0) γ x,v ( ℓ ( x, v )) , ˙ γ x,v ( ℓ ( x, v )) (cid:1) , ( x, v ) ∈ ∂ ( SM ) , is a Whitney fold with fold S ( ∂M ) .Proof. Take a smooth ρ : M → [0 , ∞ ) such that ρ − (0) = ∂M and |∇ ρ | g ≡ ∂M . Note that ∇ ρ = ν . Then, as in the proof of Lemma 4.1, it is straightforwardto check that G E ( ρ )( x, v ) = h ν ( x ) , v i g ( x ) , G E ( ρ )( x, v ) = Hess x ρ ( v, v ) + h ν ( x ) , E ( x ) i g ( x ) for ( x, v ) ∈ ∂ ( SM ). If we take ( x.v ) ∈ S ( ∂M ), then G E ( ρ )( x, v ) = 0 and G E ( ρ )( x, v ) = 0 by the strict convexity assumption. We are in the same situationas in the proof of [8, Lemma 7.5]. Repeating the arguments therein, we completethe proof. (cid:3) Now, we are ready to prove Theorem A.2.
Proof of Theorem A.2. If w ψ ∈ C ∞ α ( ∂ + SM ; C n ), then we have A w = w ψ | ∂ ( SM ) ∈ C ∞ ( ∂ ( SM ); C n ). To prove the opposite implication, suppose A w ∈ C ∞ ( ∂ ( SM ); C n ).Then Lemma A.4 and [22, Theorem C.4.4] imply the existence of a smooth function u on a neighborhood of Ψ( ∂ ( SM )) ⊂ ∂ − SM such that w = u ◦ Ψ.Now, consider the maps Φ : SM → ∂ − SM , Φ M : SM → ∂ − SM given byΦ( x, v ) : = (cid:0) γ x,v ( τ ( x, v )) , ˙ γ x,v ( τ ( x, v )) (cid:1) , ( x, v ) ∈ SM, Φ M ( x, v ) : = (cid:0) γ x,v ( ℓ ( x, v )) , ˙ γ x,v ( ℓ ( x, v )) (cid:1) , ( x, v ) ∈ SM , respectively. Since w ψ = w ◦ S − ◦ Φ, we have w ψ = u ◦ Ψ ◦ S − ◦ Φ. Observe alsothat Φ M | SM = Ψ ◦ S − ◦ Φ. Therefore, we can write w ψ = u ◦ Φ M | SM . Then thesmoothness of Φ M on SM implies that w ψ is smooth on SM . (cid:3) References [1] G. Ainsworth. The attenuated magnetic ray transform on surfaces.
Inverse Problems & Imag-ing , (1):27–46, 2013.[2] Y. M. Assylbekov and N. S. Dairbekov. Hopf type rigidity for thermostats. Ergodic Theoryand Dynamical Systems , (6):1761–1769, 2014.[3] Y. M. Assylbekov and N. S. Dairbekov. The X-ray transform on a general family of curveson Finsler surfaces. The Journal of Geometric Analysis , 28(2):1428–1455, April 2018.[4] Y. M. Assylbekov and H. Zhou. Invariant distributions and tensor tomography for Gaussianthermostats.
Communications in Analysis and Geometry , (5):895–926, 2017.[5] D. Calderbank and H. Pedersen. Einstein-Weyl geometry. Surveys in Differential Geometry ,6(1):387–423, 2001. [6] N. S. Dairbekov and G. P. Paternain. Entropy production in Gaussian thermostats.
Commu-nications in Mathematical Physics , (2):533–543, 2007.[7] N. S. Dairbekov and G. P. Paternain. Entropy production in thermostats II. Journal of Sta-tistical Physics , 127(5):887–914, 2007.[8] N. S. Dairbekov, G. P. Paternain, P. Stefanov, and G. Uhlmann. The boundary rigidityproblem in the presence of a magnetic field.
Advances in Mathematics , (2):535–609, 2007.[9] M. Dunajski. Solitons, instantons, and twistors , volume 19. Oxford University Press, 2010.[10] G. Eskin. On non-abelian radon transform.
Russian Journal of Mathematical Physics ,11(4):391–408, 2004.[11] D. Finch and G. Uhlmann. The x-ray transform for a non-abelian connection in two dimen-sions.
Inverse Problems , 17(4):695, 2001.[12] G. B. Folland et al. Weyl manifolds.
Journal of differential geometry , 4(2):145–153, 1970.[13] G. Gallavotti. New methods in nonequilibrium gases and fluids.
Open Systems & InformationDynamics , (2):101–136, 1999.[14] G. Gallavotti and D. Ruelle. SRB states and nonequilibrium statistical mechanics close toequilibrium. Communications in Mathematical Physics , (2):279–285, 1997.[15] C. Guillarmou, G. P. Paternain, M. Salo, and G. Uhlmann. The X-ray transform for connec-tions in negative curvature. Communications in Mathematical Physics , 343(1):83–127, 2016.[16] V. Guillemin and D. Kazhdan. Some inverse spectral results for negatively curved 2-manifolds.
Topology , 19(3):301–312, 1980.[17] T. Higa et al. Weyl manifolds and Einstein-Weyl manifolds.
Rikkyo Daigaku sugaku zasshi ,42(2):143–160, 1993.[18] N. Hitchin. Complex manifolds and Einstein’s equations. In
Twistor geometry and non-linearsystems , pages 73–99. Springer, 1982.[19] N. J. Hitchin. Monopoles and geodesics.
Communications in Mathematical Physics ,83(4):579–602, 1982.[20] N. J. Hitchin, R. Ward, and G. Segal. Integrable systems: Twistors, loop groups, and Riemannsurfaces.
Proceedings, Conference, Oxford, UK, September 1997 , 1999.[21] W. G. Hoover. Molecular dynamics. In
Molecular Dynamics , volume 258, 1986.[22] L. H¨ormander. The analysis of linear partial differential operators. III. volume 274 ofGrundlehren der Mathematischen Wissenschaften [Fundamental Principles of MathematicalSciences], 1985.[23] D. Jane and G. P. Paternain. On the injectivity of the x-ray transform for anosov thermostats.
Discrete & Continuous Dynamical Systems-A , 24(2):471.[24] P. Jones and K. Tod. Minitwistor spaces and Einstein-Weyl spaces.
Classical and QuantumGravity , 2(4):565, 1985.[25] A. Livˇsic. Certain properties of the homology of Y-systems.
Mat. Zametki , 10:555–564, 1971.[26] A. Livˇsic. Cohomology of dynamical systems.
Mathematics of the USSR-Izvestiya , 6(6):1278,1972.[27] N. Manton and P. Sutcliffe.
Topological solitons . Cambridge University Press, 2004.[28] L. J. Mason and N. M. J. Woodhouse.
Integrability, self-duality, and twistor theory . Num-ber 15. Oxford University Press, 1996.[29] T. Mettler and G. P. Paternain. Convex projective surfaces with compatible weyl connectionare hyperbolic. arXiv preprint arXiv:1804.04616 , 2018.[30] T. Mettler and G. P. Paternain. Holomorphic differentials, thermostats and anosov flows.
Mathematische Annalen , 373(1):553–580, 2019.[31] V. Nit¸ic˘a and A. T¨or¨ok. Regularity of the transfer map for cohomologous cocycles.
ErgodicTheory and Dynamical Systems , 18(5):1187–1209, 1998.[32] R. G. Novikov. On determination of a gauge field on R d from its non-abelian radon transformalong oriented straight lines. Journal of the Institute of Mathematics of Jussieu , 1(4):559,2002.[33] G. P. Paternain. Regularity of weak foliations for thermostats.
Nonlinearity , 20(1):87, 2006.[34] G. P. Paternain. Transparent connections over negatively curved surfaces. arXiv preprintarXiv:0809.4360 , 2008.
HE ATTENUATED THERMOSTAT RAY TRANSFORMS 23 [35] G. P. Paternain. B¨acklund transformations for transparent connections.
Journal f¨ur die reineund angewandte Mathematik , 2011(658):27–37, 2011.[36] G. P. Paternain. Transparent pairs.
Journal of Geometric Analysis , 22(4):1211–1235, 2012.[37] G. P. Paternain. Inverse problems for connections.
Inverse problems and applications: insideout. II , 60:369–409, 2013.[38] G. P. Paternain and M. Salo. Carleman estimates for geodesic X-ray transforms. preprint,arXiv:1805.02163 , 2018.[39] G. P. Paternain, M. Salo, and G. Uhlmann. The attenuated ray transform for connectionsand Higgs fields.
Geometric and Functional Analysis , (5):1460–1489, 2012.[40] G. P. Paternain, M. Salo, and G. Uhlmann. Tensor tomography on surfaces. InventionesMathematicae , (1):229–247, 2013.[41] G. P. Paternain, M. Salo, G. Uhlmann, and H. Zhou. The geodesic X-ray transform withmatrix weights-ray transform with matrix weights. preprint arXiv:1605.07894 , 2016.[42] P. Przytycki and M. P. Wojtkowski. Gaussian thermostats as geodesic flows of nonsymmetriclinear connections. Communications in Mathematical Physics , 277(3):759–769, 2008.[43] D. Ruelle. Smooth dynamics and new theoretical ideas in nonequilibrium statistical mechanics.
Journal of Statistical Physics , (1):393–468, 1999.[44] E. Scholz. The Unexpected Resurgence of Weyl Geometry in late 20th-Century Physics. In Beyond Einstein , pages 261–360. Springer, 2018.[45] V. A. Sharafutdinov. On the inverse problem of determining a connection on a vector bundle.
Journal of Inverse and Ill-posed Problems , 8(1):51–88, 2000.[46] M. Wojtkowski. Magnetic flows and Gaussian thermostats on manifolds of negative curvature.
Fundamenta Mathematicae , 163(2):177–191, 2000.[47] M. P. Wojtkowski. W-flows on Weyl manifolds and Gaussian thermostats.
Journal deMath´ematiques Pures et Appliqu´ees , 79(10):953–974, 2000.[48] H. Zhou. Generic injectivity and stability of inverse problems for connections.
Communica-tions in Partial Differential Equations , (5):780–801, 2017. Department of Computational and Applied Mathematics, Rice University, Houston,TX 77005, USA
Email address : [email protected] Department of Mathematics, University of North Carolina, Chapel Hill, NC 27514,USA
Email address ::