Boundary and scattering rigidity problems in the presence of a magnetic field and a potential

In this paper, we consider a compact Riemannian manifold with boundary, endowed with a magnetic potential $\alpha$ and a potential $U$. For brevity, this type of systems are called $\MP$-systems. On simple $\MP$-systems, we consider both the boundary rigidity problem and scattering rigidity problem, see the introduction for details. We show that these two problems are equivalent on simple $\MP$-systems. Unlike the cases of geodesic or magnetic systems, knowing boundary action functions or scattering relations for only one energy level is insufficient to uniquely determine a simple $\MP$-system, even under the assumption that we know the restriction of the system on the boundary $\p M$, and we provide some counterexamples. These problems can only be solved up to an isometry and a gauge transformations of $\alpha$ and $U$. We prove rigidity results for metrics in a given conformal class, for simple real analytic $\MP$-systems and for simple two-dimensional $\MP$-systems.


Introduction.
1.1. Posing the problems. Given a Riemannian manifold (M, g) of dimension n ≥ 2 with boundary, endowed with a magnetic field Ω, that is a closed 2-form, we consider the law of motion described by the Newton's equation where U is a smooth function on M , ∇ is the Levi-Civita connection of g and Y : T M → T M is the Lorentz force associated with Ω, i.e., the bundle map uniquely determined by for all x ∈ M and ξ, η ∈ T x M . A curve γ : [a, b] → M , satisfying (1) is called an MP-geodesic. The equation (1) defines a flow φ t on T M that we call an MP-flow. These are not standard terms in general. Note that time is not reversible on the MP-geodesics, unless Ω = 0.
When Ω = 0 the flow is called potential flow; while if U = 0 we obtain the magnetic flow. Therefore, the equation (1) describes the motion of a particle on a Riemannian manifold under the influence of a magnetic field Ω in a potential field U . Magnetic flows were first considered in [1,2] and it was shown in [3,12,16,17,18,20] that they are related to dynamical systems, symplectic geometry, classical mechanics and mathematical mechanics.
When Ω is exact, i.e. Ω = dα for some magnetic potential α, the MP-flow also arises as the Hamiltonian flow of H(x, p) = 1 2 (p + α) 2 g(x) + U (x) with respect to the canonical symplectic form of T * M .
For MP-flow the energy E(x, ξ) = 1 2 |ξ| 2 + U (x) is an integral of motion. By the Law of Conservation of Energy, for every MP-geodesic the energy is constant along it. Unlike the geodesic flow, where the flow is the same (up to time scale) on any energy levels, MP-flow depends essentially on the choice of the energy level. Throughout the paper we assume the energy level k > sup x∈M U (x) with S k M = E −1 (k), the bundle of energy k.
We define the action A(x, y) between boundary points as a minimum of the appropriate action functional, see (4) and Appendix A.1. In the case Ω = 0 and U = 0, the function A(x, y) coincides with the boundary distance function d g (x, y). In this case, we cannot recover g from d g up to isometry, unless some additional assumptions are imposed on g, see, e.g., [4]. One such assumption is the simplicity of the metric, see, e.g., [13,22,23,24]. We consider below the analog of simplicity for MP-systems.
Let Λ denotes the second fundamental form of ∂M , and ν(x) the inward unit vector normal to ∂M at x. We say that ∂M is strictly MP-convex if for all (x, ξ) ∈ S k (∂M ). For x ∈ M , we define the MP-exponential map at x to be the partial map exp MP x : T x M → M given by exp MP x (tξ) = π • φ t (ξ), t ≥ 0, ξ ∈ S k x M, where π : T M → M is the canonical projection. It is not hard to show that, for every x ∈ M , exp MP x is a C 1 -smooth partial map on T x M which is C ∞ -smooth on T x M \ {0}.
We say that M is simple w.r.t. (g, Ω, U ) if ∂M is strictly MP-convex and the MP-exponential map exp MP x : (exp MP x ) −1 (M ) → M is a diffeomorphism for every x ∈ M . In this case, M is diffeomorphic to the unit ball of R n . Therefore Ω is exact, and we let α be a magnetic potential, i.e. α is a 1-form on M such that dα = Ω.
Henceforth we call (g, α, U ) a simple MP-system on M . We will also say that (M, g, α, U ) is a simple MP-system. It is easy to see that the simplicity is stable under a small perturbation of the energy level.
First, we state the boundary rigidity problem. Given x, y ∈ M , let The time free action of a curve γ ∈ C(x, y) w.r.t. (g, α, U ) is defined as For a simple MP-system, MP-geodesics with energy k minimize the time free action (see Appendix A.1) where γ x,y : [0, T x,y ] → M is the unique MP-geodesic with constant energy k from x to y. The function A(x, y) is referred to as Mañé's action potential (of energy k), and we call the restriction A| ∂M ×∂M the boundary action function.
We say that two MP-systems (g, α, U ) and (g , α , U ) are gauge equivalent if there is a diffeomorphism f : M → M , which is the identity on the boundary, and a smooth function ϕ : M → R, vanishing on the boundary, such that g = f * g, α = f * α + dϕ and U = U • f . Observe that given two gauge equivalent MP-systems, if one is simple, then the other one is also simple. Moreover, if two simple MP-systems are gauge equivalent, then they have the same boundary action function.
The boundary rigidity problem in the presence of a magnetic field and a potential studies that to which extend an MP-system (g, α, U ) on M is determined by the boundary action functions. By the above observation, one can only expect to obtain the uniqueness up to gauge equivalence. For the zero potential, i.e. U = 0, we obtain the boundary rigidity problem for the magnetic systems that was considered by N. Dairbekov, G. Paternain, P. Stefanov and G. Uhlmann in [6]. In the absence of both magnetic fields and potentials, i.e. Ω = 0 and U = 0, we come to the ordinary boundary rigidity problem for the Riemannian metrics. For the recent surveys on the ordinary boundary rigidity problem see [5,25]. It also worths to mention that recently P. Stefanov, G. Uhlmann and A. Vasy [26] proved the boundary rigidity with partial data in dimension n ≥ 3 for metrics in a given conformal class, this is so far the only local boundary rigidity result.
Next, we define a scattering relation and state the scattering rigidity problem in the presence of a magnetic field and a potential. Let ∂ + S k M and ∂ − S k M denote the bundles of inward and outward vectors of energy k over ∂M where ν is the inward unit vector normal to ∂M . For (x, ξ) ∈ ∂ + S k M let τ (x, ξ) be the time when the MP-geodesic γ x,ξ , such that γ x,ξ (0) = x,γ x,ξ (0) = ξ, exits. By Lemma A.3 the function τ : The scattering relation S : Observe that two gauge equivalent MP-systems have the same scattering relation. Is this the only type of nonuniqueness? In other words, the scattering rigidity problem studies whether a simple MP-system (M, g, α, U ), up to gauge equivalence, is uniquely determined by the scattering relations. In the Euclidean space this problem was considered by R. G. Novikov [19], M. L. Gerver and N. S. Nadirashvili [8], in the absence of magnetic field, and by A. Jollivet [11]. On Riemannian manifolds endowed with magnetic fields, scattering rigidity problem was studied by N. Dairbekov, G. Paternain, P. Stefanov and G. Uhlmann in [6], by P. Herreros in [9], and by P. Herreros and J. Vargo in [10]. The reconstruction of both the Riemannian metrics and magnetic fields from the scattering relations was considered by N. Dairbekov and G. Uhlmann [7] for simple two-dimensional magnetic systems.
For simple MP-systems, the boundary rigidity and the scattering rigidity problems are equivalent, see Lemma 4.2 and Theorem 4.3. Therefore, we formulate all rigidity results in terms of the boundary rigidity problem. However, unlike the boundary rigidity problems for simple manifolds or simple magnetic systems (with energy 1/2), the boundary rigidity problem for simple MP-systems needs the information of the boundary action functions for two different energy levels. See the counterexamples in Section 3 and the proofs of the main results in Section 5 for details.
We consider these problems under various natural restrictions: simple MPsystems with metrics in a given conformal class, simple real-analytic MP-systems and simple two-dimensional MP-systems. The key ingredient of the proof is that by reparameterizing the curves and a conformal change of the metric, one can reduce an MP-system (g, α, U ) to a corresponding magnetic system (G, α). So the rigidity results by N. Dairbekov, G. Paternain, P. Stefanov and G. Uhlmann [6] on simple magnetic systems can be applied to the study of simple MP-systems.

1.2.
Structure of the paper. The rest of the paper is organized as follows. In Section 2, we show that by changing the metric, one can reduce a simple MP-system to a simple magnetic system with the same boundary action function. Section 3 provides counterexamples which show that knowing the boundary action function for only one energy level is insufficient for solving the boundary rigidity problem, even under the assumption that the restriction of the system on the boundary ∂M is known. In Section 4, we demonstrate the equivalence between the boundary rigidity problem and the scattering rigidity problem for a simple MP-system. Section 5 is devoted to the proofs of the boundary rigidity for various systems, namely, simple MP-systems with metrics in a given conformal class, simple real-analytic MPsystems and simple two-dimensional MP-systems. We give a final remark on the case that we only know the boundary action function for one energy level in Section 6.
2. Relation between MP-systems and magnetic systems.

2.1.
Reduction to the magnetic system. For a fixed energy level k > sup U , let σ(t) be an MP-geodesic with the constant energy k. Consider the time change Then s is the arclength of γ(s) = σ(t(s)) under the metric G = 2(k − U )g. The following version of Maupertuis' principle says that γ(s) = σ(t(s)) is a unit speed magnetic geodesic of the magnetic system (G, α). Proposition 1. Let (g, α, U ) be an MP-system on M and let k be a constant such that k > sup M U . Suppose σ(t) is an MP-geodesic of energy k. Then γ(s) = σ(t(s)) is a unit speed magnetic geodesic of the magnetic system (G, α).
Proof. It is immediate to check that γ has unit speed with respect to G. Let ρ denote the arclength of the metric g. Since we fix the energy to be k, the parameter t of σ must be proportional to the length, i.e. dt = dρ/ 2(k − U ). We denote byγ the derivative of γ with respect to s and byσ the derivative of σ with respect to t.
We define the Lagrangian L(x, ξ) = |ξ| G(x) − α x (ξ), by the Maupertuis' principle, the MP-geodesic is an extremal of the action Hence L satisfies the Euler-Lagrange equation with respect to s which has the form d ds Since s is the arclength of G, for which |γ| G = 1, this equation takes the form Taking the derivative with respect to s and multiplying by G mk we havë which is the equation of magnetic geodesics of the magnetic system (G, α).
We give an alternative proof of Proposition 1 based on the flow equation itself.
Proof. Given an MP-geodesic σ(t) with energy k and a positive smooth function φ, let G = φg, ds = 2φ(k − U )dt, so s will be the arclength of γ(s) = σ(t(s)) under the metric G. If we denote the Christoffel symbols and the covariant derivative under the new metric G by Γ i jk and D respectively, then will work for our argument), we get This indeed gives us the magnetic flow with the Lorentz force Y G = 1 2(k−U ) Y . Moreover, one can see that the magnetic potential α associated to (G, Y G ) is α too, i.e. the new magnetic system is (G, α).

2.2.
Simplicities of two systems. The next result says that the simplicity of (g, α, U ) implies the simplicity of (G, α), and vice versa. A simple magnetic system is a special case of simple MP-systems by assuming the potential U = 0 and the energy k = 1 2 , see [6] for more details. Proposition 2. The MP-system (g, α, U ) on M (of energy k) is simple if and only if so is the magnetic system (G, α) (of energy 1 2 ). Proof. Since the trajectories of these two systems coincide, for every x ∈ M the MP-exponential map exp MP x by replacing the MP-flow with a magnetic flow of energy 1 2 ). Hence, it is sufficient to prove that ∂M is strictly MP-convex if and only if it is strictly magnetic convex with respect to (G, α).
First, we introduce some notations. The inward unit vector normal to ∂M with respect to the metric G is indicated as n, thus n = (2(k − U )) − 1 2 ν (G is conformal to g). The unit sphere bundle of the metric G is denoted by SM . By Λ G we denote the second fundamental form of ∂M with respect to metric G. From the definition of the second fundamental form and using the formula for connection of G in terms of connection of g, we obtain the following formula: The Lorentz force of the magnetic field dα with respect to the metric G is indicated as Y G . The next formula is obvious, which implies that ∂M is strictly magnetic convex with respect to (G, α) by (3). By similar arguments one can show that ∂M is strictly MP-convex whenever it is strictly magnetic convex with respect to (G, α).

Boundary action functions of the two systems.
Here we show that the boundary action functions of the two simple systems (g, α, U ) and (G, α) coincide. Assuming the potential U = 0 and the energy k = 1 2 , the corresponding boundary action function of (4) is the one for a simple magnetic system. Proposition 3. Let A be the Mañé's action potential (of energy k) for a simple MP-system (g, α, U ) and A G be the Mañé's action potential (of energy 1/2) for the simple magnetic system (G, α), then A| ∂M ×∂M = A G | ∂M ×∂M .
Proof. Take x, y ∈ ∂M and consider the unique MP-geodesic σ from x to y. Then Proposition 1 implies that γ(s) = σ(t(s)) is the unit speed magnetic geodesic (from x to y) of the system (G, α) and s is the arclength of γ under the metric G. Thus, We are done.
3. Counterexamples. Before moving to the detailed study of the boundary and scattering rigidity problems of simple MP-systems, we provide some counterexamples which show that knowing the boundary action function for only one energy level is insufficient for solving the boundary rigidity problem, even under the assumption that we know the restriction of the system on the boundary ∂M .
Counterexamples: Given some simple magnetic system (g, α) on a compact manifold M with boundary, we define two MP-systems ( 1 4 g, α, U 1 ) and ( 1 2 g, α, U 2 ), where U 1 ≡ 1 on M and U 2 ≡ 2 on M . We fix the energy k = 3, then it is easy to see that these two MP-systems reduce to the same magnetic system (g, α) with energy 1 2 . Since (g, α) is simple, Proposition 2 implies that both ( 1 4 g, α, U 1 ) and ( 1 2 g, α, U 2 ) are simple MP-systems. Moreover, appling Proposition 3, we conclude that they have the same boundary action function for the energy k = 3. Obviously these two MP-systems are not gauge equivalent, since the metrics 1 4 g and 1 2 g, potentials U 1 and U 2 are even not equal on the boundary ∂M .
Next, by modifying the two MP-systems near the boundary, we can make them have the same boundary restriction. Let ϕ and ψ be two smooth functions on M , and ϕ ≡ ψ ≡ 1 for points away from a small tubular neighborhood of the boundary ∂M . We assume 1 ≤ ϕ < 3 2 in the interior of M and ϕ = 3 2 on ∂M ; 3 4 < ψ ≤ 1 in the interior of M and ψ = 3 4 on ∂M . Then ϕ = ϕU 1 < 3 2 < ψU 2 = 2ψ in the interior of M . We defineg = 1 2(3−ϕ) g andg = 1 2(3−2ψ) g. Then it is easy to check that the MP-systems (g, α, ϕ) and (g , α, 2ψ) reduce to the same magnetic system (g, α) for the energy k = 3. Applying Proposition 2 and 3 again, these two MP-systems (g, α, ϕ) and (g , α, 2ψ) are simple with the same boundary action function for energy k = 3. Moreover,g| ∂M =g | ∂M = 1 3 g, ϕ| ∂M = 2ψ| ∂M = 3 2 , i.e. the boundary restrictions of these two MP-systems are the same. However, they are still not gauge equivalent, there is no diffeomorphism f : M → M such that 2ψ = ϕ • f (since ϕ < 3 2 < 2ψ in the interior of M ). Based on above counterexamples, we reach the conclusion that there exist simple MP-systems (g, α, U ) and (g , α , U ) with the same boundary action function for some fixed energy level k, whose restrictions onto the boundary are the same (i.e. g| ∂M = g | ∂M , α| ∂M = α | ∂M , U | ∂M = U | ∂M ), but are not gauge equivalent. Roughly speaking, the Hamiltonians of these two MP-systems are conformal at the given energy level k. However, if two MP-systems (g, α, U ) and (g , α , U ) are gauge equivalent, their Hamiltonians are equivalent at all energy levels k > sup M U = sup M U . This makes one turn to considering boundary action functions of two different energy levels. 4. Boundary determination and scattering relation.

Boundary determination.
Here we show that up to gauge equivalence the boundary action functions of two different energy levels completely determine the Riemannian metric, magnetic potential and potential on the boundary of the manifold under study. As mentioned in the Section 3, the boundary action function of one energy level is insufficient for determining the restriction of the system on the boundary.
where ı : ∂M → M is the embedding map.
Proof. Let G i = 2(k i − U )g and G i = 2(k i − U )g , i = 1, 2 by Proposition 1 and Proposition 2, (G i , α) and (G i , α ) are simple magnetic systems of energy 1 2 . Let A Gi and A G i denote the Mañé's action potentials (of energy 1/2) for (G i , α) and (G i , α ) respectively. Then by Proposition 3 we have A Gi | ∂M ×∂M = A G i | ∂M ×∂M . Then [6, Theorem 2.2] implies that there is (Ḡ i ,ᾱ i ), gauge equivalent to (G i , α ), such that in any local coordinate system ∂ m G i | ∂M = ∂ mḠ i | ∂M and ∂ m α| ∂M = ∂ mᾱ i | ∂M for every multi-index m. Thus there is some diffeomorphism f i with f i | ∂M = Id, and some smooth function Now we prove the equality of derivatives on the boundary by introducing boundary normal coordinates (x , x n ) w.r.t. g near arbitrary x 0 ∈ ∂M . Since g| ∂M = g| ∂M , the same coordinates are boundary normal coordinates w.r.t.ḡ. Thus locally the metrics are of the form where i, j vary from 1 to n − 1. It suffices to prove that the normal derivatives are equal, i.e. ∂ m n g ij | x=x0 = ∂ m nḡij | x=x0 , ∂ m n U | x=x0 = ∂ m nŪ | x=x0 , m = 0, 1, · · · ; i, j = 1, · · · , n − 1. We prove above equalities by induction, the case m = 0 is granted. Assume for some nonnegative integer l and all 0 ≤ m ≤ l, ∂ m n g nḡij at x 0 . Similarly for energy k 2 , sinceḡ = f * 2 g ,Ū = U • f 2 near ∂M , we have at x 0 (9) (−∂ l+1 n U )g ij + (k 2 − U )∂ l+1 n g ij = (−∂ l+1 nŪ )ḡ ij + (k 2 −Ū )∂ l+1 nḡij . Taking difference of above two equalities, we arrive This finishes the proof.

4.2.
Scattering relation. Now we show that for simple MP-systems, the boundary rigidity problem is equivalent to the problem of restoring a Riemannian metric, a magnetic potential and a potential from the scattering relations. Thus we will formulate all rigidity results in terms of the boundary rigidity problem in the next Section. Proof. First, we introduce some notations. Let G = 2(k − U )g, G = 2(k − U )g , by Proposition 1 and Proposition 2, (G, α) and (G , α ) are simple magnetic systems of energy 1 2 . We denote by S G and S G the scattering relations of (G, α) and (G , α ) respectively (The definition of the scattering relation for a simple magnetic system is similar to that for a simple MP-system by considering the magnetic flow of energy 1 2 ). The notation ∂ + SM denotes the bundle of inward unit vectors at ∂M with respect to metric G (and also of G , since G| ∂M = G | ∂M ).
Suppose A| ∂M ×∂M = A | ∂M ×∂M , then by Proposition 3 we have Lemma 2.5] implies that S G = S G . Now we prove that this implies S = S . Since the trajectories of (g, α, U ) and (G, α) coincide, for any (x, ξ) ∈ ∂ + S k M the scattering relation S can be expressed in terms of S G in the following way where s G = π • S G ( Here we define c(x, v) .
= (x, cv) ). Exactly in the same way S can be expressed in terms of S G . Since S G = S G , these expressions imply that S = S .
Conversely, assume that S = S . Since the trajectories of these two systems coincide, for any (x, ξ) ∈ ∂ + SM the scattering relation S G can be expressed in terms of S in the following way where s = π • S. Exactly in the same way S G can be expressed in terms of S . Since S = S , these expressions imply that S G = S G . Then [6, Lemma 2.6] implies that A G | ∂M ×∂M = A G | ∂M ×∂M . Applying Proposition 3 we come to A| ∂M ×∂M = A | ∂M ×∂M . Remark 1. Theorem 4.3 together with the counterexamples of the previous section shows that for generally a simple MP-system, knowing the scattering relation of only one energy level is also insufficient for solving the scattering rigidity problem.

Rigidity in a given conformal class.
Here we give the proof of our first main result which is a rigidity theorem in a fixed conformal class of a metric. The theorem below generalizes the corresponding well-known results for the ordinary boundary rigidity problem, see [4,14,15], and for the magnetic boundary rigidity problem, see [6].
Theorem 5.1. Let (g, α, U ) and (g , α , U ) be simple MP-systems on M with the same boundary action functions for both energy k 1 and k 2 . If g is conformal to g, then g = g, α = α + dϕ and U = U for some smooth function ϕ on M vanishing on ∂M , hence (g , α , U ) is gauge equivalent to (g, α, U ).
Proof. Let G i = 2(k i − U )g, G i = 2(k i − U )g , i = 1, 2, by Proposition 1 and Proposition 2, (G i , α) and (G i , α ), for i = 1, 2, are all simple magnetic systems of energy 1 2 . Let A Gi and A G i denote the Mañé's action potentials (of energy 1/2) for (G i , α) and (G i , α ) respectively. Then by Proposition 3 we have A Gi | ∂M ×∂M = A G i | ∂M ×∂M . By the assumption g = ωg for some strictly positive function ω ∈ C ∞ (M ), therefore Applying [6, Theorem 6.1], we get G i = G i , i.e.
and that there are Thus U = U on M (since k 1 = k 2 ), together with (10) this gives ω ≡ 1.

Remark 2.
In Jollivet's paper on the scattering rigidity problem [11], the metrics g and g are the same, namely the Euclidean metric, which means ω ≡ 1 under the setting of Theorem 5.1.
That's why one fixed energy level is sufficient for Euclidean case. However, for general simple MP-systems we need the information of two energy levels, as can be seen from the counterexamples and the proof above.

5.2.
Rigidity of real-analytic systems. Our next result says that rigidity also holds in a class of real-analytic simple MP-systems. This generalizes the corresponding result for the magnetic boundary rigidity problem in [6].
Theorem 5.2. If M is a real-analytic compact manifold with boundary, (g, α, U ) and (g , α , U ) are simple real-analytic MP-systems on M with the same boundary action functions for both energy k 1 and k 2 , then these systems are gauge equivalent.
Now we consider the systems (g , α , U ) and (f * g, α , U • f ). Let A i and A i , i = 1, 2, denote the Mañé's action potentials (of energy k i ) for simple real-analytic MPsystems (g, α, U ) and (g , α , U ) respectively. By our assumption, A i | ∂M ×∂M = A i | ∂M ×∂M =Ā i | ∂M ×∂M , i = 1, 2, whereĀ i | ∂M ×∂M is the boundary action function of (f * g, α , U •f ) for energy k i (the second equality comes from the fact that (g, α, U ) and (f * g, α , U • f ) are gauge equivalent) . Then, Theorem 5.1 implies that U = U • f and g = f * g.

5.
3. Rigidity of two-dimensional systems. We show that two-dimensional simple MP-systems are always rigid. Our result generalizes the boundary rigidity theorem for simple Riemannian surfaces [21] and for simple two-dimensional magnetic systems [6].
g is conformal to f * g) and α = f * α + dϕ. Let A i and A i denote the Mañé's action potentials (of energy k i ) for simple MP-systems (g, α, U ) and (g , α , U ) respectively. By our assumption, A i | ∂M ×∂M = A i | ∂M ×∂M =Ā i | ∂M ×∂M , i = 1, 2, whereĀ i | ∂M ×∂M is the boundary action funciton of (f * g, α , U • f ) for energy k i (the second equality comes from the fact that (g, α, U ) and (f * g, α , U •f ) are gauge equivalent) . Then, Theorem 5.1 impies that U = U • f and g = f * g.

6.
Final remark. Our main results and the counterexamples have shown that it's necessary to consider two different energy levels for the boundary and scattering rigidity problems of simple MP-systems. However, assuming the boundary action functions A = A for some fixed energy k, we still can obtain some weak version of boundary rigidity.
After reviewing the proof of the main results, if under some additional assumption (fixed conformal classes, real analyticity or dimension 2) two simple MP-systems (g, α, U ) and (g , α , U ) have the same boundary action function for some energy k, then there exists a diffeomorphism f : M → M with f | ∂M = Id, and a smooth function ϕ : M → R with ϕ| ∂M = 0, such that g = (k − U ) −1 (k − U • f )f * g and α = f * α + dϕ. Thus at least we can show that the magnetic potentials of these two MP-systems are gauge equivalent, and the metrics of the two MP-systems are gauge equivalent up to some conformal factor (k − U ) −1 (k − U • f ), which is determined by the potentials of the two systems. In particular, f = Id when g is conformal to g , and for the real-analytic MP-systems, f and ϕ are both realanalytic. In some sense, this can be regarded as a weak boundary rigidity result, but the two systems may have different boundary action functions for energy levels other than k. However, if two simple MP-systems are gauge equivalent, they must have the same boundary action functions for all k > sup M U = sup M U .
Similar situation occurs in the scattering rigidity problem of simple MP-systems.
• Superlinear growth: uniformly on x ∈ M . The action of L on an absolutely continuous curve γ : For each λ ∈ R, the Mañé action potential A λ : Recall that the energy function E : T M → R for L is defined by and that the energy function is constant on every solution x(t) of the Euler-Lagrange equation Let ψ t : T M → T M be the Euler-Lagrange flow defined by where γ is the solution of (11) with γ(0) = x andγ(0) = v. For x ∈ M and k ∈ R, the exponential map at x of energy λ is defined to be the partial map exp λ x : T x M → M given by exp λ The next two propositions were proved in [6, Appendix A.1].
Proposition 5. If λ > c(L) and x, y ∈ M x = y, then there is γ ∈ C(x, y) such that A λ (x, y) = A L+λ (γ). Moreover, the energy of γ is E(γ,γ) ≡ λ. Now, we apply the above to the case of MP-systems. For a simple MP-system (M, g, α, U ), the MP-flow can also be obtained as the Euler-Lagrange flow with the corresponding Lagrangian defined by Then it is easy to see that the energy of L is E(x, v) = 1 2 |v| 2 g + U (x).
Lemma A.1. Let (g, α, U ) be a simple MP-system on M . For x, y ∈ M , x = y, where γ x,y : [0, T x,y ] → M is the MP-geodesic with constant energy k from x to y.
Proof. It is easy to see that the simplicity assumption implies that for this Lagrangian the assumptions of Proposition 4 hold for all λ sufficiently close to k. Therefore, the proposition gives k > c(L). Then Proposition 5 shows that, given x = y in M , there is γ ∈ C(x, y) with energy k such that A(x, y) = A(γ). Using simplicity, one can then prove that γ is a MP-geodesic with constant energy k, i.e., γ = γ x,y .
A.2. MP-convexity. Let M be a compact manifold with boundary, endowed with a Riemannian metric g, a closed 2-form Ω and a smooth function U . Consider a manifold M 1 such that M int 1 ⊃ M . Extend g, Ω and U to M 1 smoothly, preserving the former notation for extensions. We say that M is MP-convex at x ∈ ∂M if there is a neighborhood O of x in M 1 such that all MP-geodesics of constant energy k in O, passing through x and tangent to ∂M at x, lie in M 1 \ M int . If, in addition, these geodesics do not intersect M except for x, we say that M is strictly MP-convex at x. It is not hard to show that these definitions depend neither on the choice of M 1 nor on the way we extend g, Ω and U to M 1 . On the other hand, the MP-convexity depends on the energy level.
This implies the second statement.
Lemma A.3. For a simple MP-system, the function τ : ∂ + S k M → R is smooth.