Reconstruction of a compact Riemannian manifold from the scattering data of internal sources
Matti Lassas Teemu Saksala Hanming Zhou

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            MℳetgdR_ M]Reconstruction of a compact manifold from the scattering data of 
            internal sourcesM. Lassas]Matti Lassas ^□
            T. Saksala]Teemu Saksala ^♢, ∗
            H. Zhou]Hanming Zhou ^†
            [^□ Department of Mathematics and Statistics, University of Helsinki, Finland, 
            
            ^♢  Department of Mathematics and Statistics, University of Helsinki, Finland; Department of computational and applied mathematics, Rice University, USA
            
            ^† Department of Pure Mathematics and Mathematical Statistics, University of Cambridge, UK; Department of Mathematics, University of California Santa Barbara, Santa Barbara, CA 93106, USA, 
            
            ^∗teemu.saksala@helsinki.fi, teemu.saksala@rice.edu]
            Given a smooth non-trapping compact 
            manifold with strictly convex boundary, we consider an inverse problem of reconstructing the manifold from the scattering data initiated from internal sources. These data consist of the exit directions of geodesics that are emaneted from interior points of the manifold. We show that under certain generic assumption of the 
            metric, the scattering data 
            measured on the boundary determine the Riemannian manifold up to isometry.
            
            
            [
                [
                December 30, 2023
            =====================
            
            
            
            
            
            § INTRODUCTION, PROBLEM SETTING AND MAIN RESULT
            
            In this paper we consider an inverse problem of reconstructing a Riemannian manifold (M,g) from geodesical data that correspond to the following theoretical measurement set up. Suppose that in a domain M there is a large amount m of point sources, at points x_1,x_2,…,x_m, sending continuously light (or other high frequency waves) at different frequencies ω_j. Such point sources are observed on the boundary as bright points. Assume that on the boundary an observer records at the point z∈ M the exit direction of the light from the point source x_j, that is, an observer can see at the point z the directions of geodesics coming from x_j to z. When the observer moves along the boundary, all existing directions of geodesics coming from the point sources to the boundary are observed. We emphasize that only the directions, not the lengths (i.e., travel times) are recorded. When m becomes larger, i.e., m→∞,
            we can assume that the set of the point sources {x_j} form a dense set in M. 
            
            
            
            Let (N,g) be an n-dimensional closed smooth Riemannian manifold, where n≥ 2. Suppose that M ⊂ N is open and has a boundary M ⊂ N that is a smooth (n-1)-dimensional submanifold of N. We also assume that M is strictly convex, meaning that the second fundamental form of M (as a submanifold) is positive definite. 
            
            Let SN⊂ TN be the unit tangent bundle of the metric g. Given (p,ξ)∈ SN, it induces a unique maximal geodesic γ_p,ξ:→ N with γ_p,ξ(0)=p, γ̇_p,ξ(0)=ξ. We denote  S_pN={ξ∈ T_pN; ξ_g=1}. We will use the notation π:TN → N for the canonical projection to the base point, π(x,ξ)=x.
            
            
            
            We define the first exit time function
            
                τ_exit(p,ξ)=inf{t>0; γ_p,ξ(t)∈ N∖M},   (p,ξ) ∈ S M,
            
            where S M is the unit tangent bundle of M. We assume that 
            
                τ_exit(p,ξ)< ∞,   for all (p,ξ) ∈ S M.
            
            This means that M is non-trapping.
            
            For each point p∈M we define a scattering set of the point source p
                [                                R_ M(p)                                      = {(q,η^T)∈ T  M ;  there exist  ξ∈ S_pN;                                                                                   and  t∈ [0,τ_exit(p,ξ)]  such
                that;                                                                                     q=γ_p,ξ(t), η=γ̇_p,ξ(t)}∈ 2^T M. ]
            
            Here η^T ∈ T M is the tangential component of η∈ SM and 2^S means the power set of set S. See Figure (<ref>). 
            
            
            Let
            
            R_ M(M)={R_ M(p); p∈ M}⊂ 2 ^T M.
            
            
            
            
            
            
            
            
            
            
            
            
            
            
            
            
            
            Consider the following collection
            
            
                ( M ,R_ M(M)).
             
            We call this collection the  scattering data of internal sources that depends on the Riemannian manifold (M,g|_M). We emphasize the connection of data (<ref>) to the scattering data (see <cit.>), that is also considered in the Section <ref> of this paper.  From now on we will use a short hand notation
            
                g:=g|_M=i^∗ g,   i:M↪ N.
            
            Here i is an embedding defined by i(x)=x,   x ∈M. The inverse problem considered in this paper is the determination of the Riemannian manifold (M,g) from the data (<ref>). More precisely, this means the following: Let (N_j,g_j),   j=1,2 and  M_j be similar to (N,g) and M.  We say that the scattering data of internal sources of (M_1,g_1) is equivalent to that of (M_2,g_2), if
            
                there exists a diffeomorphism ϕ:  M_1 → M_2 such that, 
                
                      {Dϕ (R_ M_1(q)); q∈ M_1}= {R_ M_2(p); p∈ M_2}.
            
            Here Dϕ:  T  M_1 →  T M_2 is the differential of ϕ, i.e., the tangential mapping of ϕ. 
            
            
            We denote the set of all C^∞-smooth Riemannian metrics on N by ℳet(N) (we will assign the smooth Whitney topology (see <cit.>) on ℳet(N) to make it into a topological space), it turns out that there exists a generic subset (a set that contains an countable intersection of open and dense sets) G⊂ℳet(N) such that for g∈ G, the manifold (M,g) is determined by the data (<ref>) up to an isometry. 
            
            Given g∈(N),  p,q∈ N and ℓ>0, we denote the number of g-geodesics connecting p and q of length ℓ by I(g,p,q,ℓ). Define 
            
                I(g):=sup_p,q,ℓI(g,p,q,ℓ).
            
            In Theorem 1.2 of <cit.> it is shown that there exists a generic set G ⊂(N), such that for all g∈ G, 
            
                I(g)≤ 2 n+2.
            
            
            Next we define the collection of admissible Riemannian manifolds. Denote
            
                𝒢={(N,g); N  is a connected, closed, smooth Riemannian n-manifold;  g  satisfies (<ref>)}.
             Our main result is as follows,
            
            
            Let (N_i,g_i) ∈𝒢,   i=1,2 be a smooth, closed, connected Riemannian n-manifold n≥ 2, M_i ⊂ N_i be an open set with smooth strictly convex boundary with respect to g_i. Suppose that (M_i,g_i), i=1,2  is non-trapping, in the sense of (<ref>).
            
            If  properties (<ref>)–(<ref>) hold, then (M_1,g|_M_1) is Riemannian isometric to (M_2,g_2|_M_2).
            
            
            The auxiliary results needed to prove the main theorem will be stated and proven in such a way that we will use the data (<ref>) to reconstruct an isometric copy of (M,g). Therefore we formulate also the following theorem.
            
            Let (N,g) ∈𝒢 be a smooth, closed, connected Riemannian n-manifold n≥ 2, M ⊂ N be an open set with smooth strictly convex boundary with respect to g. Suppose that (M,g),  is non-trapping, in the sense of (<ref>). 
            
            If the data (<ref>)  is given we can reconstruct a isometric copy of (M,g).
            
            
            We want to underline that in the most parts  the proofs for Theorems <ref> and <ref> go side by side! Therefore we will formulate the lemmas and propositions, given in Section 2 and later, in a way having this dual goal in mind.
            
            
            
             §.§ Outline of this paper
            
            The proof of Theorem <ref> is contained in Sections 2 to 5. In Section 2 we recall some properties of the strictly convex boundaries and the first exit time function τ_exit. We will also give a generic property related to the generic property (<ref>) which guarantees that the scattering set R_ M(p) is related to a unique point p ∈ M. The main result of Section 3 is to show that the data (<ref>) determines uniquely the topological structure of the manifold M. Section 4 is devoted to the smooth structure of M and the final section shows that the data (<ref>) determines the Riemannian structure of (M, g).
            
            
             §.§ Previous literature
            
            
            The research topic of this paper is related to many other inverse problems. 
            
              §.§.§ Boundary and scattering rigidity problems
            
            
            The scattering data of internal sources (<ref>) considered in our paper takes into account the geodesic rays emitted from all the points of M, though we only know the exit positions and directions of these geodesics on the boundary. Instead of the  scattering data of internal sources, the  scattering relation is defined as follows
            
                L_g:_+SM→_-SM,   L_g(x,ξ):=(γ_x,ξ(τ_exit(x,ξ)),γ̇_x,ξ(τ_exit(x,ξ))),
            
            where 
            
                _±SM:={(x,ξ)∈ SM; ±⟨ξ,ν(x)⟩_g≤ 0}
            
            are the sets of inward and outward pointing vectors on the boundary M. Here ν is the outward pointing unit normal to M. Notice that M is non-trapping, thus L_g is well-defined. We will see later that scattering data of internal sources ( M, R_ M(M)) determines the scattering relation L_g. The scattering rigidity problem asks: if L_g=L_g̃, does it implies that g̃=ψ^* g with ψ:M→M a diffeomorphism fixing the boundary?  
            
            
            The scattering rigidity problem is closely related to the boundary rigidity problem, which is concerned with the determination of the metric g (up to a diffeomorphism fixing the boundary) from its boundary distance function d_g:  M× M→ℝ. Notice that given x,y∈ M, d_g(x,y) equals the length of the distance minimizing geodesic connecting x and y. See <cit.> for recent surveys on these topics. In particular, the scattering rigidity problem and boundary rigidity problem are equivalent on simple manifolds. A compact Riemannian manifold M is
            simple if the boundary M is strictly convex and any two points can be joined by a
            unique distance minimizing geodesic. Michel <cit.> conjectured that simple manifolds are boundary distance rigid, so far this is known for simple surfaces <cit.>. More recently, boundary rigidity results are established on manifolds of dimension 3 or larger that satisfy certain global convex foliation condition <cit.>. In the current paper, since the scattering data of internal sources contains more information than the scattering relation, we can deal with more general geometry.
            
            
            It is worth mentioning that the problem of recovering Riemannian 
            manifolds by the length data of geodesic rays emitted from internal 
            sources was considered in <cit.>.
            
            
            
              §.§.§ The rigidity of broken geodesic flows
            
            
            Another related inverse problem is concerned with determining a manifold (M ,g) from
            the broken geodesic data, which consists of the initial and
            the final points and directions, 
            and the total length, of the broken geodesics. To define the data we first set up the notations for a broken geodesic
            
                α_x,ξ_+,z,η(t)={[ γ_x,ξ_+(t),   t < s,   (x,ξ_+) ∈_+SM;   γ_z,η(t-s),   t ≥ s,   (z,η) ∈ SM, ].
            
            where z=γ_x,ξ_+(s).  Denote the length of the curve α_x,ξ_+,z,η by ℓ(α_x,ξ_+,z,η), the Broken scattering relation is 
            
                [                                                                 ℛ                                                                 =                                {((x,ξ_+),(y,ξ_-),t)∈_+SM×_-SM×_+;;                                                                                                                                     t = ℓ(α_x,ξ_+,z,η)  and  (y,ξ_-)=(α_x,ξ,z,η(t),∂_t α_x,ξ,z,η(t)),;                                                                                                               for some  (z,η)∈ SM}. ]
            
            In <cit.> the authors show that the broken scattering data ( M, ℛ) determines the manifold (M,g) uniquely up to isometry, if dim M≥ 3. In particular, there is no restrictions on the geometry of the manifold. 
            
              §.§.§ Inverse boundary value problem for the wave equations
            
            Next we will present a third well known inverse problem, where the data is given in the boundary of the manifold. Consider the following initial/boundary value problem for the Riemannian wave equation
            
                [            _t^2 w - Δ_g w = 0   in  (0, ∞) × M,;                                     w|_× M = f,; w|_t = 0 = _t w|_t= 0 = 0,   f∈ C_0^∞((0,∞)× M) ]
            
            where Δ_g is the Laplace-Beltrami operator of metric tensor g. It is well known that for every f∈ C_0^∞((0,∞)× M) there exists a unique w^f ∈ C^∞((0,∞)×M) that solves (<ref>) (see for instance <cit.> Chapter 2.3). Thus the Dircihlet-to-Neumann operator 
            
                Λ_g:C_0^∞((0,∞)× M) → C^∞((0,∞)× M),  Λ_gf(t,x)=⟨ν(x), ∇_g w^f(t,x) ⟩_g
             
            is well defined and an inverse problem to (<ref>) is to reconstruct (M, g) from the data
            
                ( M, Λ_g).
            
            One classical way to solve this problem is the Boundary Control method (BC). This method was first developed by Belishev for the acoustic wave equation on ^n with an isotropic wave speed
            <cit.>. A geometric version of the method, suitable when the wave speed is given by a Riemannian metric tensor as presented here, was introduced by Belishev and Kurylev <cit.>. The BC-method is used to determine the collection 
            
                BDF(M):={d_g(p,·)|_ M;   p ∈ M}
              
            of boundary distance functions. We emphasize the connection between our data R_ M(M) and BDF(M), that is for any p ∈ M and  for every z ∈ M, it holds that holds
            
                ∇_ M( d_g(p,·)|_ M)|_z∈ R_ M(p),
             
            if d_g(p, ·) is smooth at z. Above ∇_ M is the gradient of a Riemannian manifold ( M, g|_ M). One significant difference is that BDF(M) contains only information about the velocities of the short geodesics as R_ M(p) contains also the information from long geodesics. We refer to <cit.> for a thorough review of the related literature.
            
            Also  the case of partial data has been considered. Let 𝒮, ℛ⊂∂ M be open and nonempty. In this case the inverse problem is the following. Does the restriction operator 
            
                Λ_g,,f:=Λ_gf|__+ ×,   f∈ C^∞_0(_+ ×)
              
            with , determine (M,g)? The answer is positive if =, (see <cit.>). In <cit.> the problem has been solved in the case ∩≠∅. The general case is still an open problem.  So far the sharpest results are <cit.>.    
            
            
            
              §.§.§ Distance difference functions
            
            
            In <cit.> the collection of distance difference functions
                {(d_g(p,·)-d_g(p,·))|_N ∖ M; p∈ M}⊂ L^∞(N ∖ M),
            
            was studied. The authors show that (N ∖ M, g|_N ∖ M) with the collection (<ref>) determines the Riemannian manifold (N,g) up to an isometry, if N is a compact smooth manifold of dimension two or higher and M is open and has a smooth boundary. In the context of this paper one can assume that at the unknown area M there occurs an Earthquake at an unknown point p∈ M at an unknown time t≥ 0. This Earthquake emits a seismic wave. At the every point of the measurement area N ∖ M there is a device that records the time when the seismic wave hits the corresponding point. This way we obtain the function (z,w)↦ D_p(z,w):=d_g(p,z)-d_g(p,w),   z,w ∈ N ∖ M that is the travel time difference of the seismic wave.
            
            Fix z ∈ M. Suppose that for every p ∈ M the corresponding distance difference function D_p(·,z): M → is smooth. Then there is a close connection between R_ M(p) and
            
                R_ M^1(p):={∇_ MD_p(·,z)|_q;q∈ M},
            
            Notice that there exists non-trapping manifolds M with strictly convex boundary such that for a given p ∈ M the mapping M∋ q ↦ d_g(q,p) is not smooth.
            
            Moreover to prove the Main theorem <ref> of this paper we will use techniques that where introduced in <cit.>. These techniques are based on the Theorem 1 of <cit.>.
            
            
              §.§.§ Spherical surface data
             Finally we will present one more geometric inverse problem where a geodesical measurement data is considered.  Let (N,g) be a complete or closed Riemannian manifold of dimension n∈ and M⊂ N be an open subset of N with smooth boundary. We denote by U:=N∖M. 
            
            
            
            
            
            
            
            
            
            
            
            
            
            
            
            
            
            
            
            
            
            
            
            
            
            
            In <cit.> one considers the Spherical surface data 
            consisting of the set U  and the collection of all pairs (Σ,r) where Σ⊂ U is a smooth (n-1) dimensional submanifold
            that can be written in the form
            
                Σ=Σ_x,r,W= {exp_x(rv) ∈ N; v∈ W},
             
            where x∈ M, r>0 and W⊂ S_xN is an open and connected set. Such surfaces Σ are called spherical surfaces, or more precisely, subsets of generalised spheres of radius r. We point out that the unit normal vector field 
            
                ν_x,r,W(γ_x,v(r)):=γ̇_x,v(r),    v ∈ W
            
            of the generalized sphere Σ_x,r,W determines a data that has a natural connection to our data R_ M(M). 
            
            Also, in <cit.> one assumes that U is given with its C^∞-smooth coordinate atlas. 
            
            Notice that in general, the spherical surface Σ may be related to many centre points and radii. For instance consider the case where N is a two dimensional sphere.
            
            In <cit.> it is shown that the Spherical surface data determine uniquely the Riemannian structure of U. However these data are not sufficient to determine (N,g) uniquely. In <cit.> a counterexample is provided. In <cit.> it is shown that the Spherical surface data  determines the universal covering space of (N,g) up to an isometry. 
            
            In <cit.> a special case of problem <cit.> is considered. The authors study a setup where M ⊂^n,   n≥ 2 and the metric tensor g|_M=v^-2e, for some smooth and strictly positive function v. Let x ∈^n ∖ M and y :=γ_x,ξ(t_0) ∈ M, for some ξ∈ S_xN and t_0>0, such that y is not a conjugate point to x along γ_p,ξ. The main theorem of <cit.> is that, if the coefficient functions of the shape operators of generalized spheres Σ_y,r,W are known in a neighborhood V of γ_x,ξ([0,t_0)) and the wave speed v is known in (N ∖ M)∩ V, then the wave speed v can be determined in some neighborhood V' ⊂ V of γ_x,ξ([0,t_0]).
            
            
            
            § STRICTLY CONVEX MANIFOLDS, ABOUT THE  EXTENSION OF THE DATA AND THE GENERIC PROPERTY
            
            
             §.§ Analysis of strictly convex boundaries
            
            We will write (z(p),s(p)) for the boundary normal coordinates of M, for a point p near M. (See 2.1,  <cit.>, for the definitions). Here for any p ∈ dom((z,s)), the mapping p↦ z(p) stands for the closest point z(p)∈ M of p and p↦ s(p) is the signed distance to the boundary, i.e.,
            
                |s(p)|=_g(p, M)
            
            and
            
                s(p)<0  if  p ∈ M,   s(p)=0  if  p ∈ M  and  s(p)>0  if  p ∈ N ∖M.
            
            Thus every q ∈ M has a neighborhood U ⊂ N, such that the map U ∋ p ↦ (z(p),s(p)) is a smooth local coordinate system. We will write ν for the unit outer normal of M, therefore, ∇_g s(p)=ν for p∈ M. Therefore the distance minimizing geodesic from p to z(p) is normal to M.
            
            
            The second fundamental form of M is said to be positive definite, i.e., the boundary M as a submanifold is strictly convex, if the corresponding shape operator 
            
            
                S:T M → T M,   S(X)=∇_X ν
            
            is positive definite. Here ∇ is the Riemannian connection of the metric tensor g.
             
            Next we recall a few well known results (without proof) about manifolds with strictly convex boundary.
            
            Let (N,g) be a complete smooth Riemannian manifold and M⊂ N open subset with smooth boundary. Suppose that the second fundamental form of M is positive definite. Then for each (p,ξ) ∈ S M there exists ϵ>0 such that the geodesic γ_p,ξ(t)∉M for any t∈ (0,ϵ).
            
            Moreover if (p,ξ) ∈ SM∖ S M, then the maximal geodesic γ_p,ξ:[a,b] →M does not hit the boundary M tangentially. 
            
            
            
            
            
            
            
            
            
            
            
            
            
            
            
            
            
            
            
            
            
            
            
            Let (N,g) be a complete smooth Riemannian manifold and M⊂ N open subset with smooth boundary. Suppose that the second fundamental form of M is positive definite. Let (p,ξ) ∈ S M. Then
            
                γ_p,ξ((0,τ_exit(p,ξ)))⊂ M.
            
            The claim follows from Lemma <ref> and (<ref>).
            
            If M is strictly convex, then for any p ∈M there exists a neighborhood U ⊂ N of p such that for all z,q ∈ U∩M the unique distance minimizing unit speed geodesic γ from z to q is contained in U and γ(t)∈ M ∩ U for all t ∈ (0,d(z,q)). 
            
            
            
            
            
            
            
            
            
            
            
            
            
            
            
            
            
            
            
            
            
            
            
            
            
            
            
            
            
            
            
            
            
            
            
            
            
            
            
            
            
            
            
            
            
            
            
            If  (M,g) is compact, non-trapping and M is strictly convex, then the first exit time function τ_exit:SM→_+ given by (<ref>) is continuous and there exists L>0 such that 
            
                max{τ_exit(p,ξ);  (p,ξ)∈ SM}≤ L.
            
            Moreover, the exit time function τ_exit is smooth in S M∖ S M and in _+ SM.
            
            
            See Section 3.2. of <cit.>.
            
            
            
            
            
            
            
            
            
            
            
            
            
            
            
            
            
            
            
            
            
            
            
            
            
            
            
            
            
            
            
            
            
            
            
            
            
            
            
            
            
            
            
            
            
            
            
            
            
            
            
            
            
            
            
            
            
            
            
            
            
            
            
            
            
            
            
            
            
            
            
            
            
            
            
            
            
            
            
            
            
            
            
             §.§ Extension of the measurement data
            
            We start with showing that the data (<ref>) determines the boundary metric. This is formulated more precisely in the following lemma.
            
            Let (N,g) be a smooth, closed, connected Riemannian n-dimensional manifold n≥ 2, M⊂ N an open set with smooth boundary. Then (<ref>) determines the data
            
                (( M, g|_ M) ,R_ M(M)),
            
            where g|_ M= i^∗g and i: M ↪M. 
            
            More precisely if (N_i,g_i) and M_i are as in Theorem <ref> and  (<ref>)–(<ref>) hold, then
            
                ϕ:  M_1 → M_2 is a diffeomoprhism  such that ϕ^∗ (g_2|_ M_2)=g_1|_ M_1.
            
            We will start with showing that we can recover the metric tensor g|_ M. Choose p ∈ M, η∈ (T_p M∖{0}) and consider the set
            
                [                                                  R(η)                                                    := {ξ∈ T_p  M;  there exists  z ∈ M such that ξ∈ R_ M(z);                                                                           and  a>0,  that satisfies  ξ=aη}. ]
            
            It is easy to see that the set R(η) is not empty and by Lemma <ref> we also have
            
                {ξ_g ∈_+; ξ∈ R(η)}=(0,1) ⊂.
            
            Therefore, 
            
                η_g=inf{a>0;η/a∈ R(η)}.
            
            Since η∈ T_p M was arbitrary we have recovered the g-norm function ·_g:T_p  M →. Since norm ·_g is given by the g-inner product,  we recover ⟨·, ·⟩_g using the parallelogram rule, that is
            
                ⟨ξ, η⟩_g=ξ_g^2+η_g^2-ξ-η_g^2/2,   η, ξ∈ T_p  M.
             
            Suppose then that for manifolds (M_1,g_1) and (M_2,g_2) properties (<ref>)–(<ref>) are valid. Choose p ∈ M_1 and η∈ T_p  M_1 and denote R(η) as in (<ref>). Then by (<ref>)–(<ref>) we have
            
                Dϕ (R(η))=R(D ϕη)⊂ T_ϕ(p) M_2.
            
            and by the equations (<ref>)–(<ref>) we have η_g_1=D ϕη_g_2  and equation (<ref>) follows from the parallelogram rule. 
            
            
            Let p ∈M, we define the complete scattering set of the point sourcep as 
            
                [                          R^E_ M(p)                                  =   {(q,η)∈_- SM;   there is ξ∈ S_pN;                                    and  t∈ [0,τ_exit(p,ξ)]  such
                that;                                             q=γ_p,ξ(t), η=γ̇_p,ξ(t)}. ]
            
            We emphasize that the difference between R_ M(p) and R_ M^E(p) for p ∈M is that R_ M(p) contains the tangential components of vectors ξ∈ R^E_ M(p). We denote 
            
            R^E_ M(M):={R^E_ M(p); p ∈ M}. In the next Lemma we will give an equivalent definition for (<ref>).
            
            [Corollary of Lemma <ref>]
            
            For any p ∈M
                R^E_ M(p)={(γ_p,ξ(τ_exit(p,ξ)),γ̇_p,ξ(τ_exit(p,ξ)));  ξ∈ S_pN}.
            
            Since M is strictly convex and M is non-trapping, the claim follows from the definition of R^E_ M(p) and Corollary <ref>.
            
            
            Let (N_i,g_i) and M_i be as in Theorem <ref> and suppose that (<ref>)–(<ref>) hold. Since M_i is compact, there exists r>0 and a neighborhood K_i ⊂M_i of M_i such that the mapping 
            
                f_i: M_i × (-r,0]→ K_i ⊂M_i,   f_i(z,s):=exp_z(sν(z))
            
            is a diffeomorphism. 
            
            By (<ref>) the map
            
                Φ: M_1 × (-r,0]→ M_2 × (-r,0], Φ(z,s)=(ϕ(z),s).
            
            is a diffeomorphism. Thus the mapping
            
                Φ:K_1 → K_2,  Φ=f_2 ∘Φ∘ f_1^-1,
            
            is a diffeomorphism between K_1 and K_2. Moreover the map Φ is a local representation of Φ in the boundary normal coordinates.
            
            Suppose that (N,g) and M are as in Theorem <ref>. Then the data (<ref>) determines the complete scattering data 
            
                (( M, g|_ M), R^E_ M(M)).
            
            
            More precisely if (N_i,g_i) and M_i are as in Theorem <ref> such that (<ref>) and (<ref>) hold, then 
            
                ⟨η, ξ⟩_g_1=⟨ D Φη, DΦξ⟩_g_2,  for all ξ,η∈ TM_1
            
            and
            
                {DΦ (R^E_ M_1(q)); q∈M_1}= { R^E_ M_2(p); p∈M_2}.
            
            We will start with showing that for every q ∈ M we can recover R_ M^E(q).
            Choose q ∈ M and let (p,η)∈_- SM be such that (p,η^T) ∈ R_ M(q). Then η^T_g < 1 and in the boundary normal coordinates x ↦ (z(x),s(x)) close to p  the vector η can be written as
            
                η=η^T+(√(1-η^T_g^2))ν.
            
            Since g|_ M is known we have recovered η in the boundary normal coordinates. Since q∈ M was arbitrary, we have recovered the collection R^E_ M(M).
            
            Next we will show that for every q ∈ M we can recover R_ M^E(q). Let q ∈ M and (p,η)∈_- SM. We will define a set
            
                [                                               Σ(p,η)                                                   :=      {R^E_ M(q') ∈ R_ M^E(M);    (p,η) ∈ R^E_ M(q')};                                                                                                         = {R^E_ M(q')∈ R_ M^E(M): q' ∈γ_p,-η(0,τ_exit(p,-η))}, ]
            
            with which we can verify if (p,η)∈ R^E_ M(q) or not. Notice that by Lemma <ref>Σ(p,η)=∅ if and only if γ_p, -η([0,τ_exit(p,-η)])∩ M is empty if and only if η is tangential to the boundary. 
            
            Suppose first that Σ(p,η) ≠∅. Then (p,η)∈ R^E_ M(q) if and only if there exists (q,ξ)∈ S_qN such that Σ(p,η)=Σ(q,ξ). Suppose then that Σ(p,η)=∅ and thus η is tangential to M. By Lemma <ref> holds τ_exit(p,η)=0. Therefore, η∈ R_ M^E(q) if and only if p=q. We conclude that we can always check with data (<ref>) if (p,η)∈_- SM is included in R^E_ M(q) or not. Since q∈ M was arbitrary, we have recovered the collection R^E_ M( M). Therefore, we have recovered the set R_ M^E(M).
            
            
            Next we will verify (<ref>) and (<ref>). Let (N_i,g_i) and M_i be as in Theorem <ref> such that (<ref>) and (<ref>) hold.  Choose p ∈ M_1 and ξ, η∈ T_p  M_1. By (<ref>), (<ref>) and (<ref>) we have
            
                ⟨ DΦξ, DΦη⟩_g_2=⟨ Dϕξ^T, Dϕη^T  ⟩_g_2+√(1-Dϕη^T_g_2^2)√(1-D ϕξ^T_g_2^2)=⟨ξ,  η⟩_g_1.
            
            Thus the equation (<ref>) is valid. 
            
            Next we will prove
            
                {DΦ (R^E_ M_1(q)); q∈ M_1}= { R^E_ M_2(p); p∈ M_2}.
             
            Let q ∈ M_1. By (<ref>) there exists z ∈ M_2 such that 
            
                R_ M_2(z)=Dϕ R_ M_1(q).
            
            
            
            
            
            Let (p,η) ∈  R^E_ M_1(q). Then η_g_1=1 and η^T∈ R_ M_1(q). Since Dϕη^T∈ R_ M_2(z) we conclude by (<ref>) that DΦ(p,η) ∈ R^E_ M_2(z).
            Therefore, we have DΦ (R^E_ M_1(q)) ⊂ R^E_ M_2(z). By symmetric argument we also have R^E_ M_2(z)⊂ DΦ( R^E_ M_1(q)). Thus we have proved the left hand side inclusion in  (<ref>). Again by symmetric argument we prove the right hand side inclusion in (<ref>).
             
            Next we will show that
            
                {DΦ (R^E_ M_1(q)); q∈ M_1}= { R^E_ M_2(p); p∈ M_2}.
            
            Let q ∈ M_1. By the definition of the mapping Φ and the equation (<ref>) it is enough to prove that
            
                DΦ (R^E_ M_1(q))=R^E_ M_2(ϕ(q)).
            
            Let (p,η) ∈  R^E_ M_1(q) and denote Σ(p,η) as in (<ref>).
            By equations (<ref>) and  (<ref>) it holds that the set
            
                [                          Σ(Φ p, DΦη) :={R_ M_2(z);z∈ M_2, DΦη∈ R_ M_2(z)};                                        ={DΦ R_ M_1(q');R_ M_1(q')∈Σ(p,η)} ]
            
            is empty if and only if Σ(p,η) is empty. Suppose first that Σ(p,η) is empty and thus p=q and η is tangential to the M_1. Therefore, by equation (<ref>) we have
            
                DΦη=Dϕη∈(R_ M_2(ϕ(q)) ∩ R^E_ M_2(ϕ(q))).
             
            Suppose then that Σ(p,η) is not empty. This implies that there exists a vector ξ∈ S_qN_1 that satisfies Σ(p,η)=Σ(q,ξ). Thus Σ(Φ p, DΦη)=Σ(Φ q, DΦξ) and this implies that DΦ(p,η) ∈ R_ M_2(ϕ(q)). This completes the left hand inclusion of (<ref>). Replace Φ by Φ^-1 to prove the right hand side inclusion of (<ref>). 
            
            Therefore, (<ref>) is valid.
            
            
            
            
            
            
            
            
            
            
            
            
            
            
            
            
            
            
            
            
            
            
            
            
            
            
            
            
             §.§ A generic property
            
            We will now formulate a generic property in (N) that is related to the complete scattering data (<ref>). 
            
            Let N be a smooth manifold and M⊂ N an open set. We say that a Riemannian metric g ∈(N) separates the points of set M if for all p,q ∈ M,   p≠ q there exists ξ∈ S_pN such that
            
                q ∉γ_p,ξ([-τ_exit(p,-ξ),τ_exit(p,ξ)]).
            
            This means that there exists a geodesic segment that starts and ends at the boundary of M and contains p, but does not contain  q.
            
            
            We emphasize that there are easy examples for N, M and g such that g does not separate the points of M. For instance let M ⊂ S^2 be a polar cap strictly larger than the half sphere. Consider any two antipodal points p,q ∈ M. Then the standard round metric does not separate the points p and q.
            
            Our next goal is to show that for every (N,g)∈𝒢, (where 𝒢 is as in  (<ref>)) and  M⊂ N  that is open, non-trapping in the sense of (<ref>) and has a smooth strictly convex boundary, the metric g separates the points of M. This will be used in Section 3 to show that for two points p,q ∈ M the complete scattering sets R^E_ M(p) and R^E_ M(q) coincide if and only if p is q.
            
            Let (N,g) be a compact Riemannian manifold. Let M⊂ N be open, non-trapping in the sense of (<ref>) and have a smooth strictly convex boundary. Suppose that the metric g does not separate the points of M. Then there exist p, q∈ M, 
            p≠ q, an interval I⊂, and a C^∞-map I ∋ s ↦(ξ(s), ℓ(s))∈ S_pN×, such that ξ̇(s)≠ 0 and q=γ_p,ξ(s)(ℓ(s)).
            
            Since g does not separate the points of M there are p,q∈ M, p≠q,  such that for all ξ∈ S_pN
            we have
            
                S(ξ)={s∈ (-τ_exit(p,-ξ),τ_exit(p,ξ)); exp_p(s ξ)=q }≠∅.
            
            We note that for all ξ∈ S_pN the set S(ξ) is finite due to inequality (<ref>). 
            
            Fix η∈ S_pN and enumerate S(η)={s_1,…,s_K} for some K∈. Then 
            
                -τ_exit(p,-η)<s_1<s_2<, …, <s_K<τ_exit(p,η).
             
            For each s_k∈{s_1,…,s_K} we choose a n-1-dimensional submanifold S_k of N such that
            
                γ_p,η(s_k)=q∈ S_k  and γ̇_p,η(s_k)⊥ T_qS_k.
            
            For instance we can define
            
                S_k:={exp_q(tv)∈ N; v ∈ S_qN,   v ⊥γ̇_p,η(s_k),    t ∈ [0,inj(q))},
            
            where inj(q) is the injectivity radius at q. Choose ϵ>0 and consider a ϵ-neighborhood W of γ_p,η([-τ_exit(p,-η),τ_exit(p,η)]). We will write S_k for the component of S_k ∩ W that contains q. If ϵ is small enough, there exists δ, δ'>0 such that
            for any 
            
                t ∈ [-τ_exit(p,-η),τ_exit(p,η)] ∖(⋃_k=1^K(s_k-δ,s_k+δ))
             
            holds 
            
                dist_g(S_k,γ_p,η(t))≥ 2δ',  for every  k ∈{1,…,K}.
            
            By the continuity of the exponential mapping, we can choose a smaller ϵ>0 such that there exists an open neighborhood V⊂ S_pN of η such that for any 
            
                t ∈ [-τ_exit(p,-η),τ_exit(p,η)] ∖(⋃_k=1^K(s_k-δ,s_k+δ))  and ξ∈ V
             
            holds 
            
                dist_g(S_k,γ_p,ξ(t))≥δ',  for every  k ∈{1,…,K}.
            
            
            
            
            
            
            
            
            
            
            
            
            
            Next we define a signed distance function 
            
                ϱ_k(t,ξ)={[ -dist_g(γ_p,ξ(t),S_k), (t,ξ) ∈ (s_k-δ,s_k]× V; dist_g(γ_p,ξ(t),S_k), (t,ξ) ∈ [s_k,s_k+δ)× V. ].
            
            See Figure (<ref>). 
            
            Choosing a smaller δ and V, if needed, it follows that the function 
            ϱ_k: (s_k-δ,s_k+δ)× V → is smooth. Then we have ϱ_k(s_k,η)=0 and
            
                |/ tϱ_k(t,η)|_t=s_k|=γ̇_p,η(s_k)^2=1.
            
            Therefore, by the Implicit function theorem there exists an open neighborhood V_k⊂ V of η and a smooth function f_k:V_k→ (s_k-δ,s_k+δ) that solves the equation
            
                ϱ_k(f_k(ξ),ξ)=0.
            
            Define an open neighborhood U of η by U:= ⋂_k=1^K V_k and sets U_k={ξ∈ U: exp_p(f_k(ξ)ξ)=q}. Then sets U_k are closed in the relative topology of U.
            By (<ref>), (<ref>) and (<ref>) it must hold that
            
                U=⋃_k=1^K U_k.
            
            
            We claim that for some k ∈{1, …, K} it holds that U_k^int≠∅. If this is not true, then the sets P_j:= U∖ U_j,   j∈{1,…, K} are all open and dense in the relative topology of U. Moreover by (<ref>) we have
            
                P:=⋂_j=1^KP_j=∅.
            
            This is a contradiction since U is a locally compact Hausdorff space and thus by the Baire category theorem, it should hold that the set P is dense in the relative topology of U. Thus there exists k ∈{1, …, K} for which it holds that U_k^int≠∅. Choose an open U' ⊂ U_k. In particular U' is open in S_pN and there exists ϵ'>0 and a C^∞-path ξ(s), s∈ (-ϵ',ϵ'), in U' such that ξ̇(s)≠0. Denoting   ℓ(s)= f_k(ξ(s)), we have
             q=γ_p,ξ(s)(ℓ(s)) for  s∈ (-ϵ,ϵ). Therefore, the curve s↦ (ξ(s),ℓ(s)) satisfies the claim of this Lemma.
            
            
            
            
            
            
            
             
            Let (N,g) ∈𝒢 and M be as in Theorem <ref>. Then g separates the points of M.
            
            We prove the proposition by contradiction. Given g∈𝒢, assume there are p,q∈ M, p≠ q such that for all ξ∈ S_pN, 
            
                q ∈γ_p,ξ([-τ_exit(p,-ξ),τ_exit(p,ξ)]).
            
            In particular (see Lemma <ref>), there exists an open interval (-ϵ, ϵ), such that for s∈ (-ϵ,ϵ), (ξ(s),ℓ(s)) is a C^1-path on S_pN× such that ξ̇(s)≠ 0 and q=γ_p,ξ(s)(ℓ(s)). This implies 
            
                [                                                  0                                                  =                                / s(γ_p,ξ(s)(ℓ(s)));                                                                                                     = γ̇_p,ξ(s)(ℓ(s))·d/dsℓ(s)+ℓ(s)· D _p|_ℓ(s)ξ(s)ξ̇(s);                                                                                                    =:                                           T_1+T_2. ]
            
            Since ξ(s)=1, it implies ξ(s)⊥ξ̇(s). By Gauss Lemma, we have γ̇_p,ξ(s)(ℓ(s))⊥ D _p|_ℓ(s)ξ(s)ξ̇(s), thus T_1⊥ T_2. Applying equation  (<ref>), we obtain T_1=T_2=0, therefore, d/ds(ℓ(s))=0. This is equivalent to saying that ℓ(s)≡ const, for s∈ (-ϵ,ϵ). This implies I(g)=∞. By equation (<ref>), we arrive a contradiction.
            
            
            § THE RECONSTRUCTION OF THE TOPOLOGY
            
            We define a mapping 
            
                R^E_ M: M→ 2^ SM,   p ↦ R^E_ M(p).
            
            The aim of this section is to show that the map R^E_ M is a homeomorphism, with respect to some suitable subset of 2^ SM and topology of this subset. Recall that ( M, g|_ M) is known, by the Lemma <ref>. Therefore we can talk about topological properties of 2^ SM. Thus the goal is to reconstruct a homeomorphic copy of (M, g). We start with showing that the map R^E_ M is one-to-one, if the generic property <ref> holds.
             Suppose that (N,g) and M are as in the Theorem <ref>.
            Then the mapping R^E_ M:M→ 2^ SM is one-to-one.
            
            We will prove the claim by contradiction. Therefore, we divide the proof into two separate cases.
            
            Suppose first that there is p∈ M,   q∈M such that R^E_ M(p)=R^E_ M(q), that is
            
                {(γ_p,ξ(τ_exit(p,ξ)),γ̇_p,ξ(τ_exit(p,ξ)));  ξ∈ S_pN}
                   =   {(γ_q,η(τ_exit(q,η)),γ̇_q,η(τ_exit(q,η)));  η∈ S_qN}⊂_-SM.
             
            By the definition of R^E_ M(p) it holds that 
            
            
                S_p  M⊂ R^E_ M(p)=R^E_ M(q).
            
            Therefore, we deduce that any tangential geodesic starting at p hits q before exiting M. Since M is strictly convex, this is true only if p=q.
            
            
            Suppose then that there exist p ∈ M,  q ∈M,    p≠ q such that R^E_ M(p)=R^E_ M(q). By the first part we may assume that q ∈ M. Let ξ∈ S_pN. Then for ±ξ we have
            
            
                (γ_p,±ξ((τ_exit(p,±ξ)),γ̇_p,±ξ((τ_exit(p,±ξ)))∈ R^E_ M(q).
            
            Therefore, it holds that
            
                q ∈γ_p,ξ((-τ_exit(p,-ξ),τ_exit(p,ξ))).
            
            Since ξ∈ S_pN was arbitrary, the equation (<ref>) is valid for every ξ∈ S_pN and we have proved that the metric q does not separate the points p and q. This is a contradiction with Proposition <ref> and therefore, p=q.
            
            
            
            
            
            
            
            
            To reconstruct the topology of M from R^E_ M(M), we need to first give a topological structure to the latter. Notice that on the unit tangent bundle S M there is a natural metric induced by the underlying metric g, namely the Sasaki metric. We denote the Sasaki metric associated with g by g_S. Then on the power set 2^S M, we assign the Hausdorff distance, i.e., given A, B∈ 2^SM
                d_H(A,B):=max{sup_a∈ Ainf_b∈ Bd_g_S(a,b), sup_b∈ Binf_a∈ Ad_g_S(a,b)}.
            
            
            However, in general the Hausdorff distance on 2^S M needs not to be metrizable. If we consider the subset 𝒞(S M):={ M}⊂ 2^S M, then (𝒞(S M), d_H) is a compact metric space by Blaschke selection theorem (see for instance <cit.>, Theorem 4.4.15). The topology on 𝒞(S M) thus is induced by the Hausdorff metric d_H. 
            We turn to consider a subspace 𝒞( SM):={  S M}⊂ 2^ SM of 𝒞(SM). Since the boundary M is compact, 𝒞( S M) is a compact metric space. 
            
            
            
            For any q ∈M, the set R^E_ M(q)⊂ S M is compact.
            
            
            
            
            
            Consider a continuous mapping E_q:S_qN →_-S M given by
            
                E_q(ξ)=(γ_q,ξ(τ_exit(q,ξ)),γ̇_q,ξ(τ_exit(q,ξ))).
            
            Since S_qN is compact, and E_q is continuous it holds by Corollary <ref> that
            
                R^E_ M(q)=E_q(S_qN)  is compact.
            
            
            
            
            
            
            
            
            
            
            
            By Lemma <ref> it holds that R^E_ M(M) ⊂𝒞( S M). From now on we consider the mapping 
            
                R^E_ M:M→𝒞( S M),   p ↦ R^E_ M(p).
            
            Let (X,d) be a complete metric space and 𝒞(X) be the collection of all closed subsets of (X,d). We say that a sequence (A_j)_j=1^∞⊂𝒞(X) converges to A ∈𝒞(X) in the Kuratowski topology, if the following two conditions hold:
            
              (K1) Given any sequence (x_j)_j=1^∞, x_j∈ A_j with a convergent subsequence x_j_k→ x as k →∞, then the limit point x is contained in A.
            
              (K2) Given any x ∈ A, there exists a sequence (x_j)_j=1^∞, x_j ∈ A_j that converges to x.
            
            The mapping R^E_ M:M→𝒞( S M) is continuous. 
            
            Let q ∈M and q_j ∈M,   j∈ be a sequence that converges to q. As S M is compact it holds that the Kuratowski convergence and the Hausdorff convergence are equivalent (see e.g. <cit.> Proposition 4.4.14). Thus it suffices to show that R^E_ M(q_j) converges to R^E_ M(q) in the space 𝒞( S M) in the sense of Kuratowski.
            
            First we prove (K1). Let (p_j,η_j) ∈ R^E_ M(q_j). Changing in to subsequences, if necessary, we assume that (p_j,η_j) → (p,η) ∈ S M as j →∞. Let ξ_j∈ S_q_jN be such that for each j∈,
            
                γ_q_j,ξ_j(τ_exit(q_j,ξ_j))=p_j  and γ̇_q_j,ξ_j(τ_exit(q_j,ξ_j))=η_j.
            
            As the first exit time function is continuous and S M is compact we may  with out loss of generality assume that (q_j,ξ_j) → (q',ξ) ∈ S_q'N,   q' ∈M and  τ_exit(q_j,ξ_j) →τ_exit(q',ξ). Since q_j → q, it holds that q'=q. 
            By the continuity of the exponential mapping and the first exit time function the following holds
            
                (p,η)=lim_j →∞(p_j,η_j)=lim_j →∞(γ_q_j,ξ_j(τ_exit(q_j,ξ_j)),γ̇_q_j,ξ_j(τ_exit(q_j,ξ_j)))
            
                =(γ_q,ξ(τ_exit(q,ξ)), γ̇_q,ξ(τ_exit(q,ξ))).
            
            Thus (p,η)∈ R^E_ M(q). This proves (K1).
            
            Next we prove (K2). Let (p,η) ∈ R^E_ M(q). Let ξ∈ S_qN  be such that
            
                γ_q,ξ(τ_exit(q,ξ))=p  and γ̇_q,ξ(τ_exit(q,ξ))=η.
            
            Since  q_j → q as j→∞ we can choose  ξ_j ∈ S_q_jN,  j ∈ such that ξ_j →ξ as j →∞. 
            
            Denote
            
                γ_q_j,ξ_j(τ_exit(q_j,ξ_j))=p_j  and γ̇_q_j,ξ_j(τ_exit(q_j,ξ_j))=η_j,
            
            since the exponential map and the first exit time function are continuous, then 
            
                (p_j,η_j) ∈ R^E_ M(q_j)  and  (p_j,η_j) → (p,η).
              
            This proves (K2).
            
            We conclude that R^E_ M(q_j) converges to R^E_ M(q) in the Kuratowski topology. 
            
            The mapping R^E_ M:M→ R^E_ M(M) ⊂𝒞( S M) is a homeomorphism.
            
            Note the R^E_ M:M→ R^E_ M(M) ⊂𝒞( S M) is continuous, one-to-one and onto, and M is compact and 𝒞( SM) is a metric space and thus a topological Hausdorff space. These yield that R^E_ M:M→ R^E_ M(M) is a homeomorphism.
            
            By the Proposition <ref> the manifold topology of the data set R^E_ M(M) is determined. Thus the topological manifold R^E_ M(M)  is a homeomorphic copy of M. The rest of this section is devoted to constructing a map from M_1 onto M_2 that we later show to be a Riemannian isometry.
            
            
            Define a map 
            
                DΦ:𝒞( S M_1) →𝒞( S M_2),   F ↦ DΦ(F).
            
            The map DΦ:𝒞( S M_1) →𝒞( S M_2) is a homeomorphism.
            
            We start by noticing that, if (X,d_X) and (Y,d_Y) are compact metric spaces, and f:X → Y is continuous, then the lift f:𝒞(X) →𝒞(Y),  f(K)=f(K) is well defined. Let (K_i)_i=1^∞⊂𝒞(X) be a sequence that converges to K ∈𝒞(X) with respect to Kuratowski topology. We will show that f(K_i) also converges to f(K) with respect to Kuratowski topology, and by <cit.> Proposition 4.4.14 this will imply that the lift f is continuous.
            
            Let (y_i)_i=1^∞,   y_i ∈ f(K_i) be a sequence with a convergent subsequence (y_i_k)_k=1^∞,   y_i_k∈ f(K_i_k), we denote the limit point by y∈ Y. Thus for each i_k, there exists x_i_k∈ K_i_k such that f(x_i_k)=y_i_k. Since X is a compact metric space the  sequence (x_i_k)_k=1^∞ has a convergent subsequence in X, we denote it by (x_i_k)_k=1^∞ again. 
            
            Notice that K_i→ K in the sense of Kuratowski convergence, we get x_i_k→ x ∈ K. By the continuity of f, one has 
            
                y=lim_k →∞y_i_k=lim_k →∞f(x_i_k)=f(x)∈ f(K).
            
            
            Let y ∈ f(K) and x ∈ K such that f(x)=y. Since K_i → K in the sense of Kuratowski, there exists a sequence (x_i)_i=1^∞,   x_i ∈ K_i that converges to x. On the other hand, f is continuous, thus the sequence f(x_i)∈ f(K_i) converges to f(x)=y. This completes the proof of the convergence of f(K_i) to f(K).
            
            
            Notice that by (<ref>) the mapping D Φ: TM_1 → TM_2 is a smooth invertible bundle map. In particular D Φ is continuous with a continuous inverse. Therefore, D Φ is a well defined homeomorphism according to the first part of the proof.  
            
            
            Next we define a mapping
            
                Ψ: M_1→M_2,  Ψ:= (R_ M_2^E)^-1∘DΦ∘ R^E_ M_1.
            
            By (<ref>) and Proposition <ref> the map Ψ is well defined. Now we prove the main theorem of this section.
            
            Let (N_i,g_i) and M_i be as in Theorem <ref> such that (<ref>) and (<ref>) hold. Then the map Ψ: M_1→M_2, is a homeomorphism such that Ψ|_M_1:M_1 → M_2 and Ψ|_ M_1=ϕ.
            By (<ref>), Proposition <ref> and Lemma <ref> the map Ψ is a  homeomorphism.
            
            Let p ∈ M_1. Suppose that q:=Ψ(p)∈ M_2, it holds that S_q  M_2⊂ R^E_ M_2(q).
            
            On the other hand by the proof of Lemma <ref> there is no p' ∈ M_1 such that S_p' M_1 ⊂ R^E_ M_1(p). Since D Φ is an isomorphism, we reach a contradiction and thus q ∈ M_2. Similar argument for the inverse mapping Ψ^-1 and p ∈ M_2 proves the second claim.
            
            Let p ∈ M_1, then S_p  M_1 ⊂ R^E_ M_1(p)  and by (<ref>)
                DΦ (S_p  M_1)= S_ϕ(p) M_2 ⊂ R^E _ M_2(ϕ(p)).
            
            Therefore, by the proof of Lemma <ref> we have Ψ(p)=ϕ(p).
            
            
            
            § THE RECONSTRUCTION OF THE DIFFERENTIABLE STRUCTURE
            
            In this section we will show that the map Ψ:M_1→M_2 is a diffeomorphism. First we will introduce a suitable coordinate system of smooth manifold M_i that is compatible with the data (<ref>). We will consider separately coordinate charts for interior and boundary points. Then we will define a smooth structure on topological manifold R^E_ M_i(M_i) such that the maps R^E_ M_i: M_i → R^E_ M_i(M_i) are diffeomorphisms. The third task is to show that  the map D Φ:R^E_ M_1(M_1) → R^E_ M_2(M_2) is a diffeomorphism with respect to smooth structures of R^E_ M_i(M_i). By the formula (<ref>), that defines the map Ψ, the following diagram commutes 
            
                [            M_1      R^E_ M_1⟶  R^E_ M_1(M_1);                              ;             ↓Ψ                           ↓DΦ;                              ;            M_2      R^E_ M_2⟶ R^E_ M_2(M_2), ]
            
            and the above steps prove that the map Ψ is a diffeomorphism. Also we will  explicitly construct the smooth structure for R^E_ M(M) only using the data (<ref>). In the steps below, we will often consider only one manifold and do not use the sub-indexes, M_1 and M_2, when ever it is not necessary. 
            
            
            
             §.§ The recovery of self intersecting geodesics and conjugate points
            
            Let us choose a point p∈M for the rest of this section. The first part of the section is dedicated to finding for the point p a suitable q∈ M, a neighborhood V_q of p and to construct a map
            
                Θ_q(z):=1/exp_q^-1(z)_gexp_q^-1(z)∈ S_qN ,   z∈ V_q .
            
            using our data (<ref>). 
            
            We start with considering what does it require from q ∈ M to be "suitable“.
            
            
            Let (q,η)∈ R^E_ M(p), we say that η is a conjugate direction with respect to p, if p is a conjugate to q on the geodesic γ_q,-η. This is equivalent to the existence  s∈ (0,τ_exit(q,-η)) such that γ_q,-η(s)=p and a non-trivial Jacobi field J such that J(0)=0 and J(s)=0. 
            
            We say that a geodesic γ is self-intersecting at the point q ∈M, if there exists t_1<t_2 such that 
            
                γ(t_1)=q=γ(t_2).
            
            Notice that the geodesic γ_q,-η([0,τ_exit(q,-η)]) may be self-intersecting at p, and moreover there may be several s'∈  (0,τ_exit(q,-η)) such that γ_q,-η(s')=p and non-trivial Jacobi fields that vanish at 0 and s'.
            
            In the next Lemma we show that under the assumptions of Theorem <ref> most of the geodesics that start and end at M and hit the point p, do it only one time.
            
            
            
            
            
            
            
            
            
            
            
            
            
            
            
            
            
            
            
            
            
            
            
            
            
            
            
            
            
            
            
            
            
            
            
            
            
            
            
            
            
            
            
            
            
            
            
            Let (N,g) and M be as in the Theorem <ref>. Let p ∈M. Denote by
            
                [                                                            I                                                           := {ξ∈ S_pN;  the geodesic γ_p,ξ:[-τ_exit(p,-ξ),τ_exit(p,ξ)] →M;                                                                                              is self-intersecting at  p}. ]
            
            Then the set S_pN ∖ I is open and dense. 
            
            
            If r ∈ R_ M^E(M) is given and p∈M is the unique point for which R_ M^E(p)=r we define
            
                K(p):={(q,η) ∈ (R^E_ M (p) ∖ S_pN) ;  γ_q,-η is not self-intersecting at  p}⊂∂ SM.
            
            See Figure (<ref>). 
            
            Then K(p) is not empty and K(p) is determined by r and data (<ref>).
            
            More precisely if (N_i,g_i) and M_i are as in Theorem <ref> such that (<ref>) and (<ref>) hold, then for every  p ∈M_1 the set K(p)≠∅ and 
            
                DΦ (K(p))=K(Ψ(p)).
            
            Suppose first that p ∈ M. We start with proving that I is closed. Let ξ∈I and choose a sequence (ξ_j)_j=1^∞⊂ I that converges to ξ in S_p N. Choose a sequence (t_j)_j=1^∞⊂ that satisfies
            
                exp_p(t_jξ_j)=p  and  0<|t_j|.
            
            By (<ref>) we can without loss of generality assume that
            
                t_j ⟶ t  as  j ⟶∞.
            
            Then it must hold that 0< |t| since any geodesic starting at p cannot self-intersect at p before time inj(p)>0, where inj(p) is the injectivity radius at p. Therefore, exp_p(tξ)=p and we have proved that ξ∈ I. Thus I is closed. 
            
            From the proofs of Lemma <ref> and Proposition <ref> it follows that the set I is nowhere dense, this is that I does not contain any open sets. Thus S_pN ∖ I is open and dense. Moreover the set
            
                K(p)={(γ_p,ξ(τ_exit(p,ξ)),γ̇_p,ξ(τ_exit(p,ξ)))∈_-SM;   ξ∈ S_pN ∖ I}
             
            is not empty. 
            
            Denote r = R_ M^E(p). Next we will show that r and data (<ref>) determine the set K(p). Let (q,η) ∈ r. We write J(q,η) for the set of self-intersection points of geodesic segment γ_q,-η:(0,τ_exit(q,-η)) → M. Due to non-trapping assumption of M the set J(q,η) is finite. Since the map R_ M^E: M→ R_ M^E( M) is a homeomorphism, it holds that 
            
                z ∈γ_q,-η((0,τ_exit(q,-η))) ∖  J(q,η)
             
            if and only if R_ M^E(z) has a neighborhood V ⊂ R_ M^E(M) such that Σ(q,η) ∩ V
            is homeomorphic to interval [0,1) or (0,1), here Σ(q,η) is as in (<ref>). Therefore, for a given r=R_ M^E(p) and (q,η)∈ r the data (<ref>) determines the set R_ M^E(J(q,η)) and it holds that 
            
                (q,η) ∈ K(p)  if and only if  r ∉ R_ M^E(J(q,η)).
            
            
            Finally we will verify the equation (<ref>) in the case of p ∈ M_1. Denote r=R_ M_1^E(p). By Theorem <ref> the map Ψ is a homeomorphism. With (<ref>) this implies that for every (q, ξ)∈_-SM_1
                D Φ(Σ(q,ξ))=Σ(ϕ (q),DΦξ)
            
            and the self-intersection points of Σ(y,ξ) are mapped onto self-intersection points of Σ(ϕ(y),DΦξ) under the map D Φ. Therefore, DΦ (K(p))=K(Ψ(p)).
            
            
            Suppose next that p ∈ M.  Assume that there exists a sequence (ξ_j)_j=1^∞⊂ I that converges to a vector ξ∈ S_p M in S_pN. Choose a sequence (t_j)_j=1^∞ such that
            
                exp_p(t_jξ_j)=p  and  0<t_j.
            
            Due to Lemmas <ref> and <ref> we may assume that t_j → 0 as j →∞. But then we have a contradiction with the injectivity radius of p. Therefore, I is contained in the interior of _+ S_pM. Then by a similar argument as in the case of p ∈ M we have shown that S_pN ∖ I is open and dense. 
            
            Let (y,ξ) ∈_- SM.  We will use a short hand notation
            
                Σ(y,ξ)   :=   {R_ M^E(q); q ∈M,   (y,ξ) ∈ R_ M^E(q))}
                   =   R^E_ M(γ_y,-ξ([0,τ_exit(y,-ξ)]))⊂ 2^C(∂ SM),
            
            for the image of the geodesic segment γ_y,-ξ([0,τ_exit(y,-ξ)]) under the map R^E_ M. 
            Denote r = R_ M^E(p). Then for a vector (q,η)∈ r it holds that (q,η)∈ K(p) if and only if q ≠ p. Thus the set 
            
                K(p)={(γ_p,ξ(τ_exit(p,ξ)),γ̇_p,ξ(τ_exit(p,ξ)))∈_-SM;   ξ∈ (_+ S_pM)^int∖ I}.
              
            is not empty.
            
            Now we will verify the equation (<ref>) in the case of p ∈ M_1. Since the map Ψ is a homeomorphism we have by the data (<ref>) that
            
                D Φ (Σ(y,ξ))=Σ(ϕ (y),D Φξ)  for all  (y,ξ)∈_-SM_1.
            
            Therefore, DΦ(K(p))=K(Ψ(p)) and the equation (<ref>) is proven.
            
            
            
            In the next Lemma we will show that we can find the set of conjugate directions with respect to p from data (<ref>) and there exist lots
            of (q,η) ∈ r:=R_ M^E(p) such that η is not a conjugate direction with respect to p. If (q,η) ∈ K(p) is not a conjugate direction with respect to p, we will later construct coordinates for p such that (n-1)-coordinates are given by η.
             
            
            Let (N,g) and M be as in the Theorem <ref>. If r ∈ R_ M^E(M) is given and p∈M is the unique point for which R_ M^E(p)=r, we define the set of "good“ directions
            
                [                                                               K_G(p):= {(q,η) ∈ K(p); η is not a conjugate direction  with respect to p}⊂ SM. ]
             
            Then K_G(p) is not empty, π(K_G(p))⊂ M is open. Moreover r and the data (<ref>) determine the set K_G(p).
            
            More precisely if (N_i,g_i) and M_i are as in Theorem <ref> such that (<ref>) and (<ref>) hold, then for every  p ∈M_1 the set K_G(p)≠∅, π(K_G(p))⊂ M_1 is open and 
            
                DΦ (K_G(p))=K_G(Ψ(p)).
            
            Suppose first that p ∈ M. Let δ:SN → [0,∞] be the cut distance function, i.e.,
            
            
                δ(y,η):=sup{t>0;d_g(γ_y,η(t),y)=t}.
            
            
            Let ξ∈ S_pN be such a unit vector that γ_p,ξ is a shortest geodesic from p to the boundary M. By Lemma 2.13 of <cit.> it holds that 
            
                δ(p,ξ)>dist_g(p,  M).
             
            Therefore, (γ_p,ξ(τ_exit(p,ξ)),γ̇_p,ξ(τ_exit(p,ξ))) is not a conjugate direction with respect to p. Moreover there exists an open neighborhood V⊂ S_pN of ξ such that for any ξ' ∈ V the vector (γ_p,ξ'(τ_exit(p,ξ')),γ̇_p,ξ'(τ_exit(p,ξ'))) is not a conjugate direction. Let I ⊂ S_pN be defined as in (<ref>). Denote
            
                V_c:={ξ∈ S_pN; Dexp_p|_τ_exit(p,ξ)ξ is not singular }
            
            that is open and non-empty, since V ⊂ V_c. Therefore, by the Lemma <ref> it holds that the set V_c ∩ (S_pN ∖ I)=V_c ∖ I is open and non-empty. Thus
            
                K_G(p)={(γ_p,ξ(τ_exit(p,ξ)),γ̇_p,ξ(τ_exit(p,ξ)))∈ (_-SM∖ S_pN);  ξ∈ V_c ∖ I }
             
            is not empty and moreover π(K_G(p)) ⊂ M is open. 
            
            
            Next we show that for given r=R_ M^E(p) the data (<ref>) determines the set K_G(p), if r∈ R^E_ M(M). Fix (q,η)∈ K(p). Let t_p ∈ (0,τ_exit(q,-η)] be such that γ_q,-η(t_p)=p. Consider the collection H(q,η) of C^∞-curves σ:(-ϵ,ϵ)→_- S M,  σ(s)=(q(s),η(s)) that satisfy the following conditions
            
              (H1)σ(0)=(q,η), 
            
            
              (H2)r∈Σ(q(s),η(s)) for all s ∈ (-ϵ,ϵ)
              (H3)r ∉ R_ M^E(J(q(s),η(s))).
            
            Here we use the notations from Lemma <ref>
                Σ(q(s),η(s))=R^E_ M(γ_q(s),-η(s)((0,τ_exit(q(s),-η(s))))),
            
            and 
            J(q(s),η(s)) is the collection of self-intersection points of the geodesic segment
            
            γ_q(s),-η(s):(0,τ_exit(q(s),-η(s))) →M. Choose a C^∞-curve s ↦σ(s) on _-SM, such that (H1) holds. By (<ref>) we can check, if the property (H2) holds. By the proof of Lemma (<ref>) we can check, if property (H3) holds  for the curve σ. Therefore, for a given r∈ R_ M^E(M) and (q,η) ∈ K(p) we can check the validity of properties (H1)–(H3) for any smooth curve σ on _-SM with  the data (<ref>). 
            
            
            Next we show that the collection H(q,η) is not empty. Let ξ∈ S_pN be the vector that satisfies (γ_p,ξ(t_p),γ̇_p,ξ(t_p))=(q,η). Then τ_exit(p,ξ)=t_p. Let w ∈ T_pN, w ⊥ξ and ϵ>0 be small enough. We define
            
                w(s):=ξ+sw/ξ+sw_g and  f(t,s):=exp_p(tτ_exit(p,w(s))w(s)),   s∈ (-ϵ,ϵ), t∈ [0,1].
            
            Notice that the curve s↦ f(1,s)∈ M is smooth by Lemma <ref>. We conclude that a curve σ can be for instance defined as follows
            
                σ(s)=(q(s),η(s)):=(f(1,s),τ_exit(p,w(s))^-1/ tf(t,s)|_t=1),   s∈ (-ϵ,ϵ),
            
            where ϵ>0 is small enough. Observe that the curve σ(s) satisfies conditions (H1) and (H2) automatically. The condition (H3) is valid by the Lemma <ref>, since (q,η)∈ K(p). We also note that
            
                d/dsq(s)|_s=0=D exp_p|_t_pξ(d/dsτ_exit(p,w(s))|_s=0ξ+t_pẇ(0))
            
            and
            
                ẇ(0)=w.
            
            
            Now we show the relationship between conjugate directions and curves σ∈ H(q,η). Let us first consider some notations we will use. Let c:(a,b) → N be a smooth path and V a smooth vector field on c. By this we mean that t ↦ V(t)∈ T_c(t)N. We will write D_tV for the covariant derivative of V along the curve c. The properties of operator D_t are considered for instance in Chapter 4 of <cit.>. 
            
            We recall that locally D_t V is defined by the formula
            
                D_t V(t)=(d/dt V^k(t)+V^j(t)ċ^i(t)Γ_ji^k(c(t)))/ x^k(c(t)),
            
            where Γ_ji^k are the Christoffel symbols of the metric tensor g.
            
            
            Suppose first that γ_q,-η(t_p)=p is a conjugate point of q along γ_q,-η([0,τ_exit(q,-η)]). Then there is a non-trivial Jacobi field J on γ_p,ξ that vanishes at t=0 and at t=t_p. Then 
            
                D_t J(t)|_t=0≠ 0  and D_t J(t)|_t=0⊥ξ.
            
            Here D_t operator is defined on γ_p,ξ(t). Define a curve σ(s)=(q(s),η(s)) ∈ H(q,η) by (<ref>) and (<ref>) with w:=D_t J(t)|_t=0. We will show that 
            
                d/dsq(s)|_s=0=0  and  (D_sη(s)^T|_s=0)^T≠ 0.
            
            Here D_s is defined on q(s). Since the vector w is the velocity of J at 0 and J is a Jacobi field that vanishes at 0 and t_p we have by (<ref>) and (<ref>) that
            
                d/dsq(s)|_s=0=d/dsτ_exit(p,w(s))|_s=0η.
            
            Since s ↦ q(s) is a curve on M and η≠ 0 is not tangential to the boundary, it must hold that  
            
                d/dsτ_exit(p,w(s))|_s=0=0.
            
            Thus 
            
                d/dsq(s)|_s=0=0.
             
            Recall that the Jacobi field J satisfies due to (<ref>)
                J(t)=t Dexp_p|_tξ w=d/ds f(t',s)|_s=0,   t∈ [0,t_p],   t'=t/t_p.
             
            We will use the notation D_t' for the covariant derivative on the curve γ_p,t_pξ(t'). Then by (<ref>)–(<ref>), (<ref>)–(<ref>) and Symmetry Lemma (<cit.>, Lemma 6.3) we have
            
            
            
            
            
            
            
                D_sη(s)|_s=0=t_p^-1D_s/ t'f(t',s)|_t'=1,s=0=t_p^-1D_t'/ sf(t',s)|_t'=1,s=0
            
                = t_p^-1D_t J(t)|_t=t_p≠ 0,
            
            since J is not a zero field. Therefore, we conclude that D_sη(s)|_s=0≠ 0.  Moreover since η(s)_g≡ 1 we have η(s)⊥ D_sη(s). Suppose first that (D_sη(s)|_s=0)^T= 0, then D_sη(s)|_s=0 is normal to the boundary and thus η(0) is tangential to the boundary. This is a contradiction since η(0)=η, that is not tangential to the boundary. Thus we conclude that (D_sη(s)|_s=0)^T≠ 0. Write 
            
                η(s)=c(s)ν(q(s))+η(s)^T,  where c is some smooth function.
            
            Let x↦ (z^j(x))_j=1^n be the boundary normal coordinates  near q. Here we consider that z^n(x) represents the distance of x to M. Then the coefficient functions of ν are V^k=δ^n_k. Therefore for every k∈{1,…,n} the observation q̇(0)=0 yields
            
                (D_s ν(s)|_s=0)^k=(d/ds V^k(s)+V^j(s)q̇^i(s)Γ_ji^k(q(s)))|_s=0=0,
            
            here Γ_ji^k are the Christoffel symbols of metric g in boundary normal coordinates. Thus
            
                (D_s η(s)|_s=0)^T=
                c(0)D_s ν(q(s))|_s=0+(D_s η(s)^T|_s=0)^T=(D_s η(s)^T|_s=0)^T.
            
            Therefore, (<ref>) is valid.
            
            
            Next we consider curves σ(s)=(q(s),η(s))∈ H(q,η). Notice that for every σ(s) there exists ϵ>0 and a smooth function s↦ a(s)∈ (0,∞),   s∈ (-ϵ, ϵ) such that a(s)→ t_p as s→ 0 and 
            
                Γ(s,t):=exp_q(s)(-ta(s)η(s)),   t∈ [0,1],   s∈ (-ϵ, ϵ)
            
            is a geodesic variation of γ_q,-η([0,t_p]) that satisfies Γ(s,1)=p. Let us consider s ↦(q(s),-ta(s)η(s)), t∈ [0,1] as a smooth curve on TN and exp:TN → N. We use a short hand notation -tW(s):=-ta(s)η(s). We use coordinates (z^j,v^j)_j=1^n for π^-1U ⊂ TM, where U is the domain of coordinates (z^i)_i=1^n. Let (y^i)_i=1^n be coordinates at exp((q(0),-tV(0). 
            Then the variation field 
            
                [                                                        V(t)                                                          := / sΓ(s,t)|_s=0=(exp^j / z^iq̇^i(0)-texp^j / v_iẆ^i(0))/ y^j;                                                                                                                       =              (exp^j / z^iq̇^i(0)-tDexp_p|_-tW(0)Ẇ(0))/ y^j, ]
            
            is a Jacobi field that vanishes at t=1. By (<ref>) we have
            
                V^j(0)=q̇^j(0).
             
            Therefore by (<ref>) the point p is a conjugate point of q along γ_q,-η([0,τ_exit(q,-η)]), if there exists a curve σ∈ H(q,η) that satisfies 
            
            
                q̇(0)=0  and Ẇ(0)≠ 0.
            
            By the definition (<ref>)  of the operator D_s  and the vector field W(s) this is equivalent to
            
                q̇(0)=0  and  D_s(a(s)η(s))|_s=0≠ 0.
            
            
            Suppose that there exists such a curve σ(s)=(q(s),η(s))∈ H(q,η) for which (<ref>) is valid. Since V is a Jacobi field that vanishes at 1, we have by  (<ref>),  (<ref>) and the definition of σ(s) that
            
                0=V(1)=-Dexp_q|_-t_pη(D_s(a(s)η(s))|_s=0)
            
                =-ȧ(0)ξ- Dexp_q|_-t_pη(t_pD_sη(s)|_s=0).
            
            Thus ȧ(0)=0 and by (<ref>) it holds that D_sη(s)|_s=0≠ 0, which implies that (D_sη(s)|_s=0)^T≠ 0. Therefore, by similar computations as in (<ref>) we have 
            
            (D_sη(s)^T|_s=0)^T≠ 0. We conclude that, if there exists a curve σ∈ H(q,η) such that (<ref>) holds then, p is a conjugate point to q on γ_q,-η and moreover, 
            
                (D_s η(s)^T|_s=0)^T≠ 0.
            
            
            To summarize, for a given r ∈ R_ M^E(M) a vector (q,η) ∈ K(p) is a conjugate direction of p along geodesic γ_q,-η if and only if there exists σ∈ H(q,η) such that (<ref>) is valid. Moreover since s ↦ q(s) is a curve on the boundary M we can check the validity of (<ref>) for any σ∈ H(q,η) by data (<ref>). Therefore for given r∈ R_ M^E(M) the data (<ref>) determines the set K_G(p). 
            
             
            Next we will prove equation (<ref>) in the case of p ∈ M_1. Let (q,η)∈ K(p). Suppose that D Φη∈ K(Ψ(p)) is not in K_G(Ψ(p)). Then there exists a smooth curve s ↦σ(s)=(q(s),η(s)) ∈_- SM_2 that satisfies the properties (H1)–(H3) and for which (<ref>) is valid. 
            
            As it holds that Ψ is a homeomorphism and Φ is a diffeomorphism that preserves the boundary metric (in the sense of (<ref>)) the curve 
            
                s ↦ (ϕ^-1(q(s)), D Φ^-1η(s)):=(q(s),η(s))
             
            satisfies also the conditions (H1)–(H3) and moreover 
            
                q̇(0)=0  and (D_s η(s)^T|_s=0)^T=D Φ^-1 (D_s η(s)^T|_s=0)^T ≠ 0.
            
            Thus (<ref>) is valid for the curve (q(s),η(s)), which implies that (q,η) ∉ K_G(p). Thus (<ref>) is valid in the case of p ∈ M_1.
            
            
            Suppose then that p ∈ M. It follows from the Lemma <ref> that the set K_G(p) is not empty and
            
                K_G(p)={(γ_p,ξ(τ_exit(p,ξ)),γ̇_p,ξ(τ_exit(p,ξ)))∈_-SM ;  ξ∈(V_c ∖ (I ∪ S M) )}.
             
            Let (q,η) ∈ K_G(p). Let ξ∈ S_pN be the unique vector that satisfies η =γ̇_p,ξ(τ_exit(p,ξ)). Since Dexp_p is not singular at τ_exit(p,ξ)ξ, we can choose an open neigborhood U of τ_exit(p,ξ)ξ such that for every v ∈ U the differential map D exp_p|_v is not singular and exp_p(U) is an open neighborhood of q. Since q ≠ p it holds that τ_exit(p,ξ) >0. Therefore we may assume that U∩ T M = ∅ and U ⊂ V_c ∖ I, since I is closed. 
            Thus,
            
                exp_p(U)∩ M={γ_p,v(τ_exit(p,v));   v∈ h(U)}⊂π(K_G(p)),
             
            where h(v):=v/v_g and also in this case we deduce that π(K_G(p)) ⊂ M is open. 
            
            Next we show that K_G(p) is determined by r and data (<ref>), if r ∈ R_ M^E( M). Fix (q,η)∈ K(p). We note that q≠ p and by Lemma (<ref>) the vector η is not tangential to the boundary. Let t_p ∈ (0,τ_exit(q,-η)] be such that γ_q,-η(t_p)=p. Consider the collection H'(q,η) of C^∞-curves σ:(-ϵ,ϵ)→_- S M,  σ(s)=(q(s),η(s)) that satisfy the following conditions
            
              (H1')σ(0)=(q,η), q(s) ≠ p,  for any  s ∈ (-ϵ,ϵ)
              (H2')r∈Σ(q(s),η(s)) for all s ∈ (-ϵ,ϵ).
            
            
            Then by similar proof as in the case of p ∈ M we can show that H'(q,η) ≠∅, H'(q,η) is determined by r and data (<ref>) and (q,η) is a conjugate direction with respect to p if and only if there exists a curve σ∈ H'(q,η) for which equation (<ref>) is valid. 
            
            The equation (<ref>) is also valid in the case of p ∈ M_1 by a similar argument as in the case of p ∈ M_1.
            
            
            
            
            
            
            
            
            
            
            
            
            
            
            
            
            
            
            
            
            
            
            
            
            
            
            
            
            
            
            
            
            
            
            
            
            
            
            
            
            
            
            
            
            
            
            Now we are ready to consider the map defined in (<ref>). Choose (q,η) ∈ K_G(p) and let t_p<0 be such that exp_q(t_p η)=p. Let V_q⊂ M be a neighborhood of p. 
            Assuming that this neighborhood is small enough, there is a neighborhood U_q⊂ T_qN of t_p η such that the exponential map 
            
                exp_q:U_q→ V_q
            
            is a diffeomorphism.
            
            
            We emphasize that the mapping 
            
                Θ_q(z)=1/exp_q^-1(z)_gexp_q^-1(z)∈ S_qN ,   z∈ V_q .
            
            depends on the neighborhood U_q of t_p η.
            
            
            Let p,q ∈ N,   p≠ q. Let η∈ S_qN and t>0 be such that exp_q(tη)=p. Suppose that Dexp_q is not singular at tη and denote Dexp_q|_tηη=:ξ∈ S_pN. Then ker(DΘ_q(p)) =span(ξ).
            
            Let v ∈ T_pN. Then 
            
                DΘ_q(p)v=Dexp_q^-1v/t-η⟨ Dexp_q^-1v, η⟩_g/t,
            
            and the claim follows.
            
            In the next Lemma we will show that for given r=R^E_ M(p) and (q,η) ∈ K_G(p) the data (<ref>) determine the map Θ_q.
            
            Let (N,g) and M be as in the Theorem <ref>. If r ∈ R_ M^E(M) is given and p∈M is the unique point for which R_ M^E(p)=r, then for any (q,η) ∈ K_G(p) there exists a neighborhood V_q of p and a neighborhood of U_q of t_pη, where exp_q(t_pη)=p, such that exp_q^-1:V_q → U_q is well defined. Moreover, the map Θ_q:V_q → S_qN is smooth and well defined.
            
            The set R^E_ M(V_q) and the map Θ_q ∘ (R^E_ M)^-1:R^E_ M(V_q) → S_qN are determined from the data (<ref>) for given r ∈ R_ M^E(M) and (q,η)∈ K_G(p).
            
            More precisely, if (N_i,g_i) and M_i are as in Theorem <ref> such that (<ref>) and (<ref>) hold. Then for a given r=R^E_ M(p) ∈ R_ M^E(M_1) and (q,η)∈ K_G(p) it holds that
            
                Θ_ϕ(q)(Ψ(z))= DΦ (Θ_q(z)),   z ∈ V_q.
            
            Assume first that p ∈ M. Let (q,η)∈ K_G(p), then the existence of sets V_q and U_q follows. The map Θ_q is well defined and smooth since Θ_q=h∘exp_q^-1. Here h:T_qN∖{0}→ S_qN,   h(v):= v/v_g is smooth and well defined since q≠ p.
            
            
            Next we will show that the set V_q and the map Θ_q are determined from the data (<ref>), if (q,η)∈ K_G(p) is given. Let V ⊂ M be a neighborhood of p and U⊂ T_qN a neighborhood of t_pη. Denote U:=h(U) ⊂ S_qN. Since η∈ U, we may assume that for all z∈ V the set R^E_ M(z) ∩ U⊂ S_qN is not empty.  We define a set valued mapping P_q:V → 2^S_qN by formula
            
                P_q(z)= R^E_ M(z) ∩ U.
            
            Then P_q is well defined and for given r and (q,η)∈ K_G(p) we can recover P_q from (<ref>). We claim that, if V and U are small enough, then
            
                for all z ∈ V the set  R^E_ M(z) ∩ U⊂ S_qN  has a cardinality of  1.
            
            Therefore, P_q, coincides with Θ_q in V. We will show that if (<ref>) is not valid, then we end up in a  contradiction with the assumption (q,η)∈ K_G(p). We will divide the proof into two parts.
            
            Assume first that there exists a sequence (η_j)_j=1^∞⊂ R^E_ M(p) ∩ S_qN that converges to η and η_j ≠η for any j ∈. Choose t_j<0 such that exp_q(t_jη_j)=p. Due to (<ref>) we may assume that t_j converges to some t. Therefore, by the continuity of the exponential map, we have 
            
                p=lim_j→∞exp_q(t_jη_j)=exp_q(t η).
            
            Since (q,η) ∈ K(p) it must hold that t=t_p. As p=exp_q(t_jη_j) for every j ∈ and t_jη_j → tη in T_qN, we end up with a contradiction to the assumption (q,η) ∈ K_G(p). 
            
            
            Suppose next that there exist sequences (z_j)_j=1^∞⊂M and ξ^1_j,ξ_j^2 ∈ R^E_ M(z_j) ∩ S_qN,  ξ_j ^1 ≠ξ_j^2 for every j ∈ such that for i∈{1,2} holds
            
                z_j ⟶ p,  and ξ_j ^i ⟶η as  j ⟶∞.
            
            Choose t_j^1∈ (-τ_exit(q,-ξ_j^1),0), t_j^2∈ (-τ_exit(q,-ξ_j^2),0),   j ∈ such that for i∈{1,2}, holds exp_q(t_j^iξ_j^i)=z_j. Again by (<ref>) we may assume that t_j^i converges to t^i ∈ (-τ_exit(q,-ξ),0) as j →∞ for i ∈{1,2}, respectively. Therefore, we have 
            
                p=lim_j →∞z_j=lim_j→∞exp_q(t_j^iξ_j^i)=exp_q(t^iη).
            
            Since (q,η) ∈ K(p) it must hold that t^i=t_p for both i∈{1,2}. Therefore, t_j^iξ_j^i → t_pη in T_qN as j →∞ for both i∈{1,2}. Since we assumed that ξ_j^1≠ξ_j^2 for any j ∈, we have shown that mapping exp_q is not one–to–one near t_pη. Hence we end up again in a contradiction with the assumption (q,η)∈ K_G(p). 
            
            Thus we have proven the existence of such sets V and U for which (<ref>) is valid. Using data (<ref>) we can verify for a given r and the neighborhoods U of η and R^E_ M(V) of r whether the property (<ref>) is valid. We will denote by U_1 ⊂ S_qN a neighborhood of η and V_q a neighborhood of p that satisfy (<ref>). 
            
            The next step is to prove that for any r=R_ M_1^E(p)∈ R_ M_1^E(M_1) and (q,η)∈ K_G(p) the equation (<ref>) is valid. Choose neighborhoods U_1 ⊂ S_qN of η and V_q ⊂ M_1 of p for which (<ref>) is valid. By (<ref>) it holds that (ϕ(q),DΦη)∈ K_G(Ψ(p)). Since Ψ is a homeomorphism Ψ(V_q) is a neighborhood of Ψ(p) and due to (<ref>)DΦ(U_1)⊂ S_ϕ(q)N_2 is a neighborhood of DΦη. Therefore (<ref>) is valid for Ψ(V_q) and DΦ(U_1). Let z ∈ V_q and  {ξ}=  R^E_ M(z) ∩ U_1. Then {DΦξ} =R^E_ M(Ψ (z)) ∩ D Φ(U_1) and the equation (<ref>) follows.
            
             
            Finally we assume that p ∈ M. We notice that by Lemma (<ref>) it holds K_G(p) ∩ T M =∅ and by the definition of K_G(p) we have p ∉π(K_G(p)). Therefore, we can choose  for every (q,η) ∈ K_G(p) such a neighborhood V_q⊂M of p that q ∉ V_q. The rest of the proof is similar to the case p ∈ M.
            
            
             §.§ Interior coordinates
            
            For this subsection we assume that p ∈ M. Let (q, η) ∈ K_G(p) be such that a similar map Θ_q:V_q̃→ S_qN exists for some neighborhood V_q of p. Let v∈ T_qN and define a smooth map
            Θ_q,q, v(z)=(Θ_q(z),⟨v|,Θ_q̃(z)⟩_g )∈ S_qN× ,   z∈ V_q∩ V_q̃.
            
            See Figure (<ref>). 
            
            
            In the next Lemma we will show that this map is a smooth coordinate map near p. By Inverse function theorem it holds that a smooth map f:N →^n is a coordinate map near p ∈ N if and only if Df(p)≠ 0. Here Df(p) is the Jacobian matrix of f at p.
            
            
            
            Let (N,g) and M be as in the Theorem <ref>. If r ∈ R_ M^E(M) is given and p∈ M is the unique point for which R_ M^E(p)=r, then there exist (q,η),(q, η) ∈ K_G(p) and v∈ T_qN such that Θ_q,q,v is a smooth coordinate mapping from a neighborhood V_q,q̃,v of  p. 
            
            
            
            Moreover, for given r = R^E_ M(p) ∈ R_ M^E(M) and (q,η),(q, η) ∈ K_G(p) the data (<ref>) determines v∈ T_q̃N, a neighborhood R_ M^E(V_q,q̃,v)  of r and the map Θ_q,q,v∘ (R_ M^E)^-1: R_ M^E(V_q,q̃,v) → (S_qN × (∖{0})) such that Θ_q,q,v a smooth coordinate map from a neighborhood V_q,q̃,v of p onto some open set U ⊂ (S_qN × (∖{0})).
            
            
            
            
            More precisely, let (N_i,g_i) and M_i be as in Theorem <ref> such that (<ref>) and (<ref>) hold. Then for any p ∈ M_1 and  (q,η),(q, η) ∈ K_G(p) and v∈ T_qN the following holds: Let ξ∈ S_qN be outward pointing and s∈. Then
            
                Θ_ϕ(q),ϕ(q),DΦ v(Ψ(z))=(DΦξ,s)  if and only if Θ_q, q,v(z)=(ξ,s)
            
            and
            
                D Θ_q, q,v(p)≠ 0  if and only if   DΘ_ϕ(q),ϕ(q),DΦ v(Ψ(p)) ≠ 0.
            
            Choose  (q,η)∈ K_G(p). By Lemma <ref> the set π(K_G(p)) is open and, there exists (q, η) ∈ K_G(p), such that 
            
                q∉γ_q,-η([0,τ_exit(q,-η)]).
            
            By Lemma <ref> there exists a neighborhood V_q∩ V_q of p such that the mapping Θ_q,q,v is well defined and smooth for any v ∈ T_q N.   
            
            
            Next we show that there exists such a vector v∈ T_q N that the Jacobian DΘ_q,q,v is invertible at p. By the choice of (q,η) and (q, η) there exist t,t >0 and ξ,ξ∈ S_pN such that ξ and ξ are not parallel and  (γ_p,ξ(t),γ̇_p,ξ(t))=(q,η) and (γ_p,ξ(t),γ̇_p,ξ(t))=(q,η).
            Choose linearly independent vectors w_1, …, w_n-1∈ S_qN that are all perpendicular to η. Then vectors 
            
                ξ_i:=Dexp_q|_-t ηw_i ∈ T_pN,   i ∈{1, …, n-1}
            
            are linearly independent and perpendicular to ξ. Therefore, the vectors ξ_1, …, ξ_n-1,ξ span T_pN. 
            
            Let v ∈ T_qN be such that ⟨ v , Θ_q(p)⟩_g ≠ 0 and ⟨ v , DΘ_q(p)ξ⟩_g≠ 0. Notice that by Lemma <ref> such a v exists. Choose coordinates (x^1,…, x^n) at p such that
            
                ∂/ x^i(p)=ξ_i;   i ∈{1,…, n-1} and ∂/ x^n(p)=ξ.
            
            In these coordinates the Jacobian matrix of Θ_q,q,v at p is
            
                DΘ_q,q,v(p)=( [ V 0; a c ]),
            
            where V is the Jacobian  matrix of Θ_q at p with respect to (n-1) first coordinates, a, 0∈^n-1 and c = / x_n⟨ v , Θ_q(x)⟩_g|_x=p. Then by the choice of basis ξ_1,…, ξ_n-1, ξ and Lemma <ref> it holds that V is invertible. 
            
            
            
            
            Moreover
            
                c = / x^n⟨ v , Θ_q(x)⟩_g|_x=p=d/dt⟨ v , Θ_q(γ_p,ξ(t))⟩_g|_t=0=⟨ v , DΘ_q(p)ξ⟩_g≠ 0,
            
            
            
            
            and we conclude that
            
                DΘ_q,q,v(p)=(± 1) c  V≠ 0.
            
            This shows that there exist (q,η),(q, η) ∈ K_G(p) and v∈ T_q N such that the corresponding map Θ_q,q,v is a coordinate map in some neighborhood of p. Since the invertibility of the Jacobian DΘ_q,q,v(p) is invariant to the choice coordinates, we conclude that the map Θ_q,q,v is a smooth coordinate map at p.
            
             
            Now we will check that for a given r=R^E_ M(p) and (q,η),(q, η) ∈ K_G(p) the data (<ref>) determines v∈ T_q̃N, a neighborhood R_ M^E(V_q,q̃,v) of r and the map Θ_q,q,v∘ (R_ M^E)^-1:R_ M^E(V_q,q̃,v)→ (S_qN ×). By Lemma <ref>, the data (<ref>) determines the set K_G(p) and it is not empty. Choose  any (q,η)∈ K_G(p). By equation (<ref>) we can choose (q,η) ∈ K_G(p) such that (<ref>) is valid. By Lemmas <ref> and <ref> we can construct the neighborhoods V_q, V_q of p and the map Θ_q,q,v:V_q,q→ S_qN ×, where V_q,q:=V_ q∩ V_q, for any v ∈ T_qN. Suppose now on that the points (q,η) and (q, η) are given. By (<ref>) and (<ref>) we notice that a sufficient and necessary condition for the map Θ_q,q,v to be a smooth coordinate map at p is that vector v  is contained in A_q:=T_qN ∖ ((DΘ_q(p)ξ)^⊥∪Θ_q(p)^⊥) that is open and dense.
            
            Lastly we will introduce a method to test, if v ∈ T_qN is admissible, that is, DΘ_q,q,v(p)≠ 0. By the Propostion <ref> the map R_ M ^E:M→ R_ M ^E(M) is a homeomorphism. Thus we can determine the set 
            
                [                                                           B_q                                                             =   {w ∈ T_qN; the map Θ_q,q,w∘ (R_ M ^E)^-1 is a homeomorphism;                                                               from a neighborhood of R_ M ^E(p)  onto some open set V ⊂^n}. ]
            
            For all w,w' ∈ B_q we define a homeomorphism
            
                H_w,w':U_w,w'→ V_w,w',    H_w,w':=Θ_q,q,w∘Θ_q,q,w'^-1,
            
            where U_w,w', V_w,w'⊂^n are open. Let 
            
                C^2_q:={(w,w')∈ B_q× B_q;   H_w,w' is smooth and  (D H_w,w'(Θ_q,q,w(p))≠ 0)}.
            
            Thus A_q× A_q⊂  C^2_q and the data (<ref>), with R^E_ M(p),(q,η) and (q, η) determine C^2_q. Let (w,w')∈ C^2_q. Then by the Inverse function theorem also (w',w) is included in C^2_q. Thus  
            
                ((T_qN ∖ A_q ) × A_q  )∩ C^2_q=∅.
            
            Write C_q for the projection of C^2_q into T_qN with respect to first component. 
            
            Choose v ∈ C_q. We claim that v is admissible if and only if there exists an open and dense set A ⊂ T_qN such that
            
                {v}× A ⊂ C^2_q.
            
            If v is admissible, then for every w ∈ A_q holds (v,w)∈ C^2_q. Therefore, {v}× A_q⊂  C^2_q and (<ref>) follows. Suppose then that (<ref>) is valid for some open and dense set A. Then A∩ A_q is also open, dense and {v}×(A∩ A_q)⊂ C^2_q. Suppose that v ∈ T_qN ∖ A_q. Then it follows that
            
                ({v}×(A∩ A_q))∩ C^2_q≠∅.
             
            But this contradicts (<ref>) and it must hold that v ∈ A_q, which means that v is admissible. 
            
            If v, w ∈ T_qN,   v ≠ w are chosen and they both are admissible, then the map H_v,w: U_v,w→ V_v,w is determined by r, (q,η), (q, η) ∈ K_G(p) and data (<ref>). Let U'_v,w⊂ U_v,w be an open neighborhood of Θ_q,q,w(p) such that DH_v,w(z)≠ 0 for every z ∈ U'_v,w. We conclude that a neighborhood V_q,q, v⊂ V_q,q of p such that the map Θ_q,q,v:V_q,q, v→ S_qN × is a smooth coordinate map, can be defined for instance by formula
            
                V_q,q, v:=Θ_q,q, v^-1(U'_v,w).
            
            
            Finally we will prove the equations  (<ref>) and  (<ref>). Let p ∈ M_1. By (<ref>) and (<ref>)  we have that (q,η),(q, η) ∈ K_G(p) and (<ref>) holds if and only if the same is true for (ϕ (q),D Φη),(ϕ(q), DΦη). If  (q,η),(q, η) ∈ K_G(p) and (<ref>) holds then by (<ref>) and (<ref>) the equation (<ref>) follows. 
            
            Lastly we notice that B_ϕ(q)=D Φ (B_q). Therefore, we conclude that for given (q,η),(q, η) ∈ K_G(p) for which (<ref>) is valid the equation (<ref>) holds for v ∈ T_qN_1 if and only if the same holds for  D Φ v ∈ T_ϕ (q)N_2. Therefore, the equation (<ref>) follows from (<ref>). 
            
            
             §.§ Boundary coordinates
            
            
            Next we construct the coordinates near M. Let p ∈ M. Let I be the set  of self-intersection directions of p (see (<ref>)). By Lemma <ref> the set S_pN ∖ I is open and dense in S_pN. As p ∈ M the data (<ref>) determines the set I, since it holds that
            
                I={(p,ξ) ∈_-S_pM:  there exists η∈_-S_pM,  η≠ξ such that Σ(p,ξ)=Σ(p,η)}.
            
             
            Note that the usual boundary normal coordinates might not work well with our data, since it might happen that p is a self-intersecting point of the boundary normal geodesic γ_p,ν. In this case it is difficult to reconstruct the map U ∋ w ↦ z(w) ∈ M, from the data (<ref>). Here U is the domain of boundary normal coordinates and z(w) is the closest boundary point. To remedy this issue, we will prove in the next lemma that any non-vanishing, non-tangential, inward pointing vectorfield on M determines a similar coordinate system as the boundary normal coordinates.
            
            
            
            Let (M, g) be a smooth compact manifold with strictly convex  boundary and let W be a non-vanishing, non-tangential and inward pointing smooth vector field on M. Let p ∈ M.  Then there exists T>0  such that the map
            
                E_W:  M  × [0,T) →M,   E_W(z,t)=exp_z(tW(z))
            
            is well defined and moreover, there exists T_p ∈ (0,T) and ϵ>0 such that the restriction 
            
                E_W: [0,T_p) × B_ M(p,ϵ) →M
            
            is a diffeomorphism. Here 
            
                B_ M(p,ϵ):={z ∈ M: d_ M(p,z) < ϵ}
            
            and d_ M(p,z) is the distance from p to z along the boundary. 
            
            Since W(z) ∈ (_+ SM)^int for every z ∈ M and  M is compact, the Lemma <ref>  guarantees that there exists T>0 such that for every t ∈ [0,T] and z ∈ M it holds that E_W(t,p)∈M.
            
            Let (z^j)_j=1^n-1 be local coordinates at p on the boundary.  Since the map E_W is smooth it suffices to show that the Jacobian matrix DE_W is invertible at (p,0). 
            By straightforward computation we have
            
                / tE_W(p,t)|_t=0=W(p)  and / z^jE_W(z,0)|_z=p=  / z^j,   j ∈{1,…, n-1}.
            
            Since W(p) is not tangential to the boundary, it holds by (<ref>) that DE_W at (p,0) is invertible.
            
            
            Choose a non-vanishing, non-tangential inward pointing smooth vector field W on M such that W(p)∉ I=I_p. 
            By Lemma <ref> there exists such a neighborhood V of p such that the map  E_W^-1:V → M × [0,T_p) is a smooth coordinate map. Let w ∈ V. We denote 
            
                Π_W(w):=z∈ M  if and only if E_W(z,t)=w for some unique  t∈[0,T_p).
            
            See Figure (<ref>). We note that the map Π_W is smooth.
            
            
            
            In the next Lemma we show that the data (<ref>) determines a continuous map Π_W^E that coincides with Π_W near p. Later we will use the map Π_W^E to construct (n-1)-coordinates for points near p.  
            
            Let (N,g) and M be as in the Theorem <ref>. For given r ∈ R_ M^E(  M) where p∈ M is the unique point for which R_ M^E(p)=r and let W be a non-vanishing, non-tangential inward pointing smooth vector field such that W(p)∉ I,  the data (<ref>) with r and W,  determine such a (q,η) ∈ K_G(p) and a neighborhood V_q,W⊂M of p, q ∉ V_q,W for which the map Θ_q:V_q,W→ S_qN is well defined. Moreover, 
            
            
                γ̇_q,-Θ_q(p)(τ_exit(q,-Θ_q(p))) ∦ W(p)
            
            and
            
                a map Π^E_q,W:V_q,W→ M∩ V_q,W,   is determined by (<ref>).
            
            Here Π^E_q,W is a continuous map which coincides with Π_W near p.
            
            More precisely, let (N_i,g_i) and M_i be as in Theorem <ref> and suppose that (<ref>) and (<ref>) hold. 
            Let W be a smooth non-vanishing, non-tangential, inward pointing vector field on M_1 such that W(p)∉ I.
            If 
            
              (i)r=R^E_ M_1(p)∈ R^E_ M_1( M_1)
              (ii)(q,η) ∈ K_G(p)
              (iii)V_q,W is neighborhood of p such that q ∉ V_q,W and for every z ∈ V_q,W the property (<ref>) holds.
            
            are given. Then 
            
                [                                                                                                  γ̇_q,-Θ_q(p)(τ_exit(q,-Θ_q(p))) ∦ W(p);                                                      if and only if                                                                    ;                                                                     γ̇_ϕ (q),-Θ_ϕ(q)(ϕ(p)))(τ_exit(ϕ (q),-Θ_ϕ (q)(ϕ(p)))) ∦ DΦ W(ϕ(p)), ]
            
            and
            
                ϕ ( Π^E_q,W(z))= Π^E_ϕ(q), D Φ W(Ψ(z)),   for all  z ∈ V_q,W.
            
            We start by showing how property (<ref>) can be verified. Let R_ M^E(U) ⊂ R_ M^E(M) be a neighborhood of r=R_ M^E(p). For every  r' ∈ (R_ M^E(U)  ∩ R_ M^E( M)),   r' ≠ r we define the number N(r') to be the cardinality of the set 
            
                A(r,r'):={ξ∈ (S_zN) ∩ r;  Σ(z,ξ) ⊂ R^E_ M(U),   (z,ξ) ∈ K_G(p)},
            
            here z=(R^E_ M)^-1(r'). By Lemma <ref> there exists a neighborhood R_ M^E(U') ⊂ R_ M^E(U) of r such that N(r')=1 for every r' ∈ (R_ M^E(U')  ∩ R_ M^E( M)),   r'≠ r. Since the sets A(r,r') are determined by r and (<ref>) we can find the set R_ M^E(U'). We denote by  U' ⊂ U the unique neighborhoods of p that are related to R_ M^E(U') and  R_ M^E(U). We write z=(R^E_ M)^-1(r') ∈ U', for given r' ∈ R_ M^E(U'). Therefore, by Lemma <ref> for the vector ξ∈ A(r,r'), there exists a neighborhood V_z⊂ U' of p such that z ∉ V_z and the map Θ_z:V_z → S_zN is smooth and well defined. Moreover there exists a unique (p,v_z)∈ S_pN so that 
            
                γ_p,v_z(τ_exit(p,v_z))=z  and γ̇_p,v_z(τ_exit(p,v_z))=Θ_z(p)∈ A(r,r')
            
            and
            
                Σ(z,Θ_z(p))=Σ(p,v_z).
            
            Then v_z= γ̇_z,-Θ_z(p)(τ_exit(z,-Θ_z(p))) and thus the data (<ref>) determines, if v_z and W(p) are parallel. It follows from Lemma <ref> that there exists  q ∈ U' such that v_q is not parallel to W(p). Therefore we can use data (<ref>) to determine a point q that satisfies the property  (<ref>).
            
            Let (q,η) ∈ A(r,R_ M^E(q)). Write V_q⊂ U' for a neighborhood of p, q ∉ V_q such that the map Θ_q: V_q → S_qN is well defined. 
            
            
            
            Next we will construct the map Π_W^E, see (<ref>). Since γ_p,-W is not self-intersecting at p there exists T>0 such that for any t∈ [0,T] the point γ_p,-W(t) is not a self-intersecting point on geodesic γ_p,-W. We will prove that there exists 0<T'≤ T and ϵ>0 such that for any t ∈ [0,T'] and z ∈ B_ M(p,ϵ) the point γ_z,-W(t) is not a self-intersecting point of the geodesic γ_z,-W. 
            
            If that is not true, then there exists a sequence (p_j)_j=1^∞⊂ M that converges to p as j →∞ and moreover, there exist sequences (t_j)_j=1^∞, (T_j)_j=1^∞⊂ such that for every j ∈, we have T_j>t_j≥ 0, T_j≤τ_exit(p_j,-W), 
            
                γ_p_j,-W(t_j)=γ_p_j,-W(T_j)  and  t_j → 0  as  j →∞.
            
            By Lemma <ref> we may assume that T_j → T_0∈_+ as j →∞. Then
            
                p=lim_j →∞γ_p_j,-W(t_j)=lim_j →∞γ_p_j,-W(T_j)=γ_p,-W(T_0).
            
            Notice that T_0>0 since otherwice geodesic loops γ_p_j,-W([t_j,T_j]) would be short and this would contradict Lemma <ref>. Therefore due to Corollary <ref> it must hold that T_0=τ_exit(p,-W). But this contradicts the assumption that p is not a self-intersecting point of γ_p,-W.
            
            Consider the set
            
                O(p,T^(1),ϵ^(1)):={exp_z(sW(z))∈M;   z ∈ B_ M(p,ϵ^(1)),   s ∈(0,T^(1))}⊂ V_q,
            
            where T^(1)>0 and ϵ^(1)>0  are so small that  for any t ∈ [0,T^(1)] and z ∈ B_ M(p,ϵ^(1)) the point γ_z,-W(t) is not a self-intersecting point on the geodesic γ_z,-W.  For every w ∈ M ∩ O(p,T^(1),ϵ^(1)) we denote the component of Σ(w,W) ∩ R_ M^E(O(p,T^(1),ϵ^(1))) that contains R_ M^E(w) by C(w,T^(1),ϵ^(1)). Moreover, due to the choice of T^(1) and ϵ^(1) it holds that R_ M^E(γ_w,W([0,T^(1)])) = C(w,T^(1),ϵ^(1)), for every w ∈ M ∩ O(p,T^(1),ϵ^(1)).  
            
            
            By Lemma <ref>  there exists T^(2)∈ (0, T^(1)) and ϵ^(2)∈ (0,ϵ^(1)) such that, the set O(p,T^(2),ϵ^(2)) is open and for all  w,z ∈ ( M ∩ O(p,T^(2),ϵ^(2))),   w≠ z it holds that
            
                γ_w,W([0,T^(2)]) ∩γ_z,W([0,T^(2)])=∅.
            
            Therefore
            
                C(w,T^(2),ϵ^(2)) ∩ C(z,T^(2),ϵ^(2))=∅.
            
            Finally we notice that 
            
                R_ M^E(O(p,T^(2),ϵ^(2)))=⋃_w ∈ ( M ∩ O(p,T^(2),ϵ^(2)))C(w,T^(2),ϵ^(2)).
            
            Then for a point z ∈ O(p,T^(2),ϵ^(2)) there exists a unique w ∈ ( M ∩ O(p,T^(2),ϵ^(2))) such that R_ M^E(z) ∈ C(w,T,^(2), ϵ^(2)).
            
            
            Next we will show how we can determine a set V_q,W⊂ V_q that is similar to O(p,T^(2),ϵ^(2)) just using the data (<ref>). Let ϵ >0 be so small that B_ M(p,ϵ) ⊂ V_q. For every w ∈ B_ M(p,ϵ) we write C(w) for the maximal connected subset of Σ(w,W) that contains R_ M^E(w) and satisfies 
            
              (P1) The sets C(w) are homeomorphic to [0,1).
            
              (P2) For all w,z ∈ B_ M(p,ϵ),   w≠ z the sets C(w) and C(z) do not intersect.
            
              (P3) The set V:=⋃_w ∈ B_ M(p,ϵ) C(w) ⊂ R_ M^E(M) is open and contained in R_ M^E(V_q).
            
            We emphasize that properties (P1)–(P3) hold by Lemma <ref>, if ϵ>0 is small enough. Moreover since the map R_ M^E is a homeomorphism, for any given value of ϵ we can verify,  by using data (<ref>), if conditions (P1)–(P3) ave valid. Thus, let us choose  ϵ>0 so that (P1)–(P3) are valid and define V_q,W:=(R^E_ M)^-1(V). 
            
            Finally we are ready to define the map Π^E_q,W by the equation
            
                Π^E_q,W:V_q,W→ M,  Π^E_q,W(z):={w ∈ ( M ∩ V_q,W) ;   R_ M^E(z) ∈ C(w)}.
            
            By the continuity of the map E_W (as in Lemma <ref>) and the properties (P1)–(P3) it holds that the map Π^E_q,W is continuous on V_q,W. Moreover due to Lemma <ref> the map Π^E_q,W coincides with Π_W near the point p.
            
            
            Let p ∈ M_1 and let vector field W ∈ (_+SM)^int be such that W(p)∉ I.  Choose (q,η) ∈ K_G(p) and V_q,W, a neighborhood of p such that q ∉ V_q,W for which (<ref>) holds.  Finally we will verify that properties (<ref>)–(<ref>)  are valid for p if and only if they are valid for Ψ(V_q,W)⊂M_2.  
            
            The property (<ref>) follows from  (<ref>) and (<ref>).
            
            By the definiton (<ref>) of the mapping Ψ the properties (P1)–(P3) are valid in V_q,W if and only if the same holds for Ψ(V_q,W). Therefore (<ref>) follows from (<ref>).
            
            
            
            Given p∈ M and a non-vanishing, non-tangential inward pointing smooth vector field W on M such that W(p)∉ I, we choose (q,η) ∈ K_G(p) and a neighborhood V_q,W of p such that q ∉ V_q,W as in Lemma <ref>. Let v ∈ T_qN and define 
            
                Q_q,v: V_q,W→,  Q_q,v(x):=⟨ v,Θ_q(x) ⟩_g.
             
            Now we are ready to construct a boundary coordinate system near p∈ M.
            
            
            
            
            
            
            
            
            
            
            
            
            
            Let (N,g) and M be as in the Theorem <ref>. Let r ∈ R_ M^E(  M) and let p∈ M to be the unique point for which R_ M^E(p)=r.  Let W be a non-vanishing, non-tangential inward pointing smooth vector field on M such that W(p)∉ I, we choose (q,η) ∈ K_G(p) and a neighborhood V_q,W of p such that q ∉ V_q as in Lemma <ref>. Then there exists v ∈ T_qN and a neighborhood V_q,v,W of p such that (Q_q,v, Π^E_q,W):V_q,v,W→ (× M) defines a smooth boundary coordinate system near p. Here
            
                Q_q,v(x):= ± (Q_q,v(Π^E_q, W(x))-Q_q,v(x)),
            
            where the sign is chosen such that Q_q,v≥ 0. See Figure (<ref>).
            
            
            
            Moreover, for given r=R_ M^E(p) ∈ R_ M^E(M), W such that W(p)∉ I and (q,η) ∈ K_G(p) the data (<ref>) determines v ∈ T_qN and a neighborhood R_ M^E(V_q,v,W) of r and the map (Q_q,v, Π^E_q,W)∘ (R_ M^E)^-1:R_ M^E(V_q,v,W) → (× M), such that (Q_q,v, Π^E_q,W):V_q,v,W→ (× M) is a smooth coordinate map from a neighborhood V_q,v, W of p onto some open set U ⊂ (× M).
            
            
            
            
            
            More precisely, let (N_i,g_i) and M_i be as in Theorem <ref> such that (<ref>) and (<ref>) hold. Then for a given p ∈ M_1,  (q,η)∈ K_G(p), W, that is a non-vanishing, non-tangential inward pointing smooth vector field on M_1 such that W(p)∉ I, neighborhood V_q,W of p as in Lemma <ref> and v∈ T_qN_1  we have that (<ref>) holds. If s∈, w ∈ M_1 and z ∈ V_q then
            
                Q_q,v(z)=(s,w)  if and only if Q_ϕ(q),DΦ v(Ψ(z))=(s,ϕ(w)),
            
            and
            
                D (Q_q,v, Π^E_q, W)(p)≠ 0,  if and only  D(Q_ϕ(q),DΦ v, Π^E_ϕ(q), D Φ W)(Ψ(p)) ≠ 0.
            
            Recall that by Lemma <ref> and (<ref>) it holds that W(p)is not contained in ker(D Θ_q). Therefore there exists v ∈ T_qN such that
            
            
            
            
                ⟨ v, DΘ_qW(p) ⟩_g ≠ 0.
            
            Choose the local coordinates at p with the map E_W (as in Lemma <ref>). Then with respect to these coordinates we have
            
                D(Q_q,v, Π^E_q, W)(p)=([           0^T -(W Q_q,v)(p);        Id_n-1             0 ]),
            
            since Π^E_q,W coincides with Π_W near p. Also it holds that
            
                (W Q_q,v)(p)=⟨ v, DΘ_q W(p) ⟩_g ≠ 0.
            
            Therefore D(Q_q,v, Π^E_q, W)(p) is invertible. Moreover, (Q_q,v, Π^E_q, W)(z)=(0,z) if z ∈ M and due to  (<ref>), Lemma <ref> and the Taylor expansion of Q(q,v) it holds that Q_q,v(w)≠ 0 when w ∉ M∩ V_q,W is close to p. 
            
            We choose so small neighborhood V_q, v, W for p such that Q_q,v: V_q, v, W→ vanishes only in V_q, v, W∩ M. Therefore we have proven that (Q_q,v, Π^E_q, W): V_q, v, W→^n is a smooth boundary coordinate map at p.
            
            
            
            Notice that for given r ∈ R_ M^E( M), W, (q,η), ∈ K_G(p) and v∈ T_qN the data (<ref>) determines a neighborhood R^E_ M(V_q, v, W) of r and the map (Q_q,v, Π_q, W) ∘ (R^E_ M)^-1 by a similar argument as in the proof of Lemma <ref>.
            
            
            The equations  (<ref>) and (<ref>) can be proven by a similar argument as in the proof of Lemma <ref>.
            
            Let A be such an index set that collections (V_q,q, v^α,Θ_q,q,v^α)_α∈ A and
            
            (V^β_q,v,W,(Q_q,v, Π^E_q,W)^β)_β∈ A, that are as in Lemmas <ref> and <ref>, form a smooth atlas of (M, g). 
            
            We define an atlas for R^E_ M(M) using the following charts
            
                (R^E_ M(V_q,q, v^α),Θ_q,q,v^α∘ (R^E_ M)^-1)_α∈ A and  (R^E_ M(V^β_q,v,W),(Q_q,v, Π^E_q,W)^β∘ (R^E_ M)^-1)_β∈ A.
            
            
            By Lemmas   <ref> and <ref> the charts given in (<ref>) define a smooth structure for R^E_ M(M). 
            
            Let (M, g) be as in Theorem <ref>. If data (<ref>) is given, we can find the smooth charts of (<ref>) using only the data  (<ref>). Moreover with respect to this smooth structure the map R^E_ M:M→ R^E_ M(M)  is a diffeomorphism.
            
            The claim follows from Lemmas  <ref>, <ref> and the definition (<ref>). 
            
            Now we are ready to present the main theorem of this section.
            
            The mapping Ψ:M_1 →M_2 is a diffeomorphism.
            
            Since  maps R_ M_i^E, i ∈{1,2} are diffeomorphisms it suffices to prove that 
            
                D Φ: R_ M_1^E (M_1)→ R_ M_2^E(M_2)
            
            is a diffeomorphism. 
            
            Let p ∈ M_1. By Lemma <ref> there exist (q,η), (q, η) ∈ K_G(p), v ∈ T_qM_1 and a neighborhood V=V_q,q, v of p such that the map Θ_q,q, v:V → (S_qM_1 ×) is a smooth coordinate map. Then it follows from (<ref>) and (<ref>) that Θ_ϕ(q),ϕ(q), DΦ v:Ψ(V) → (S_qM_2 × (∖{0})) is a smooth coordinate map. Therefore the local representation of D Φ, with respect to the smooth structures as in (<ref>), on R^E_ M(V) is given by
            
                (_-S_qM_1 ×  (∖{0}))∋(ξ,s) ↦ (DΦ ξ,s) ∈ (_- S_ϕ(q)M_2 ×  (∖{0})).
            
            Since Φ:K_1 → K_2 (see (<ref>)) is a diffeomorphism it holds that  D Φ is  diffeomorphic on R^E_ M_1(V).
            
            
            Let p ∈ M_1. By Lemma <ref> there exist (q,η) ∈ K_G(p), v ∈ T_qM_1, a smooth vector field W on M_1, and a neigborhood U=V_q,v,W of p such that the map (Q_q,v, Π^E_q, W) :U →× M_1 defines smooth local boundary coondinates. Moreover by (<ref>) and (<ref>) the map (Q_ϕ(q), DΦ v, Π^E_ϕ(q), DΦ W) :Ψ(U) →× M_2 defines smooth local boundary coordinates near ϕ(p).  Therefore the  local representation of D Φ on R^E_ M_1(U), with respect to the smooth structures as in (<ref>),  is given by 
            
                (× M_1)∋ (s,z)↦ (s,ϕ(z)) ∈× M_2  .
            
            Since ϕ: M_1 → M_2 is a diffeomorphims it holds that  D Φ on R^E_ M_1(U) is diffeomorphic to its image.
            
            Thus we have proved that Ψ is a local diffeomorphism. By Theorem <ref> the map Ψ is one-to-one and therefore it is a global diffeomorphism.
            
            
            
            
            
            
            
            
            
            
            
            
            
            
            
            
            
            
            
            
            
            
            
            
            
            
            
            
            
            
            
            
            
            
            
            
            
            
            
            
            
            
            
            
            
            
            
            
            
            
            
            
            
            
            § THE RECONSTRUCTION OF THE RIEMANNIAN METRIC
            
            So far we have shown that the map Ψ:M_1 →M_2 is a diffeomorphism. The aim of this section is to show that Ψ is a Riemannian isometry. We start with showing that Ψ preserves the boundary metric. This is formulated precisely in the following Lemma.
            
            The map Ψ:M_1→M_2 has a property
            
                (Ψ^∗ g_2)|_ M_1=g_1|_ M_1.
             
            Since Ψ is smooth, the metric tensor Ψ^∗ g_2 is well defined.  By  Theorem <ref> and (<ref>) we have
            
                (Ψ^∗ g_2)|_ M_1
                =ϕ^∗(g_2|_ M_2)=g_1|_ M_1.
            
            
            
            
            
            
            
            
            
            
            
            
            
            
            
            
            
            
            
            
            
            
            
            
            
            
            
            
            
            
            
            
            
            
            
            
            
            
            
            
            
            We denote g:=Ψ^∗g_2 the pullback metric on M_1 and g:=g_1. We will also denote from now on M_1=M. By Lemma  <ref>g and g coincide on the boundary M. Next we will show that they also have the same Taylor expansion at the boundary.
            
            
            
            
            Suppose that N, M, g and  g are as in Theorem <ref> such that data (<ref>)  and (<ref>) is valid with ϕ=id. 
            
            
            
            
            
            
            Let (x^1,…, x^n) be any coordinate system near the boundary and α∈^n any multi-index. Write g=(g_ij)_i,j=1^n and g=(g_ij)_i,j=1^n. Then for all i,j ∈{1, …,n} holds
            
                ∂^α g_ij |_ M= ∂^αg_ij |_ M,  ^α:=∏_k=1^n (/ x^k)^α_k
            
            Let (p,η) ∈_- SM. Suppose that η is not tangential, since M is non-trapping there exists a unique vector (q,ξ) ∈_- SM such that (q,ξ)≠ (p,η) and  
            
                Σ(p,η)=Σ(q,ξ),
             
            where set Σ(p,η) is defined as in (<ref>). Therefore, the data (<ref>) determines the scattering relation
                L_g: _+ SM→_- SM,   L(p,-η)=(γ_p,-η(τ_exit(p,-η)),γ̇_p,-η(τ_exit(p,-η)))=(q,ξ).
            
            Thus by (<ref>) it holds that L_g=L_g=:L, where L_g is the scattering relation of metric tensor g.
            
            It is shown in <cit.> Proposition 2.1 that the scattering relation L_g with Lemma <ref> determine the first exit time function τ_exit for ξ close to S_p M. More precisely, for every p ∈ M there exists a neighborhood 𝒱_p ⊂_+ S_pM of S_p  M such that for all (p,ξ) ∈𝒱_p
                τ_exit(p,ξ)=τ_exit(p,ξ),
            
            where τ_exit is the first exit time function of g. Denote 𝒱:=∪_p ∈ M𝒱_p.
            
            
            We call the pair (L|_𝒱, τ_exit|_𝒱) the local lens data. In Theorem 2.1. of <cit.> it is shown that the local lens data implies (<ref>).
            
            
            
            Lemma <ref> and Proposition <ref> imply that we can smoothly extend g onto N such that g=g̃ in N∖M. Next we will show that the geodesics of g and g are the same.
            
            
            
            Metrics g and g are geodesically equivalent, i.e., for every geodesic γ:→ N of metric g, there exists a reparametrisation α:→ such that the curve γ∘α is a geodesic of metric g, and vice versa.
            
            Since M is non-trapping, any geodesic arc γ of metric g that is contained in M can be parametrized with a set Σ(p,η) for some (p,η)∈_-SM =_-  SM. Thus by (<ref>) any geodesic γ of g can be re-parametrized to be a geodesic γ of metric g and vice versa. This means that there exists a smooth one-to-one and onto function  α:[a',b']→ [a,b] such that
            
                α̇>0,  γ(t)=(γ∘α)(t).
             
            See Subsection 2.0.5 of <cit.> for more details.
            
            
            Now we are ready to prove the main result of this section. The proof is similar to the one given in <cit.> which works for more general settings. The key ingredient of the proof is the Theorem 1 of <cit.>.
            
            
            The metrics g and g coincide in M.
            
            Define a smooth mapping I_0:TN → as 
            
            
                I_0(p,v)=( (g)|_p/(g)|_p)^2/n+1g(v,v),
            
            note that the function 
            
                f(p):= (g)|_p/(g)|_p
             
            is coordinate invariant, since in any smooth local coordinates (U,(x^j)_j=1^n) the function f coincides with the function p ↦ (g^ik(p)g_jk(p)) in U where (g^ij)_i,j=1^n is the inverse matrix of (g_ij)_i,j=1^n and the (1,1)-tensor field
            
                g^ikg_jk/ x^i⊗ dx^j
             
            can be considered as a linear automorphism on TU.
            
            Let γ_g be a geodesic of metric g. Define a smooth path β in TN as β(t)=(γ_g(t), γ̇_g(t)). Then β is an integral curve of the geodesic flow of metric g.
            Theorem 1 of <cit.> states that, if g and g are geodesically equivalent, 
            
            
            
            
            
            
            then function t↦ I_0(β(t)) is a constant for all geodesics γ_g of metric g.
            
            Since g and g coincide in the set N ∖ M, we have that for any  point z ∈ N ∖ M and vector v ∈ T_zN
                I_0(z,v)=g_z(v,v)=g_z(v,v).
            
            Let p ∈ M. Choose γ(t):=γ_z, ξ(t), ξ∈ S_zN,z∈ N ∖ M  be a g-geodesic such that p=γ(t_p) for some t_p> 0. Then I_0(z,ξ)=1
            and  by Theorem 1 of <cit.>I_0 is constant along the geodesic flow (γ(t),γ̇(t)), i.e.,
            
            
                I_0(p,γ̇(t_p))=I_0(z,ξ)=1.
            
            
            Since M is non-trapping, (<ref>) implies that, in local coordinates, for any ξ∈ S_pN
                g_ij(p)ξ^iξ^j=1=I_0(p,ξ)=(f(p))^2/n+1g_ij(p)ξ^iξ^j.
            
            Then the equation (<ref>) holds for all ξ∈ T_pN and both sides of equation (<ref>) are smooth in ξ, we obtain 
            
                g_ij(p)=(f(p))^2/n+1g_ij(p),  for all  i,j∈{1,…,n}.
             
            Moreover
            
                (f(p))^2n/n+1-1=1,
            
            which implies that f(p)=1. By (<ref>) we conclude that g= g at p. Since p ∈ M was arbitrary the claim follows. 
            
            The proof of Theorem <ref> follows from Theorems <ref>, <ref> and Proposition <ref>.
            
            We will give a Riemannian metric G to smooth manifold R^E_ M(M) as a pullback of g, that is G:= ((R^E_ M)^-1)^∗g. A priori we don't know g, thus  we don't  know G. However as the smooth structure is known we can recover the collection (R^E_ M(M)), that is the collection of all Riemannian metrics of the smooth manifold R^E_ M(M). 
            
            Since (M,g) is non-trapping we can use the sets Σ(p,η),   (p,η) ∈_-SM to recover the images of the geodesics of G. 
            
            Thus by the proof of Proposition <ref> the following set contains precisely one element  that is (G,G),
            
                [                                       {(h,h') ∈(R^E_ M(M))^2: ∂^α h_ij |_R^E_ M( M)= ∂^α h'_ij |_R^E_ M( M) for every α∈^n,;                                                                              metrics  h and h' are geodesically equivalent,;                                                                                   sets Σ(p,η),   (p,η) ∈_-SM are the images;                                                                                                        of their geodesics}. ]
             
            We conclude that we have proven the following Proposition.
            
            
            The data (<ref>) determine the metric tensor G:= ((R^E_ M)^-1)^∗g.
            
            
            
            
            Finally we note that the proof of Theorem <ref> follows from the Propositions <ref>, <ref> and <ref>.
            Acknowledgements. The research of ML, and TS was partly supported by  the Finnish Centre of Excellence in Inverse Problems Research
            and Academy of Finland. In particular, ML was supported by projects 284715 and 303754, TS by projects 273979 and 263235. TS would like to express his gratitude to the Department of Pure Mathematics and Mathematical Statistics of the University of Cambridge for hosting him during two research visits when part of this work was carried out. TS would like to  also thank Prof. Gunther Uhlmann for hosting him in University of Washington where part of this work was carried out. HZ is supported by EPSRC grant 
            EP/M023842/1. He would like to thank Prof. Gunther Uhlmann and the 
            Department of Mathematics and Statistics of the University of Helsinki 
            for the generous support of his visit to Helsinki during his PhD when 
            part of the work was carried out.
            
            
            abbrv
