A foliated and reversible Finsler manifold is determined by its broken scattering relation
Maarten V. de Hoop Joonas Ilmavirta Matti Lassas Teemu Saksala
math.DG 86A22, 53Z05, 53C60





The broken scattering relation consists of the total lengths of broken geodesics that start from the boundary, change direction once inside the manifold, and propagate to the boundary. We show that if two reversible Finsler manifolds satisfying a convex foliation condition have the same broken scattering relation, then they are isometric. This implies that some anisotropic material parameters of the Earth can be in principle reconstructed from single scattering measurements at the surface.


[NO \title GIVEN]
    [NO \author GIVEN]
    December 30, 2023
======================








§ INTRODUCTION









The broken scattering relation of a Finsler manifold with boundary describes all the scenarios where a geodesic starts inward at the boundary, changes direction at some point in the interior, and makes its way back to the boundary.  

The objective in this paper is to reconstruct the Finsler manifold from such data, provided that some assumptions are met.

We have two key assumptions. One is that the Finsler geometry is reversible, meaning that the Minkowski norm on each tangent space is symmetric (-v=v) or, equivalently, that the reverse of a geodesic is a geodesic.

We also assume that the manifold has a strictly convex foliation with a family of smooth hyper surfaces. 



In this paper we will also describe an important relationship between Finsler geometry and elastic waves. 





 §.§ Statement of the result

Let M be a smooth, compact manifold of dimension 3 or higher, with smooth boundary M. We use the notation TM for the tangent bundle of M. Recall that a Finsler metric F TM → is a continuous positive function such that on each fiber T_xM of TM the function F is a Minkowski norm meaning:

    
  * F is smooth outside zero section.
    
  * F is positively homogeneous of order one, that is F(x,av)=aF(x,v) for any a>0, x ∈ M and v ∈ T_xM
    
  * F is convex in the sense that the local Riemannian metric
    
    g_ij(x,v):=1/2^2/ v_i v_j[F^2](x,v),    i,j ∈{1,…,  M}

    is positive definite for any (x,v)∈ TM, v≠ 0. 

If F is a Finsler metric we call the pair (M,F) a Finsler manifold. If the Finsler metric is symmetric with respect to directional variable v in the sense of F(x,v)=F(x,-v) we call F reversible Finsler metric. 




A Finsler manifold with boundary is said to have a strictly convex foliation if there is a smooth function f M→ so that


  * f^-1{0}=∂ M,

f^-1(0,S]=(M),

f^-1(S) has empty interior

  * for each s∈[0,S) the set Σ_s f^-1(s) is a strictly convex smooth surface in the sense that f≠0 and any geodesic γ, having initial conditions in TΣ_s, satisfies ∂_t^2f(γ(t))|_t=0<0.



In a similar fashion, we may define a strictly convex surface without a foliation.
For any surface Σ there is a real-valued function f defined in its neighborhood so that f=0 and f≠0 on Σ.
The surface is called strictly convex if any geodesic γ(t) tangent to Σ at t=0 satisfies ∂_t^2f(γ(t))|_t=0<0.


In particularly we say that a reversible Finsler manifold (M,F) has a strictly convex boundary M if 

    f(x)=d(x,  M),    x ∈ U,
  
satisfies the former conditions near the boundary. In this case any pair of points in M can be connected by a distance minimizing geodesic <cit.>.








A manifold with a strictly convex foliation enjoys a certain weak non-trapping condition given in Lemma <ref>.
If there is a trapped geodesic, it has to be asymptotically in the set f^-1(S) where the foliating function f attains its maximum value, and it depends on the geometry of this set whether this is at all possible.


Next we describe what it means for two manifolds to have the same boundary data.
The Finsler function F TM→ describes the geometry, and boundary information requires knowledge of F|_∂ TM.
If we know the geometry of ∂ M, we can deduce F|_T∂ M.
However, T∂ M⊂∂ TM is a subbundle with fiberwise codimension one.
In Riemannian geometry the knowledge of the metric on T∂ M determines uniquely the metric on ∂ TM in boundary normal coordinates.
In Finsler geometry it does not, because the geometry does not factorize as a product of tangential and normal directions of the boundary.


Also a diffeomorphism ∂ M_1→∂ M_2 only induces a diffeomorphism T∂ M_1→ T∂ M_2, and that is not enough for our purpose. Thus we introduces the following compatibility condition.



Let M_1 and M_2 be two smooth manifolds with boundary.
We say that a diffeomorphism ∂ TM_1→∂ TM_2 is compatible with a diffeomorphism ∂ M_1→∂ M_2 if  is a linear isomorphism on every fiber and satisfies (T∂ M_1)=T∂ M_2 and |_T∂ M_1=.


Before showing that the broken scattering relation determines a Finsler manifold, we define what the relation is.
To this end, let ϕ_t SM→ SM 

be the geodesic flow.
The natural projection of the tangent bundle will be denoted by π TM→ M.
We will write elements of the tangent bundle either as sole vectors v (which is in T_xM) or as pairs (x,v), whichever is more convenient.

The boundary of the sphere bundle is

    ∂ SM
    =
    {v∈ SM;π(v)∈∂ M}.

We identify the inward-pointing part of this boundary,

    =
    {v∈∂ SM;vν_ν>0},

where ν is the inward pointing normal vector field and 
vν_ν:=g_ij(ν)ν^iv^j.
Similarly, we define the outward-pointing part

    =
    {v∈∂ SM;vν_ν<0}.




Let (M,F) be a Finsler manifold with boundary.
For each t>0 we define a relation R_t on  so that v R_t w if and only if there exist two numbers t_1,t_2>0 for which t_1+t_2=t and π(ϕ_t_1(v))=π(ϕ_t_2(w)).
We call this relation the broken scattering relation.


Our main result is the following theorem stating that the  broken scattering relation, that is, the lengths of the broken geodesics, determine uniquely the isometry type of a Finsler manifold.



Let (M_i,F_i),   i ∈{1,2} be two compact Finsler manifolds of dimension larger or equal to 3, with boundary.
We assume the following:


  * Both Finsler functions F_1 and F_2 are reversible.

  * The manifolds (M_i,F_i),    i ∈{1,2} have strictly convex foliations in the sense of definition <ref>.

  * There are diffeomorphisms ∂ M_1→∂ M_2 and ∂ TM_1→∂ TM_2 that are compatible in the sense of definition <ref>.


  * F_1=F_2∘ on ∂ TM_1. 


  * For any two vectors v,w∈1 and t>0 we have vR_t^(1)w if and only if (v)R_t^(2)(w), where R^(i)_t is the broken scattering relation of definition <ref> on (M_i,F_i).


Then there is a diffeomorphism M_1→ M_2 that is an isometry in the sense of F_1=F_2∘, which satisfies |_∂ M_1=, and |_∂ TM_1=.




We note that to prove theorem <ref> we will actually only need that (M_1,F_1) has strictly convex foliation and (M_2,F_2) has strictly convex boundary.

The requirement for the dimension n∈ of M_i being larger than 2 is a technical one. This is however a vital component of the proof of lemma <ref>. In the proof we need to assume that the dimension of a trace of geodesic, that is one, is strictly smaller than the dimension of the boundary M_i, which is n-1.   


There are several well known examples of compact Riemannian manifolds satisfying a convex foliation condition of definition <ref>. One particularly interesting example to us is the closed unit ball B̅⊂^n with a radial metric g(x)=c^-2(r)e, where r=|x| ∈ [0,1].  Here c is a strictly positive smooth function on [0,1] satisfying the Herglotz condition

    / r(r/c(r))>0

and c'(0)=0.

The condition does not forbid the existence of conjugate points but there are no trapped geodesics <cit.>.

Consider the smooth function f(x)=|x|^2-1 on B̅.  It is well known that the radial Riemannian metric

g(x)=c^-2(|x|)e satisfies the Herglotz condition if and only if the function f satisfies the condition of definition <ref>.
Due to convexity, in terms of definition <ref>, being an open property we have that the function f is a strictly convex foliation of a Finsler manifold (B̅,F) for any Finsler function F on B̅ that is close enough to g in the C^2-topology.


Rotation symmetric Riemannian manifolds satisfying the Herglotz condition can have conjugate points.
Therefore some small Finslerian variations of such manifolds, at least those that preserve rotation symmetry, also have conjugate points.
Therefore our conditions do allow for conjugate points on non-Riemannian manifolds.







 §.§ Comparision of inverse problems for Riemannian and Finsler manifolds and the Riemannian counterpart of theorem <ref>

 Several inverse problems studied in literature have a different nature in Riemannian and Finslerian settings. 
For example, recall that a Riemannian surface is simple if it is 
 simply connected, its geodesic have no conjugate points and ∂ M is strictly convex. For the travel time inverse problem, discussed in detail below, it is shown in  <cit.> that
 the isometry type of a simple Riemannian manifold 
 is uniquely determined by the distances d(x,y) of the boundary points x,y∈ M, but there are counterexamples for this problem for simple Finsler manifolds <cit.>. Another example of the different nature of Riemannian and Finslerian settings is that the blow-up maps have been used to construct non-smooth counterexamples <cit.> for inverse problems for the Laplace-Beltrami operator, corresponding to a Riemannian metric. However, these maps can not be used for elastic equations, corresponding to a Finsler structure, as elastic media has to satisfy physical symmetries <cit.>.


The Riemannian counterpart of Theorem <ref>, presented in <cit.>, needs less assumptions. This is due to a certain rigidity of Riemannian geometry.
Heuristically, in Finsler geometry the metric in different directions at the same point are independent. This implies that in order to reconstruct a Finsler manifold in full, one needs access to the entire tangent bundle. To prove theorem <ref> the first step is to show that the broken scattering relation determines the collection of distance functions at the boundary (see proposition <ref>). However these distances only give an access for a certain open set of directions. 

The important feature of a Finsler function arising from a Riemannian metric tensor is that it is real analytic on every punctured tangent space.
If one makes this additional assumption, then the distance functions at the boundary determine the geometry at all points in all directions as proven in <cit.>. By lemma <ref> the distance function determines the Finsler function in directions where the geodesic is minimal to its endpoint on the boundary.
We call this the good subset of the tangent bundle.
These good directions exist on every tangent space, which is why analyticity gives access to the entire bundle.

The main result of <cit.> also asserts that the collection of the distance functions does not determine the Finsler function outside the good set of the tangent bundle.

Therefore on a general Finsler manifold one needs to study also long geodesics, and we have to go beyond the scope of <cit.>.
Reversibility and strict convexity give access to all directions near the boundary.
To go further into the manifold, we use the foliation: every point in the interior is close to the boundary when one goes deep enough in the foliation. 






 §.§ Elasticity and Finsler geometry










One can define elastic geometry in terms of distance:
The distance between two points can be declared to be the shortest time it takes for the support of a solution to the elastic wave equation, originally supported at one point, to reach the other point.

A more tractable description is obtained by studying the propagation of singularities of the elastic wave equation. 
There is a concrete way to pass from the stiffness tensor to a Minkowski norm on each tangent space, and this is described in <cit.>.  
Microlocal analysis indicates <cit.> that singularities follow the geodesic flow on the cosphere bundle of a Finsler metric, and cospheres are known as slowness surfaces in the physical literature. Elastic waves have three different polarizations in three spatial dimensions. The singularities used in the derivation of the Finsler geometry correspond to the fastest polarization known as quasi-pressure or qP.



In seismology the commonly used Preliminary Reference Earth Model <cit.> is spherically symmetric, and satisfies a foliation condition, the Herglotz condition, to great accuracy.







A typical inverse problem in elasticity is to reconstruct some elastic properties from boundary data <cit.>.
Given this geometric point of view, the task is to find the Finsler geometry corresponding to the parameters.  

It was shown in <cit.> that Finsler metrics arising from elasticity are always reversible

and fiberwise real-analytic. The following result is an adaptation of theorem <ref> in the elastic setting:


Let (M_i,F_i),   i ∈{1,2} be two compact Finsler manifolds of dimension larger or equal to 3, with strictly convex boundaries.  
We assume the following:


  * Both Finsler functions F_1 and F_2 are reversible.

  * The manifolds (M_i,F_i),    i ∈{1,2} are fiberwise real analytic. That is for every x ∈ M_i the function

    F_i(x,·)  T_xM_i →

is real analytic.

  * There are diffeomorphisms ∂ M_1→∂ M_2 and ∂ TM_1→∂ TM_2 that are compatible in the sense of definition <ref>.

  * F_1=F_2∘ on ∂ TM_1. 

  * For any two vectors v,w∈1 and t>0 we have vR_t^(1)w if and only if (v)R_t^(2)(w), where R^(i)_t is the broken scattering relation of definition <ref> on (M_i,F_i).

Then there is a diffeomorphism M_1→ M_2 that is an isometry in the sense of F_1=F_2∘, which satisfies |_∂ M_1=, and |_∂ TM_1=.



In the elastic setting no foliation condition is needed; it can be replaced with fiberwise analyticity.
But we do point out that with the foliation condition we have direct access to the Finsler function on all of the tangent bundle, whereas without the foliation condition we have to resort to analytic continuation which may be problematic for applications.
If our methods are used outside Hookean elasticity, the analyticity property 
and reversibility
may no longer be available.










 §.§ Related problems


Let us consider a compact Riemannian manifold (M,g) with boundary. The classical boundary rigidity problem is the following: We assume that we are given the distances d(x, y), through M, of all boundary points x,y ∈ M. Can we determine the isometry type of the manifold (M,g)? Michel <cit.> observed that in the case of simple manifolds these distance functions also determine the values of the geodesic flow at the boundary, this is the scattering relation or lens relation:

    L = {(v,w,t)∈××: ϕ_t(v)=w  for some  t ≤ 0}.

Thus L carries the information when and where and in which direction a geodesic, sent from the boundary, hits the boundary again. 

The natural conjecture is that for simple manifolds the scattering relation determines the isometry type of the manifold (see <cit.>). If the manifold is trapping one cannot determine the metric up to isometry if only the scattering relation is known  <cit.>.   
 



On the other hand if a Riemannian manifold admits a suitable convex foliation condition, then a local version of the scattering rigidity problem, studied in <cit.>, imply the global boundary rigidity result. See for instance <cit.> for a survey of different types of foliation conditions and geometric properties that imply their existence.  




Microlocal analysis connects singularities of solutions and solution operators to (hyperbolic) partial differential equations (PDEs) to geometry. The PDE related to our problem is the elastic wave equation. Its principal symbol is a matrix, which largest
eigenvalue is directly related to the Finsler metric <cit.>. Single scattering can be modeled by introducing, in the
elastic wave equation, a right-hand side representing a contrast source. The coefficients in this source, identified with the contrast in stiffness tensor, have a nonempty wavefront set. In the inverse problem studied in this paper, this wavefront set is assumed to be dense on the cosphere bundle associated with M. Thus in the case of very heterogeneous media with many scattering points inside the manifold one can obtain further information by looking at the propagation of singularities of waves going through the manifold. This is the broken scattering relation. 
Kurylev-Lassas-Uhlmann showed in <cit.> that this relation determines Riemannian manifold (M,g), upto an isometry. They reduce the problem to the setup of <cit.>, where it is shown that the collection of distance functions at the boundary determine the isometry class. In a sense they turn a data given by boundary sources, to one given by interior point sources. For inverse problems related to interior point sources and Riemannian wave equation see for instance <cit.>, and <cit.> for an interior source problem in Finsler geometry. 


  


 


§ OUTLINE OF THE PROOF


We prove theorem <ref> in this section.
The proofs of the key lemmas are postponed to subsequent sections.
The rough plan of the proof is as follows:




  * The first step is to verify that the broken scattering relation determines the boundary distance functions {d(x,·) M →: x ∈int  M}. Then we show that the boundary distance function determines the topological and smooth structures. This is based on earlier work <cit.>. 

  * By the previous result the boundary distance function determines F on the part of TM from where the geodesic flow reaches the boundary in a sufficiently short time. This part is known as the “good set” G⊂ TM∖0.

  * If a point x∈ M is sufficiently close to the strictly convex boundary, then more than half of the directions on T_xM belong to G, so F is determined there. 

  * By reversibility F is determined on all of T_xM. This implies that we have found the Finsler geometry in all directions in a neighborhood of the boundary. Using the foliation, we write the neighborhood as  f^-1([0,]) for some >0.

  * We may then ask how far in the foliation the metric is uniquely determined. By the previous argument it holds at least for a little bit. If it only holds up to some s<S, then we may reiterate the argument on the smaller manifold f^-1([s,S]). To do so, we must propagate the data from the original boundary inward to the new one. Thus the uniqueness extends beyond the alleged limit s, proving uniqueness on the whole manifold.

















We first give the lemmas and definitions needed to make the proof precise, then finish the proof, and finally complete the proof by proving the lemmas.



 §.§ Auxiliary results


The boundary distance function of a point x∈ M is the function r_x∂ M→ defined by r_x(y)=d(x,y). 
Notice that we assumed the Finsler metrics to be reversible, so it does not matter which direction we measure the distance in. 

In the cases where we are considering multiple manifolds, the boundary distance function of x∈ M_i is denoted by r_x^(i).



Let (M_i,F_i), i=1,2, be two compact reversible Finsler manifolds, with strictly convex boundaries, whose broken scattering relations agree in the sense of conditions (<ref>)–(<ref>) in theorem <ref>.



Then the boundary distance functions agree in the sense that

    {
    r_x^(1);x∈ M_1
    }
    =
    {
    r_y^(2)∘;y∈ M_2
    }.




We note that in (<ref>) we are given non-indexed sets. That is for a function r_x M→ we know whether it is included in the set (<ref>), but we do not know the indexing point x ∈ M. The map ∂ M_1→∂ M_2 is the same as in theorem <ref>.


We say that a direction v∈ TM∖0 is minimizing if the maximal geodesic starting at v reaches ∂ M in finite time and is a shortest curve joining its endpoints.
We denote the set of minimizing directions by G (for “good”) — obviously with G^(i)⊂ TM_i∖0.
The fibers are denoted by G_x^(i).



Let (M_1,F_1) and (M_2,F_2) be two compact Finsler manifolds with boundary.
Suppose there is a diffeomorphism ∂ M_1→∂ M_2 so that (<ref>) holds.









Then there is a diffeomorphism M_1→ M_2 so that |_∂ M_1=.
In addition, F_1(v)=F_2((v)) for all v∈ G^(1).


For a set A⊂ T_xM we denote by -A the reflection and by σ(A)=A∪ -A the symmetrization.
We also use the same notation on the whole bundle, applied fiberwise.



If a Finsler manifold (M,F) has a strictly convex boundary, then the set

    U
    =
    {x∈ M;σ(G_x)=T_xM∖0}

is a neighborhood of the boundary M ⊂ M.




Let F_1 and F_2 be two Finsler functions on a compact smooth manifold M with boundary.
Suppose (M,F_1) has a strictly convex foliation with a function f M→[0,S].
Let s∈(0,S) and denote M=f^-1([s,S]).
Assume the following:


  * The two metrics coincide below s in the sense that F_1=F_2 on T(M∖M).

  * For any two v,w∈_inSM (these bundles coincide for the two metrics) and t>0 we have vR_t^(1)w if and only if vR_t^(2)w, where R_t^(i) is the broken scattering relation of (M,F_i).

Then the scattering relations R_t^(i) on (M,F_i) coincide in the sense that the assumptions of theorem <ref> are valid with  and  being the identity maps.
Most importantly, F_1=F_2 on ∂ TM and for all v,w∈M we have vR_t^(1)w if and only if vR_t^(2)w.


Finally we recall the second main result of <cit.>:


Let (M_1,F_1) and (M_2,F_2) be two compact Finsler manifolds with boundary.
Suppose there is a diffeomorphism ∂ M_1→∂ M_2 so that (<ref>) holds.
If Finsler functions F_1 and F_2 are fiberwise real analytic, then there
exists a Finslerian isometry Ψ (M_1, F_1) → (M_2,F_2) so that Ψ|  M_1 = ϕ.







 §.§ Proofs of the theorems



Now we are ready to present the detailed proofs of theorems <ref> and  <ref> respectively.
We use the notations introduced in the previous subsection.


It follows from proposition <ref> that the boundary distance functions of the two manifolds agree up to identifying the boundaries with .
By lemma <ref> there is a diffeomorphism



M_1→ M_2, and we use this diffeomorphism to identify the two manifolds as M=M_1=M_2.
This manifold inherits the foliation from (M_1,F_1).



This smooth manifold M has two Finsler functions F_1 and F_2, and the goal is to show that they are equal.
Lemma <ref> shows that F_1=F_2 in the good set G:=G^(1)⊂ TM∖0.
By reversibility we conclude that F_1=F_2 in σ(G).
Lemma <ref> guarantees that σ(G) contains the punctured tangent bundle of some neighborhood of ∂ M.
Therefore there is >0 so that F_1=F_2 in tangent bundle T(f^-1([0,])).





Now define

    I
    =
    {
    h∈[0,S];
    F_1(x,v)=F_2(x,v)
     whenever f(x)≤ h and v∈ T_xM∖0}.

We showed that [0,)⊂ I, and it is clear that I is an interval.
By continuity of the two Finsler functions, it is a closed interval and therefore I=[0,s] for some s∈(0,S].
If s=S, the two Finsler functions coincide on the whole tangent bundle and the proof is complete.

Suppose then that s∈(0,S).
We are now in the setting of lemma <ref>.
The two metrics coincide in the strip f^-1([0,s]), so the data may be propagated through it.
We have now two metrics on the shrunk manifold M=f^-1([s,S]) and their broken scattering relations coincide by lemma <ref>.

Let us denote the good set of (M,F_i) by G^(i).
Repeating the argument obtained above, we find that there is a diffeomorphism

M→M so that F_1=^*F_2 on G^(1), which again has full fibers when the base point is in some neighborhood of the boundary.
It is straightforward to check that if (x,v)∈ G and x∈M, then (x,v)∈G^(1).
We also note that for any x ∈M the intersection T_x M∩ G^int is non-empty, since any small perturbation of an initial direction of the geodesic connecting x to a closest boundary point in M, is contained in G. Take (x,v)∈ G^int and let γ be the forward-maximal geodesic with respect to F_1 with such initial data. Due to <cit.> it follows that the lift of γ is contained in G^int. Since the geodesic coefficients G^i(w),   i ∈{1,…,n},   w ∈ TM (see for instance <cit.> for the definition) of F_1 and F_2 agree on G^int it follows that

γ is also a geodesic with respect to F_2.

But as F_1 and ^*F_2 agree in a neighborhood of the lift of γ, we have that the F_2-geodesic γ starts at (x).
Thus (x)=x and one could choose any starting point x∈M, and so =𝕀.
Therefore F_1(x,v)=F_2(x,v) for x∈M close enough to ∂M and all v∈ T_xM.
This means that the two metrics coincide in f^-1([0,s+]) for some >0, contradicting the maximality of s.

Finally we verify that
 and 
coincide on TM_1.








First we note that definition <ref>, proposition <ref> and lemma <ref> imply

    |_T  M_1== |_T  M_1.






We recall that   and  are linear on the fibers, and therefore to conclude the proof it suffices to show that  preserves the inward pointing  unit normal vector field to the boundary. By taking the directional differential _v of condition (<ref>) of theorem <ref>, and using equation (<ref>), we get
⟨_v(F_2)(ν), w⟩=0,   w ∈ T M_1.

Since M_1 → M_2 is a diffemorphism we have shown that the Legendre transform of ν is co-normal to T M_2. This implies that ν is normal to the boundary M_2. Due to condition (<ref>) of theorem <ref> it holds that  maps _1 onto _2. Therefore we have verified
ν
is the inward pointing unit normal to M_2. The proof is complete.



Theorem <ref> is a direct consequence of proposition <ref> and lemma <ref>.








§ SHORT AND LONG GEODESICS



In this section we analyze short geodesics near the boundary to prove lemma <ref>.
To do so, we first make an observation concerning the lengths of geodesics of arbitrary — even infinite — length.

For any Finsler manifold (M,F) with boundary we define the exit time function SM→[0,∞] so that (v) is the first time t>0 for which π(ϕ_t(v))∈∂ M.
If there is no such time, the value is taken to be infinity.
If v is based at a boundary point and points tangentially or outward, we set (v)=0. 


























































On any complete Finsler manifold M with strictly convex boundary the exit time function SM→[0,∞] is continuous.

Without assuming completeness, continuity holds where <∞, including a neighborhood of .



We will check continuity separately at different kinds of points v∈ SM, depending on the value (v).

If (v)=0, then v is based on the boundary and points tangentially or outwards.
(That is, v∈.)
By the assumption of strict convexity, any geodesic starting near v is short due to <cit.>, establishing continuity of  at v.

If (v)∈(0,∞), then the geodesic γ_v starting at v reaches ∂ M in finite time.
By strict convexity γ̇_v((v)) is transverse to ∂ M, and it follows from the implicit function theorem that  is in fact smooth in a neighborhood of v.

If (v)=∞, then the geodesic γ_v is trapped.
Continuity at v can only fail if there is a sequence of vectors v_k∈ SM so that v_k→ v as k→∞ and (v_k)≤ T for some T∈(0,∞).
There is some r>0 so that for all k∈ we have d(π(v_k),π(v))<r.
All the forward-maximal geodesics γ_v_k are thus contained in the metric closed ball K=B̅(π(v),r+T).
By completeness of M the set K is compact. Up to extracting a subsequence, the endpoints γ_v_k((v_k))∈∂ M∩ K converge to a point z∈∂ M and (v_k) converges to T'≤ T.
By continuity of the geodesic flow, we have that ϕ_T'(v)=lim_k→∞ϕ_(v_k)(v_k)∈ S_zM.
This means that γ_v meets the boundary at time T'<∞, which is a contradiction.

We conclude that  is continuous at all points of SM.
Completeness was only needed to prove continuity where =∞, and that cannot happen near  when ∂ M is strictly convex.



If the manifold is not compact, we can restrict the analysis to a bounded neighborhood of a given boundary point and do everything on a compact submanifold.
On a compact Finsler manifold there is some >0 so that any geodesic with length less than  is minimizing.

Consider any point p∈∂ M, and let ν_p be the inward unit normal at p and γ_p the geodesic starting in the direction of ν_p.
For any t>0 small enough so that the geodesic γ_p up to time t exists we define the map P^p_t S_pM→ S_γ_p(t)M by parallel transport.

Fix some T>0 small enough.
We define a function Q[0,T)×∂ SM→ SM so that Q(t,v)=P^π(v)_t(v).
This function is continuous, so by lemma <ref> the composed function ∘ Q[0,T)×∂ SM→[0,∞] is also continuous.
(Continuity will only be needed near directions where =0.) 
This composed function vanishes on {0}×, so there is an open neighborhood V⊂[0,T)× SM of {0}× so that ∘ Q|_V<.
Since all geodesics shorter than  are minimizing, we have Q(V)⊂ G.

Any boundary point p∈∂ M has a neighborhood W⊂∂ M and two numbers a,h>0 so that [0,h)× V_p⊂ V with

    V_p
    {v∈ SM;π(v)∈ W and vν_ν<a}
    .

We note that V_p contains larger portion of S_pM than the Southern hemisphere. We have that Q([0,h)× V_p)⊂ G and

    Q([0,h)×σ(V_p))
    =
    σ(Q([0,h)× V_p))
    ⊂σ(G).

But we have
σ(V_p)
=
{v∈ SM;π(v)∈ W},
so that the set
U'
=
{γ_p(t);t∈[0,h) and p∈ W
}
is a subset of the set U defined in the claim of the lemma.
Now U' is a neighborhood of the arbitrary boundary point p, so we have established that U does indeed contain a neighborhood of the boundary.




§ PROPAGATION OF DATA THROUGH A LAYER



In order to prove lemma <ref> to propagate data from ∂ M to ∂M, we first observe that a foliated manifold enjoys a certain non-trapping property. 




Let (M,F) be a compact Finsler manifold which is foliated by a smooth function f M→ in the sense of definition <ref>.
If a maximal geodesic γ satisfies ∂_t(f(γ(t))<0 at t=0, then γ reaches ∂ M in finite time.



Consider the function h(t)=f(γ(t)), defined over some maximal interval [0,T] or [0,∞).
The goal is to show that h obtains the value zero in finite time.
Given that h'(0)<0 and h is smooth, this can only fail if one of the following happen:


  (i)

The derivative h'(t) vanishes for some t.

  (ii)

The function has a limit: h is strictly decreasing and lim_t→∞ h(t)=H∈[0,∞).


Let us first exclude case (i).

If h' vanishes somewhere, there is a smallest t>0 for which h'(t)=0.
We have that h'<0 on [0,t).
The geodesic γ is tangent to the level set f^-1(h(t)), so by definition <ref> we have h”(t)<0.
This implies that for some >0 we have h'(t-)>0, which is a contradiction.

Let us then move to case (ii).

The curve γ now approaches the surface Σ f^-1(H).
By monotonicity there is an increasing sequence of times t_k→∞ so that h'(t_k)→ 0.
Upon extracting a subsequence, there exists a point x∈Σ so that γ(t_k)→ x and γ̇(t_k) converges to some v∈ S_xM.
As h'(t_k)→ 0, it follows that v is tangent to Σ.

Let γ̃ be the geodesic starting with initial conditions (x,v).
We denote h̃=f∘γ̃.
Since v is tangent to Σ, we have h̃'(0)=0, so definition <ref> implies h̃”(0)<0.






















By smoothness of f and the time additive property of geodesic flow we have

    lim_k→∞
    h”(t_k+s)
    
    
    
    
    
    = 
    
    h̃”(s)

for any s∈.

Because h̃”(0)<0, there are >0 and δ>0 so that h”(t_k+s)≤- for all s∈[-δ,δ] when k is large enough.
This together with h'≤0 (which is due to monotonicity) and the fundamental theorem of calculus gives

    h(t_k+δ)
    =
    h(t_k-δ)
    +
    2δ h'(t_k-δ)
    +
    ∫_-δ^δ∫_-δ^s
    h”(t_k+r)
     r
     s
    ≤
    h(t_k-δ)
    -2δ^2

for large enough k.
As -2δ^2 is independent of k and h is monotonous, it follows that h cannot converge.
This is a contradiction.


Therefore there is indeed some finite t so that h(t)=0.





Equality of the Finsler functions on ∂M to all directions on the bundle — which amounts to  and  being identities — follows from the assumed equality of the Finsler functions on the strip f^-1([0,s]) and continuity.

For any v∈ SM, we denote by γ_v the unique geodesic with γ̇_v(0)=v.
Take any two v,w∈M and t>0.

By lemma <ref> there is a>0 so that the geodesic segment γ_v|_[-a,0]⊂ f^-1([0,s]) connects a point on ∂ M to the point π(v)∈∂M.
Because the two Finsler metrics agree in f^-1([0,s]), this same curve is a geodesic in both geometries.
By strict convexity this geodesic meets ∂ M transversely, so that v'γ̇_v(-a) belongs to .
Similarly, there are b>0 and w'∈ corresponding to w.

Because the two geodesic flows agree from v' to v and from w' to w, we have that vR_t^(i)w if and only if v'R_t+a+b^(i)w' for both i ∈{1,2}.
The broken scattering relations R^(i)_t+a+b agree on ∂ M, and so the broken scattering relations R_t^(i) agree on ∂M.






§ FROM BROKEN SCATTERING RELATION TO BOUNDARY DISTANCE FUNCTIONS



In this section we will prove proposition <ref>. We start with considering only one Finsler manifold (M,F) with strictly convex boundary whose broken scattering relation is known.



 §.§ Critical distance functions



We begin with studying classical critical distance functions. The first one is the cut distance function S M→ (0,∞] given by


    (x,v) sup{t>0; γ_x,v(t)  exists and   d(x,γ_x,v(t))=t}.

Since the boundary of the M is strictly convex it holds that any distance minimizing curve is a geodesic. Therefore function  is well defined and moreover it is continuous. 

Next we formulate two auxilliary lemmas related to  function.



For any (x,v) ∈  and 
s_2 > s_1 ≥ 0,
which satisfy

    γ_x,v(s_1)=γ_x,v(s_2),
      we have    s_1+s_2 > 2(x,v).




Since F is reversible this claim can be proven like <cit.>.




































Let z∈ M, t∈ (0,(z,ν(z))) and U ⊂ TM be a neighborhood of ν(z) that is diffeomorphic to some open set of ^2n-1× [0,∞). For any ϵ> 0 we can choose δ=δ(z,t,ϵ)>0 such that the following holds: If v_i ∈ U,   i∈{1,2} satisfy


    v_1 R_2tv_2,    and   v_i-ν(z)_e< δ,

then there exist t_1,t_2>0, so that  

    γ_v_1(t_1)=γ_v_2(t_2),      t_1+t_2 =  2t,    and    |t_i-t| <ϵ.

Here ·_e is the Euclidean norm on U.


The proof is analogous to the proof of <cit.>.












































The second critical distance functions is the boundary cut distance function

    (z):=sup{t>0:  d(z,γ_z,ν(z)(t))=t= d(∂ M,γ_z,ν(z)(t))},    z ∈ M.













These two critical distance functions satisfy the following:



For any z∈∂ M it holds that 

    (z,ν(z)) > (z).









































































 §.§ Family of focusing directions


Let (M,F) be a compact Finsler manifold of dimension 3 or higher with reversible Finsler function F and strictly convex boundary. Let x ∈ M^int and v ∈ S_xM be a direction such that the corresponding geodesic γ_v is the distance minimizer from x to z_x∈∂ M a closest boundary point to x. In the proof of lemma <ref> we showed that the exit time function  is smooth in some neighborhood V ⊂ S_x M of v, and moreover implicit function theorem implies that the map

    H V ∋η↦
    π(ϕ_(η)(η))∈ M

is a diffeomorphism to some neighborhood U⊂ M of z_x. Using this identification we define a smooth inward pointing unit length vector field on U

    V(z):=-ϕ_(η)(η),   η:=H^-1(z).

This vector field satisfies 

    V(z_1)R_T(z_1,z_2)V(z_2),  and  V(z_x)=ν(z_x),

where 

    T(z_1,z_2):=(H^-1(z_1))+((H^-1(z_2)).

Moreover, the function


    t U →,    t(z):=1/2T(z,z)=((H^-1(z))

is smooth and its differential vanishes at z_x




which is a closest boundary point to x. Thus the geodesics given by initial conditions (z,V(z)),   z ∈ U focus at the common interior point x at time t(z).  

We change the point of view and set the following definition:


Let z_0∈∂ M and t_0 ∈ (0,(ν(z_0))). We say that a collection 

    F(z_0,t_0):={U, V(·),t(·)}

where U⊂ M is a neighborhood of z_0, V U→∂_inSM is a smooth vector field and  t U → (0,∞) is a smooth  function, is called a family of focusing directions around (z_0,t_0) if  

    V(z_1)R_t(z_1)+t(z_2)V(z_2),   for all  z_1,z_2 ∈ U, 
    
       V(z_0)=ν(z_0),    t(z_0)=t_0,     and    t(z)|_z=z_0= 0.



We note that the broken scattering relation determine all the families of focusing directions, but it is possible that not all of them focus in the sense of

    π(ϕ_t(z)(V(z)))=π(ϕ_t_0(ν(z_0))),    for all  z ∈ U.

Next we give the following result that guarantees the actual focusing if the focusing time t_0 is small enough. 


Let z_0∈∂ M, t_0 ∈ (0,(ν(z_0))) and F(z_0,t_0)


 ={U', V(·),t(·)} be a family of focusing directions around (z_0,t_0). If

    t_0<  (z_0,ν(z_0))=:τ_M(z_0),

then there exists a neighborhood U ⊂ U' of z_0 such that (<ref>) holds true.


For the proof of the lemma we need the following auxiliary result.



Let z_0 ∈ M,  t_0>0 and the family of focusing directions F(z_0,t_0)={U,V(·),t(·)} around (z_0,t_0) be as in lemma <ref>. Let γ be some geodesic that intersects γ_V(z_0) at r_0 transversely. If there exist functions r,ρ U →  for which hold

    ρ(z_0)=0,   r(z_0)=r_0,   γ(ρ(z))=γ_V(z)(r(z)),

and 

    0≤ r(z)≤ r_1< τ_M(z_0),    |ρ(z)|≤ρ_1< inj(M).

Then

t_0=r_0.




The proof consists of several steps.

(I) Since F(z_0,t_0) is a family of focusing directions the equation (<ref>) implies the existence of functions s,s U → [0,∞) that satisfy

    γ_V(z)(s(z))=γ_V(z_0)(s(z)),    s(z)+s(z)=t(z)+t_0.

Since vector field V is continuous, equation (<ref>) and lemma <ref> imply

    s(z)→ t_0,   s(z) → t_0,    as  z → z_0.

Therefore 
s(z_0)=s(z_0)=t_0,
and by an analogous proof to one given in <cit.>
we show that the functions s,  s are smooth near z_0 and satisfy

    s(z_0)=s(z_0)=0.
















































(II) 
Let W⊂ M be a neighborhood of z_0 where functions s,s are smooth. We consider a map

    E W → SM,    E(z):=-ϕ_s(z)(V(z)).

We begin with studying the differential
E|_z_0 T_z_0 M → T_(x,η)SM,   (x,η):=E(z_0).
Recall that the tangent bundle T(SM) has a canonical decomposition to  Horizontal H(SM) and Vertical V(SM) sub-bundles. We denote the projections from T(SM) to these bundles by P_H and P_V respectively. 
We note first that

    x(z):=(π∘ E)(z)=γ_V(z)(s(z))=γ_ν(s(z)).

Also a simple computation shows that after identifying H_(x,η)(SM) to T_xM we have
x|_z_0=(P_H∘( E))|_z_0.

Therefore

    (P_H∘( E))|_z_0=[γ_V(z)(s(z))] |_z=z_0  =[γ_ν(s(z))] |_z=z_0   =γ̇_ν(t_0)⊗s(z_0)=0,

implies that

    E|_z_0 T_z_0 M→ T_(x,η) SM =H_(x,η)(SM)⊕ V_(x,η)(SM),    E|_z_0 v=(0,Θ v),

where 

Θ T_z_0 M → V_(x,η)(SM)

is a linear map. Recall that V_(x,η)(SM) is isomorphic to T_η(S_xM), which is (n-1)-dimensional as also T_z_0 M. Therefore 


to show that Θ is a linear isomorphism it suffices to prove that it is injective. We define a map

    G(z):=exp_x(z)(s(z)E(z))=z,    for all  z ∈ W.

Thus the differential G at x_0 is an identity operator on T_z_0 M. Since s(z_0)=t_0 is less than the cut distance (z_0,ν(z_0)) and s(z)|_z=z_0= x(z)|_z=z_0=0 we have

    G|_z=z_0v=(exp_x)|_t_0E(z_0)t_0 E|_z=z_0v=(exp_x)|_t_0 E(z_0)t_0Θ v=v.

This implies that Θ is an injection.



(III)
Now we prove that the function r given in the claim of this lemma is continuous at z_0. If this is not true there exist ϵ>0 and a sequence z_k∈ M which converges to z_0 but for which hold

    |r_k-r_0|>ϵ,    r_k:=r(z_k).

Since functions r and ρ are bounded we can without loss of generality assume that 
r_k→ r' <τ_M(z_0),   |r'-r_0| ≥ϵ,  ρ_k:=ρ(z_k)→ρ'∈,  such that  |ρ'|< inj(M).
Thus

    γ(ρ')=lim_k →∞γ(ρ_k)=lim_k →∞γ_V(z_k)(r_k)=γ_ν(r').

Since r_0,r' <τ_M(z_0) we have that
x:=γ_ν(r_0)=γ(0),  and  x':=γ_ν(r')=γ(ρ')
are two different points where γ and γ_ν intersect. Since
r_0,r'< τ_M(z_0), and 
|ρ'|< inj(M),
there are two different distance minimizing geodesics connecting x to x', which is not possible.

(IV) Let us then assume that r_0< t_0. We study a map

    Φ U ×→ M,   Φ(z,λ)=exp_z(λ V(z)),

and show that it is a local diffeomorphism near (z_0,r_0). Since 0<t_0-r_0< τ_M(z_0) and the Finsler function F is reversible, the map exp_x_0 is a local diffeomorphism near (t_0-r_0)η, where
x_0=γ_ν(t_0-r_0),  η_0 =-γ̇_ν(t_0-r_0).
Thus

    exp_x_0|_(t_0-r_0)η_0 T_(t_0-r_0)η(T_x_0M) → T_z_0M

is a linear isomorphism. Next we note that 

    Φ(z,λ)=γ_E(z)(s(z)-λ)=exp_x(z)((s(z)-λ)E(z)),

where E and s(·) are the same maps as in parts (I) and (II) of the proof. Since the differentials of x and s vanish at z_0 we have

    _λΦ|_(z_0,r_0)=-exp_x_0|_(t_0-r_0)E(z_0)E(z_0):=-AE(z_0),

and

    _zΦ|_(z_0,r_0)=(t_0-r_0)A  E|_z_0.

Therefore for any (v,λ)∈ T_z_0 M × holds

    Φ|_(z_0,r_0)(v,λ)=A((t_0-r_0)Θ v-λ E(z_0)).

Since A is invertible it holds that 

    Φ|_(z_0,r_0)(v,λ)=0   if and only if   (t_0-r_0)Θ v-λ E(z_0)=0.

However this can only happen if (v,λ)=0 as Θ is injective and 

    Θ v ∈ T_E(z_0)S_x_0M   implies   g_E(z_0)(E(z_0), Θ v)=0.

Due to the inverse function theorem Φ is a local diffeomoprhism near (z_0,r_0).

(V) Let Σ be a (n-1)-dimensional surface in M such that γ is a curve on Σ and γ_ν is transverse to Σ at r_0. 
Due to implicit function theorem there exists a smooth function r U → so that

    Φ(z,λ)∈Σ,   if and only if  λ=r(z)  for all  z∈ U,    and   r(z_0)=r_0.

Since r(·) is continuous and satisfies 
Φ(z,r(z))=γ(ρ(z))∈Σ
it holds that near z_0 functions r and r coincide. Therefore the map

    Φ U ∋ z ↦Φ(z,r(z)) ∈Σ

is smooth and its image is contained in the image of γ. However we have proven that Φ is a local diffeomorphism  near z_0. Thus we arrive into a contradiction since Φ maps (n-1)-dimensional surface (n-1≥ 2) onto 1-dimensional surface. Thus r_0 < t_0 is false. By an analogous argument we can prove that r_0> t_0 is also false and therefore it must hold that r_0=t_0.


We are ready to present the proof for lemma <ref>.


The proof is an adaptation of <cit.> where one uses lemmas <ref> and <ref>.




































 §.§ Boundary distance functions

We start with giving a stronger formulation of lemma <ref>.



Let z_0 ∈ M, 0<t_0<τ_M(z_0), and F(z_0,t_0)={U,V(·),t(·)} be a family of focusing directions around (z_0,t_0). Let γ be some geodesic that intersects every geodesic of F(z_0,t_0) in the sense that 

    γ(ρ(z))=γ_V(z)(r(z)),

for some functions ρ, r U → satisfying

    0≤ r(z)≤ r_1<  τ_M(z_0),    |ρ(z)|< L,  for some  L>0.

If in addition h(z):=r(z)+ρ(z) is continuous then for any z ∈ M which is close to z_0 holds that

    γ(h(z_0)-t_0)=γ_V(z)(t(z)).

That is all geodesics of F(z_0,t_0) meet at the same point γ(h(z_0)-t_0).


If γ and γ_V(z_0) are the same geodesic the result follows from lemma <ref>.

If we assume that γ is a different geodesic to γ_V(z_0) it the holds due to the definition of injectivity radius of M that the geodesic segments γ([-L,L]) and γ_V(z_0)([0,r_1]) can intersect at most finitely many times. Thus there exists N ∈ such that
γ(ρ)=γ_V(z_0)(r), for (ρ,r) ∈ [-L,L]×  (0,r_1) if and only if  (ρ,r)=(ρ_0^k, r^k_0), for some ρ_0^k∈{ρ_0^1,…, ρ_0^N}⊂ [-L,L], and r_0^k∈{r^1_0,…, r_0^N}⊂ (0,r_1).

Let 0< ϵ < 1/2inj(M). We claim that there exists R_0>0 such that for all 0<R<R_0 and 

    z ∈ U(R):=B(z_0,R) ∩ M,    it holds that min_j∈{1,…, N }|ρ(z)-ρ_0^j|<ϵ.

If this is not true, the boundedness of functions r(·) and ρ(·) imply the existence of a sequence (z_k)_k=1^∞⊂ M that converges to z_0 and which satisfy


    lim_k→∞ r(z_k)=r∈ [0,r_1] ,   lim_k→∞ρ(z_k)=ρ∈ [-L,L]∖{ρ_0^1,…, ρ_0^N}.

Therefore we arrive in a contradiction  γ(ρ)=γ_V(z_0)(r).

We set 



    W_j(R)
    :=   {z∈ U(R): ∃ r∈ [0,r_1],  ∃ρ∈ [-L,L], 
    
        s.t. γ_V(z)(r)=γ(ρ),  r+ρ=h(z),   |ρ-ρ_0^j| ≤ϵ},   j ∈{1,…, N},

and show that these sets are relatively closed in U(R). Choose a sequence (z_k)_k=1^∞⊂ W_j(R) that converges to z ∈ U(R). For any k ∈ we choose r_k ∈ [0,r_1] and ρ_k ∈ [ρ_0^j-ϵ, ρ_0^j+ϵ] for which

    γ_V(z_k)(r_k)=γ(ρ_k),    and    r_k+ρ_k=h(z_k).

After choosing a sub-sequence of (z_k)_k=1^∞ we may without loss of generality assume that lim_k→∞r_k= r∈ [0,r_1], lim_k→∞ρ_k=ρ∈ [ρ_0^j-ϵ, ρ_0^j+ϵ],  

    γ_V(z)(r)=γ(ρ),   and   h(z)=r +ρ.

Here we used the continuity of the geodesic flow and the function h(·) to obtain the last two equations. Thus we have verified that z ∈ W_j(R). Therefore W_j(R) is closed and measurable. 

Equation (<ref>) implies that

    U(R)=⋃_j=1^N W_j(R).

Thus there exists j ∈{1,…, N} for which W_j(R) has a strictly positive (n-1)-dimensional measure. Choose k ∈{1,…, N} and suppose that r_0^k≠ t_0. By following the proof of lemma <ref> we show that after choosing a smaller R>0 there exist 0<T<1/2inj(M), (n-1)-dimensional surface Σ⊂ M and a diffeomorphism 

    Φ U(R) →Σ,    for which Φ(z)⊂γ([ρ_0^k-T,ρ_0^k+T]),  if  z ∈ W_k(R).

Thus the (n-1)-dimensional measure of W_k(R) is zero. We have proven that the set K ⊂{1,…, N} that contains all those k ∈{1,…, N} for which r_0^k=t_0, is not empty and moreover

    U(R)∖( ⋃_k ∈ K W_k(R)),  is of measure zero.

Therefore ⋃_k ∈ K W_k(R) is dense in U(R). Since ϵ>0 was arbitrary we can choose   k ∈ K, sequences (z_j)_j=1^∞⊂ W_k(R), (r_j)_j=1^∞,   (ρ_j)_j=1^∞⊂, such that
z_j → z_0,  ρ_j →ρ_0^k,   r_j →r∈ [0,r_1]
and
γ(ρ_j)=γ_V(z_j)(r_j),  for every  j ∈.
Since r_0^k=t_0 the continuity of geodesic flow implies

    γ_V(z_0)(r_0^k)=γ(ρ_0^k)=γ_V(z_0)(r).

Thus we must have t_0=r. Since the function h(·) is continuous we also have ρ_0^k=h(z_0)-t_0 and moreover

γ(h(z_0)-t_0)=γ_V(z_0)(t_0).

This and lemma <ref> complete the proof.





The broken scattering relation determines the boundary cut distance function .




Let z_0 ∈ M. By the definition of (z_0) it holds that for t_0 ∈(0,(z_0)) the point z_0 is the closest boundary point to x_0:=γ_ν(z_0)(t_0). However if t_0 > (z_0) then there exists w∈ M that is closer to x_0 than z_0 in the sense that: 

    There exists s<t_0 such that γ_ν(w)(s)=x_0.

By lemma <ref> we know that (z_0,ν(z_0))>(z_0). If
t_0∈((z_0), τ_M(z_0)) then according to discussion at the beginning of subsection <ref> there exist w ∈ M and a family of focusing directions F(z_0,t_0):={U, V(·),t(·)}  around (z_0,t_0) such that 

    ν(w)R_s+t(z)V(z),   and   s<t(z)   for all  z ∈ U.


First we note that the broken scattering realtion determines the exit time function since for any v ∈_inSM holds

    (v)=sup{t≥ 0:  such that vR_2tv}.

We claim that 

    (z_0)=   inf{t_0 ∈ (0,(ν(z_0))):  There exist F(z_0,t_0), s < t_0,
        and w∈ M∖{z_0} such that(<ref>) is valid}.

For the proof of this equation see in <cit.>.






















We recall that the exponential map at the boundary is 

    exp_ M M × [0,∞) → M,   exp_ M(z,t)=γ_ν(z)(t).




Let x∈ M. The broken scattering relation determines the set of boundary distance functions

    r_x M →,    r_x(z)=d(x,z),   z ∈ M,   x ∈ M.



By the definition of boundary cut distance function we have that 

    M={exp_ M(z,t) ∈ M: z ∈ M,   t ∈ [0, (z)]  }.


Therefore exp^-1_ MM ⊂ ( M ×) is a way to represent M, although points in the boundary cut locus 

    ω_ M:={x∈ M: x=exp_ M(z,(z)),   z ∈ M}

may have several representatives. As we are interested in determining the boundary distance functions the possible ambiguity does not concern us since it is proven in <cit.> that 

    r_x=r_y    if and only if    x=y ∈ M.


Let us fix z_0 ∈ M and t_0 ∈ [0,(z_0)]. Then we choose w ∈ M. As M is strictly convex any distance minimizing curve from w to x:=exp_ M(z_0,t_0) is a geodesic γ_w,η:[0,s]→ M, where s=d(w,x) and η∈ S_w M. Due to lemma <ref> there exist a family of focusing directions F(z_0,t_0)={U, V(·),t(·)} such that
V(z)R_s+t(z)η,   for any  z ∈ U.
We set 

    S:={s>0:  There exist η∈ S_wM, and F(z_0,t_0) s.t. V(z)R_s+t(z)η holds}

and claim that 

    d(x,w)=inf S.

The proof of this claim is an adaptation of the proof of an analogous statement in <cit.>.  



























In the following lemma we give an invariant definition for families of focusing directions using the invariant definition of broken scattering relation given in theorem <ref>. The claim of the lemma is a direct implication of the definition <ref>.



Let (M_i,F_i) be two compact Finsler manifolds that satisfy the conditions of theorem <ref>. Let z_0 ∈ M_1 and t_0∈ (0,(ν(z_0))). Then 
ℱ(z_0,t_0):={U,V(·),t(·)} 
is a family of focusing directions around (z_0,t_0) if and only if ℱ(z_0,t_0):={U,V(·),t(·)} is a family of focusing directions around ((z_0),t_0), where

    U:= (U)⊂ M_2,   V:=∘ V ∘^-1,    and   t:=t∘^-1.



We are ready to prove proposition <ref>.



Let x ∈intM_1. Choose z_0 ∈ M_1 and


 t_0 ∈ (0,τ_ M_1(z)] such that x=exp_ M_1(z_0,t_0). Due to lemmas <ref> and <ref> it holds that τ_ M_2( (z))=τ_ M_1(z) and therefore
x̃: =exp_ M_2((z_0),t_0) ∈intM_2.
We aim to verify  

    d_1(z,x)=d_2((z),x),    for all  z ∈ M_1.

This implies the left hand side inclusion in (<ref>). By reversing the roles of M_1 and M_2, the analogous argument verifies the right hand side inclusion in (<ref>). Choose z ∈ M_1. We denote
s_1:=d_1(x,z),  and  s_2:=d_2(x,  (z)).
Since M_1 is strictly convex there exists a distance minimizing geodesics  γ_z,η of M_1 from z to x. 

Choose a family of focusing directions ℱ(z_0,t_0)={U,V(·),t(·)} such that
V(z)R^(1)_s_1+t(z)η,  for every  z ∈ U.
Then lemma <ref> implies that ℱ(z_0,t_0)={U, V(·),t(·)} is a focusing family at ((z_0),t_0) that satisfy
V(w)R^(2)_s_1+t(w)η,  for every  w ∈U.
The proof of proposition <ref> implies s_2≤ s_1. After reversing the roles of x and x, we use the analogous argument to verify s_1≤ s_2. We have proven s_1=s_2. Since z ∈ M_1 was arbitrary the equation (<ref>) is valid.




§ FOURIER INTEGRAL OPERATORS AND ANNIHILATORS


We consider v, w ∈∂ S M and introduce their duals, v, w∈∂ (S^∗ M), via the Legendre transform <cit.>. Their tangential components in T^* ∂ M are written as v_∂, w_∂. Instead of the geodesic flow, ϕ_t, we consider the co-geodesic or Hamiltonian flow, ϕ_t on the co-sphere bundle and the canonical projection π S^∗ M → M. We note that the broken scattering relation, given in definition <ref>, is closely related to the following canonical relation

    Λ := {(π(v),
           π(w), t,
           v_∂,
           w_∂,
           τ) ;
           (x, ξ) |π(ϕ_t_1(v))
           = π(ϕ_t_2(w))
           = x ,. 
    .
           t = t_1 + t_2 ,τ = F^∗(ϕ_t_1(v))
                = F^∗(ϕ_t_2(w))=1 ,ξ = ϕ_t_1(v) +
                           ϕ_t_2(w)
                 } .

We write Y = ∂ M ×∂ M × (0,T) and X =
M. Then Λ⊂ T^*Y × T^*X. If the Finsler metric on TM is known, as we assume in theorem <ref>, the vectors
v_∂, w_∂ determine
v, w, respectively. The canonical relation
connects an element (x,ξ) in the wavefront set of scatterers in T^*X to an element in the wavefront set of scattered waves generated by sources, and detected by receivers, in ∂ M in
T^*Y. That is, there is a Fourier integral operator
(FIO) that maps scatterers to scattered waves restricted to ∂
M and propagates singularities according to the mentioned canonical
relation Λ.

The Bolker condition states that the natural projection from Λ
to T^*Y is injective. If the Bolker condition is
satisfied, the range of the FIO can be characterized by
pseudodifferential annihilators <cit.>,
which yields the canonical relation. Thus, within the framework of the
results of this paper, vanishing annihilators determine the underlying
reversible Finsler manifold, if  both Bolker condition and the foliation condition of definition <ref> hold.

Although the Bolker condition should be fairly easy to satisfy, it
does not follow from the foliation condition. To see this, we consider
the Euclidean unit disc M with a radial metric c(r)e,   r=|x|. This manifold
can satisfy the Herglotz condition, 
/ r(r/c(r))>0,
and have a geodesic whose opening
angle is between π and 2π. Informally, this corresponds to a
geodesic going around the center. Such a geodesic will meet its
reflection across the center two times at some points p,q ∈M. If we remove a segment of the
geodesic between the second intersection point q and the nearest endpoint at the
boundary and the mirror image of this segment, we obtain two broken
rays c_1 and c_2 with exactly the same total length t ∈ and boundary
data v,w ∈_inSM. This situation violates the Bolker condition. We illustrate this setup in the following picture. 






























 §.§ Acknowledgements


MVdH was supported by the Simons Foundation under the MATH + X program, the National Science Foundation under grant DMS-1815143, and the corporate members of the Geo-Mathematical Imaging Group at Rice University USA.
JI was supported by the Academy of Finland (project 295853). 

ML was supported by Academy of Finland (projects 284715 and 303754).
TS was supported by the Simons Foundation under the MATH + X program and the corporate members of the Geo-Mathematical Imaging Group at Rice University.  

Part of this work was carried out during JI's and TS's visit to University of Washington, USA, and they are grateful for Prof. Gunther Uhlmann for hospitality and support. 

abbrv


