Reconstruction of a compact Riemannian manifold from the scattering data of internal sources

Given a smooth non-trapping compact manifold with strictly con- vex boundary, we consider an inverse problem of reconstructing the manifold from the scattering data initiated from internal sources. This 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, one can reconstruct an isometric copy of the manifold from such scattering data measured on the boundary.


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.
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 p N and t ∈ [0, τ exit (p, ξ)] such that q = γ p,ξ (t), η =γ 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 ( 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 (4) to the ∂M p scattering data (see [18]), that is also considered in the Section 1.2.1 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 (4). 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, (5) {Dφ(R ∂M1 (q)); q ∈ M 1 } = {R ∂M2 (p); p ∈ M 2 }.
We denote the set of all C ∞ -smooth Riemannian metrics on N by Met(N ) (we will assign the smooth Whitney topology (see [7]) 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 (4) up to an isometry.
Given g ∈ Met(N ), p, q ∈ N and > 0, we denote the number of g-geodesics connecting p and q of length by I(g, p, q, ). Define I(g) := sup p,q, I(g, p, q, ).
In Theorem 1.2 of [10] it is shown that there exists a generic set G ⊂ Met(N ), such that for all g ∈ G, I(g) ≤ 2n + 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 (7)}. (8) Our main result is as follows, Theorem 1.1. 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 (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 (4) to reconstruct an isometric copy of (M , g). Therefore we formulate also the following theorem.
Theorem 1.2. Let (N, g) ∈ G be a smooth, closed, connected Riemannian nmanifold 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 (2).
If the data (4) is given we can reconstruct a isometric copy of (M , g).
We want to underline that in the most parts the proofs for Theorems 1.1 and 1.2 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.
1.1. Outline of this paper. The proof of Theorem 1.1 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 (7) 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 (4) 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 (4) determines the Riemannian structure of (M , g).

Previous literature.
The research topic of this paper is related to many other inverse problems.
1.2.1. Boundary and scattering rigidity problems. The scattering data of internal sources (4) 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)),γ 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 g , does it implies that g = 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 [4,24,28] 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 [18] conjectured that simple manifolds are boundary distance rigid, so far this is known for simple surfaces [20]. More recently, boundary rigidity results are established on manifolds of dimension 3 or larger that satisfy certain global convex foliation condition [25,26]. 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 [9,11,21].

1.2.2.
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 where z = γ x,ξ+ (s). Denote the length of the curve α x,ξ+,z,η by (α x,ξ+,z,η ), the Broken scattering relation is 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 [12] 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.

1.2.3.
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 where ∆ 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 (9) (see for instance [9] Chapter 2.3). Thus the Dircihlet-to-Neumann operator x) g is well defined and an inverse problem to (9) 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 R n with an isotropic wave speed [2]. 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 [3]. 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 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 [9] 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,S,R f : The answer is positive if S = R, (see [8]). In [13] the problem has been solved in the case R ∩ S = ∅. The general case is still an open problem. So far the sharpest results are [14,19]. 1.2.4. Distance difference functions. In [15] the collection of distance difference functions was studied. The authors show that (N \ M, g| N \M ) with the collection (11) 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 → R is smooth. Then there is a close connection between R ∂M (p) and 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 1.1 of this paper we will use techniques that where introduced in [15]. These techniques are based on the Theorem 1 of [27].
1.2.5. 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 ∈ N and M ⊂ N be an open subset of N with smooth boundary. We denote by U := N \ M .
In [6] 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 where x ∈ M , r > 0 and W ⊂ S x N 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)) :=γ 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 [6] 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 [6] 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 [6] a counterexample is provided. In [6] it is shown that the Spherical surface data determines the universal covering space of (N, g) up to an isometry.
In [5] a special case of problem [6] is considered. The authors study a setup where M ⊂ R n , n ≥ 2 and the metric tensor g| M = v −2 e, for some smooth and strictly positive function v. Let x ∈ R n \ M and y := γ x,ξ (t 0 ) ∈ M , for some ξ ∈ S x N and t 0 > 0, such that y is not a conjugate point to x along γ p,ξ . The main theorem of [5] 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 ]).

2.
Strictly convex manifolds, about the extension of the data and the generic property 2.1. 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, [9], 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)| = dist 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 .
Definition 2.1. 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 (12) 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.   Lemma 2.4. 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)). Lemma 2.5. If (M , g) is compact, non-trapping and ∂M is strictly convex, then the first exit time function τ exit : SM → R + given by (1) is continuous and there exists L > 0 such that Moreover, the exit time function τ exit is smooth in SM \ S∂M and in ∂ + SM .

2.2.
Extension of the measurement data. We start with showing that the data (4) determines the boundary metric. This is formulated more precisely in the following lemma.
where g| ∂M = i * g and i : ∂M → M . More precisely if (N i , g i ) and M i are as in Theorem 1.1 and (5)-(6) hold, then Proof. 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 and a > 0, that satisfies ξ = aη}.
It is easy to see that the set R(η) is not empty and by Lemma 2.2 we also have Therefore, Since η ∈ T p ∂M was arbitrary we have recovered the g-norm function · g : T p ∂M → R. Since norm · g is given by the g-inner product, we recover ·, · g using the parallelogram rule, that is Suppose then that for manifolds (M 1 , g 1 ) and (M 2 , g 2 ) properties (5)-(6) are valid. Choose p ∈ ∂M 1 and η ∈ T p ∂M 1 and denote R(η) as in (16). Then by (5)-(6) we have Dφ(R(η)) = R(Dφη) ⊂ T φ(p) ∂M 2 . and by the equations (17)- (18) we have η g1 = Dφη g2 and equation (15) follows from the parallelogram rule.
Let p ∈ M , we define the complete scattering set of the point source p as We emphasize that the difference between R ∂M (p) and R E ∂M (p) for p ∈ M is that R ∂M (p) contains the tangential components of vectors ξ ∈ R E ∂M (p). We denote In the next Lemma we will give an equivalent definition for (19).
Let (N i , g i ) and M i be as in Theorem 1.1 and suppose that (5)-(6) hold. Since M i is compact, there exists r > 0 and a neighborhood K i ⊂ M i of ∂M i such that the mapping is a diffeomorphism. Thus the mapping is a diffeomorphism between K 1 and K 2 . Moreover the map Φ is a local representation of Φ in the boundary normal coordinates.
More precisely if (N i , g i ) and M i are as in Theorem 1.1 such that (6) and (15) hold, then η, ξ g1 = DΦη, DΦξ g2 , for all ξ, η ∈ ∂T M 1 (22) and Proof. We will start with showing that for every q ∈ M we can recover R E ∂M (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 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 E ∂M (q). Let q ∈ ∂M and (p, η) ∈ ∂ − SM . We will define a set with which we can verify if (p, η) ∈ R E ∂M (q) or not. Notice that by Lemma 2.2 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.
Next we will prove Let q ∈ M 1 . By (6) there exists z ∈ M 2 such that ). Thus we have proved the left hand side inclusion in (26). Again by symmetric argument we prove the right hand side inclusion in (26).
Next we will show that By the definition of the mapping Φ and the equation (22) it is enough to prove that Let (p, η) ∈ R E ∂M1 (q) and denote Σ(p, η) as in (25). By equations (22) and (26) it holds that the set 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 (15) we have Suppose then that Σ(p, η) is not empty. This implies that there exists a vector ξ ∈ S q N 1 that satisfies Σ(p, η) = Σ(q, ξ). Thus Σ(Φp, DΦη) = Σ(Φq, DΦξ) and this implies that DΦ(p, η) ∈ R ∂M2 (φ(q)). This completes the left hand inclusion of (27). Replace Φ by Φ −1 to prove the right hand side inclusion of (27).

2.3.
A generic property. We will now formulate a generic property in Met(N ) that is related to the complete scattering data (21).
Definition 2.9. Let N be a smooth manifold and M ⊂ N an open set. We say that a Riemannian metric g ∈ Met(N ) separates the points of set M if for all p, q ∈ M, p = q there exists ξ ∈ S p N such that 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 (8)) and M ⊂ N that is open, non-trapping in the sense of (2) 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. Lemma 2.10. Let (N, g) be a compact Riemannian manifold. Let M ⊂ N be open, non-trapping in the sense of (2) 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 ⊂ R, and a C ∞ -map I s → (ξ(s), (s)) ∈ S p N × R, such thatξ(s) = 0 and q = γ p,ξ(s) ( (s)).
Proof. Since g does not separate the points of M there are p, q ∈ M , p = q, such that for all ξ ∈ S p N we have We note that for all ξ ∈ S p N the set S(ξ) is finite due to inequality (13). Fix η ∈ S p N and enumerate S(η) = {s 1 , . . . , s K } for some K ∈ N. 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 where inj(q) is the injectivity radius at q. Choose > 0 and consider a -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 is small enough, there exists δ, δ > 0 such that for any By the continuity of the exponential mapping, we can choose a smaller > 0 such that there exists an open neighborhood V ⊂ S p N of η such that for any Next we define a signed distance function See Figure (2). Choosing a smaller δ and V , if needed, it follows that the function k : Therefore, by the Implicit function theorem there exists an open neighborhood Define an open neighborhood U of η by U := K k=1 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 (29), (30) and (31) it must hold that We claim that for some k ∈ {1, . . . , K} it holds that U int k = ∅. 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 (32) we have 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 int Choose an open U ⊂ U k . In particular U is open in S p N and there exists > 0 and a C ∞ -path ξ(s), s ∈ (− , ), in U such thatξ(s) = 0. Denoting (s) = f k (ξ(s)), we have q = γ p,ξ(s) ( (s)) for s ∈ (− , ). Therefore, the curve s → (ξ(s), (s)) satisfies the claim of this Lemma. Proof. We prove the proposition by contradiction. Given g ∈ G, assume there are p, q ∈ M, p = q such that for all ξ ∈ S p N , In particular (see Lemma 2.10), there exists an open interval (− , ), such that for s ∈ (− , ), (ξ(s), (s)) is a C 1 -path on S p N × R such thatξ(s) = 0 and q = γ p,ξ(s) ( (s)). This implies Since ξ(s) = 1, it implies ξ(s) ⊥ξ(s). By Gauss Lemma, we haveγ p,ξ(s) ( (s)) ⊥ D exp p | (s)ξ(s)ξ (s), thus T 1 ⊥ T 2 . Applying equation (33), we obtain T 1 = T 2 = 0, therefore, d ds ( (s)) = 0. This is equivalent to saying that (s) ≡ const, for s ∈ (− , ). This implies I(g) = ∞. By equation (7), we arrive a contradiction.

The reconstruction of the topology
We define a mapping . 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 2.8. 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 2.9 holds.
Lemma 3.1. Suppose that (N, g) and M are as in the Theorem 1.1. Then the mapping R E ∂M : M → 2 ∂SM is one-to-one. Proof. We will prove the claim by contradiction. Therefore, we divide the proof into two separate cases.
Suppose first that there is . 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 p N . Then for ±ξ we have (γ p,±ξ ((τ exit (p, ±ξ)),γ 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))). (34) Since ξ ∈ S p N was arbitrary, the equation (34) is valid for every ξ ∈ S p N and we have proved that the metric q does not separate the points p and q. This is a contradiction with Proposition 2.11 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 SM 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 SM , we assign the Hausdorff distance, i.e., given A, However, in general the Hausdorff distance on 2 SM needs not to be metrizable. If we consider the subset C(SM ) := {closed subsets of SM } ⊂ 2 SM , then (C(SM ), d H ) is a compact metric space by Blaschke selection theorem (see for instance [1], Theorem 4.4.15). The topology on C(SM ) thus is induced by the Hausdorff metric d H . We turn to consider a subspace C(∂SM ) := {closed subsets of ∂SM } ⊂ 2 ∂SM of C(SM ). Since the boundary ∂M is compact, C(∂SM ) is a compact metric space.
Proof. Consider a continuous mapping E q : S q N → ∂ − SM given by Since S q N is compact, and E q is continuous it holds by Corollary 2.7 that . From now on we consider the mapping Definition 3.3. 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: Let q ∈ M and q j ∈ M , j ∈ N be a sequence that converges to q. As ∂SM is compact it holds that the Kuratowski convergence and the Hausdorff convergence are equivalent (see e.g. [1] Proposition 4.4.14). Thus it suffices to show that . Changing in to subsequences, if necessary, we assume that (p j , η j ) → (p, η) ∈ ∂SM as j → ∞. Let ξ j ∈ S qj N be such that for each j ∈ N, γ qj ,ξj (τ exit (q j , ξ j )) = p j andγ qj ,ξj (τ exit (q j , ξ j )) = η j .
As the first exit time function is continuous and SM 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 Since q j → q as j → ∞ we can choose ξ j ∈ S qj N, j ∈ N such that ξ j → ξ as j → ∞. Denote γ qj ,ξj (τ exit (q j , ξ j )) = p j andγ qj ,ξj (τ exit (q j , ξ j )) = η j , since the exponential map and the first exit time function are continuous, then This proves (K2). We conclude that R E ∂M (q j ) converges to R E ∂M (q) in the Kuratowski topology.
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 3.5 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
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 [1] Proposition 4.4.14 this will imply that the lift f is continuous. Let , 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 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 ( This completes the proof of the convergence of f (K i ) to f (K).
Notice that by (20) the mapping DΦ : ∂T M 1 → ∂T M 2 is a smooth invertible bundle map. In particular DΦ is continuous with a continuous inverse. Therefore, DΦ is a well defined homeomorphism according to the first part of the proof.
Next we define a mapping By (23) and Proposition 3.5 the map Ψ is well defined. Now we prove the main theorem of this section.
Proof. By (23), Proposition 3.5 and Lemma 3.6 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 ∂M2 (q). On the other hand by the proof of Lemma 3.1 there is no p ∈ ∂M 1 such that S p ∂M 1 ⊂ R E ∂M1 (p). Since DΦ 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.

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 (21). We will consider separately coordinate charts for interior and boundary points. Then we will define a smooth structure on topological By the formula (37), that defines the map Ψ, the following diagram commutes 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 (21). 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.
4.1. 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 using our data (21). 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.
In the next Lemma we show that under the assumptions of Theorem 1.1 most of the geodesics that start and end at ∂M and hit the point p, do it only one time.
Then the set S p N \ I is open and dense.
If r ∈ R E ∂M (M ) is given and p ∈ M is the unique point for which R E ∂M (p) = r we define Then K(p) is not empty and K(p) is determined by r and data (21). More precisely if (N i , g i ) and M i are as in Theorem 1.1 such that (15) and (23) hold, then for every p ∈ M 1 the set K(p) = ∅ and DΦ(K(p)) = K(Ψ(p)). (42) Proof. 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 ⊂ R that satisfies exp p (t j ξ j ) = p and 0 < |t j |.
By (13) we can without loss of generality assume that 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 2.10 and Proposition 2.11 it follows that the set I is nowhere dense, this is that I does not contain any open sets. Thus S p N \ I is open and dense. Moreover the set Here is a schematic picture about K(p), where the point p ∈ M is the blue dot. The black curves represent the geodesics γ z,ξ , and γ w,η respectively, where vectors (z, ξ), (w, η) ∈ R E ∂M (p). Notice that only (w, η) ∈ K(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 p N . Choose a sequence (t j ) ∞ j=1 such that exp p (t j ξ j ) = p and 0 < t j .
Due to Lemmas 2.2 and 2.5 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 p M . Then by a similar argument as in the case of p ∈ M we have shown that S p N \ I is open and dense.
Let (y, ξ) ∈ ∂ − SM . We will use a short hand notation , for the image of the geodesic segment γ y,−ξ ([0, τ exit (y, −ξ)]) under the map R E ∂M . Denote r = R E ∂M (p). Then for a vector (q, η) ∈ r it holds that (q, η) ∈ K(p) if and only if q = p. Thus the set is not empty. Now we will verify the equation (42) in the case of p ∈ ∂M 1 . Since the map Ψ is a homeomorphism we have by the data (23) that DΦ(Σ(y, ξ)) = Σ(φ(y), DΦξ) for all (y, ξ) ∈ ∂ − SM 1 .
In the next Lemma we will show that we can find the set of conjugate directions with respect to p from data (21) and there exist lots of (q, η) ∈ r := R E ∂M (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. (45) Then K G (p) is not empty, π(K G (p)) ⊂ ∂M is open. Moreover r and the data (21) determine the set K G (p).
Let ξ ∈ S p N be such a unit vector that γ p,ξ is a shortest geodesic from p to the boundary ∂M . By Lemma 2.13 of [9] it holds that δ(p, ξ) > dist g (p, ∂M ). Therefore, (γ p,ξ (τ exit (p, ξ)),γ p,ξ (τ exit (p, ξ))) is not a conjugate direction with respect to p. Moreover there exists an open neighborhood V ⊂ S p N of ξ such that for any ξ ∈ V the vector (γ p,ξ (τ exit (p, ξ )),γ p,ξ (τ exit (p, ξ ))) is not a conjugate direction. Let I ⊂ S p N be defined as in (40). Denote that is open and non-empty, since V ⊂ V c . Therefore, by the Lemma 4.1 it holds that the set V c ∩ (S p N \ I) = V c \ I is open and non-empty. Thus is not empty and moreover π(K G (p)) ⊂ ∂M is open.
(47) Notice that the curve s → f (1, s) ∈ ∂M is smooth by Lemma 2.5. We conclude that a curve σ can be for instance defined as follows where > 0 is small enough. Observe that the curve σ(s) satisfies conditions (H1) and (H2) automatically. The condition (H3) is valid by the Lemma 4.1, since (q, η) ∈ K(p). We also note that 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 t V for the covariant derivative of V along the curve c. The properties of operator D t are considered for instance in Chapter 4 of [17]. We recall that locally D t V is defined by the formula where Γ k ji 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 .
Here D t operator is defined on γ p,ξ (t). Define a curve σ(s) = (q(s), η(s)) ∈ H(q, η) by (47) 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 (49) and (50)  η.
Since s → q(s) is a curve on ∂M and η = 0 is not tangential to the boundary, it must hold that d ds τ exit (p, w(s)) s=0 = 0.
Thus d ds q(s) Recall that the Jacobi field J satisfies due to (53) We will use the notation D t for the covariant derivative on the curve γ p,tpξ (t ). Then by (48) since J is not a zero field. Therefore, we conclude that D s η(s) Let x → (z j (x)) n j=1 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 observationq(0) = 0 yields here Γ k ji are the Christoffel symbols of metric g in boundary normal coordinates. Thus Therefore, (52) is valid.
Next we consider curves σ(s) = (q(s), η(s)) ∈ H(q, η). Notice that for every σ(s) there exists > 0 and a smooth function s → a(s) ∈ (0, ∞), s ∈ (− , ) such that a(s) → t p as s → 0 and 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 T N and exp : T N → N . We use a short hand notation −tW (s) := −ta(s)η(s). We use coordinates (z j , v j ) n j=1 for π −1 U ⊂ T M , where U is the domain of coordinates (z i ) n i=1 . Let (y i ) n i=1 be coordinates at exp((q(0), −tV (0). Then the variation field is a Jacobi field that vanishes at t = 1. By (56) we have Therefore by (57)  Suppose that there exists such a curve σ(s) = (q(s), η(s)) ∈ H(q, η) for which (58) is valid. Since V is a Jacobi field that vanishes at 1, we have by (57), (58) and the definition of σ(s) that Thusȧ ( ) T = 0. We conclude that, if there exists a curve σ ∈ H(q, η) such that (58) holds then, p is a conjugate point to q on γ q,−η and moreover, To summarize, for a given r ∈ R E ∂M (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 (52) is valid. Moreover since s → q(s) is a curve on the boundary ∂M we can check the validity of (52) for any σ ∈ H(q, η) by data (21). Therefore for given r ∈ R E ∂M (M ) the data (21) determines the set K G (p). Next we will prove equation (46) in the case of p ∈ M 1 . Let (q, η) ∈ K(p). Suppose that DΦη ∈ 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 (52) is valid.
As it holds that Ψ is a homeomorphism and Φ is a diffeomorphism that preserves the boundary metric (in the sense of (22) Thus (52) is valid for the curve (q(s), η(s)), which implies that (q, η) / ∈ K G (p). Thus (46) is valid in the case of p ∈ M 1 .
Suppose then that p ∈ ∂M . It follows from the Lemma 2.4 that the set K G (p) is not empty and Let (q, η) ∈ K G (p). Let ξ ∈ S p N be the unique vector that satisfies η =γ p,ξ (τ exit (p, ξ)). Since D exp 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.
The equation (46) 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 (39). 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 q N of t p η such that the exponential map We emphasize that the mapping depends on the neighborhood U q of t p η.
Lemma 4.3. Let p, q ∈ N, p = q. Let η ∈ S q N and t > 0 be such that exp q (tη) = p. Suppose that D exp q is not singular at tη and denote D exp q | tη η =: ξ ∈ S p N . Then ker(DΘ q (p)) = span(ξ).
Proof. Let v ∈ T p N . Then 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 (21) determine the map Θ q .  Let (N, g) and M be as in the Theorem 1.1. If r ∈ R E ∂M (M ) is given and p ∈ M is the unique point for which R E ∂M (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 p η, where exp q (t p η) = p, such that exp −1 q : V q → U q is well defined. Moreover, the map Θ q : V q → S q N 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 q N are determined from the data (21) for given r ∈ R E ∂M (M ) and (q, η) ∈ K G (p).
More precisely, if (N i , g i ) and M i are as in Theorem 1.1 such that (15) and (23) hold. Then for a given r = R E ∂M (p) ∈ R E ∂M (M 1 ) and (q, η) ∈ K G (p) it holds that Θ φ(q) (Ψ(z)) = DΦ(Θ q (z)), z ∈ V q . (59) Proof. 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 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 (21), if (q, η) ∈ K G (p) is given. Let V ⊂ M be a neighborhood of p and U ⊂ T q N a neighborhood of t p η. Denote U := h( U ) ⊂ S q N . Since η ∈ U , we may assume that for all z ∈ V the set R E ∂M (z) ∩ U ⊂ S q N is not empty. We define a set valued mapping P q : V → 2 SqN by formula Then P q is well defined and for given r and (q, η) ∈ K G (p) we can recover P q from (21). We claim that, if V and U are small enough, then for all z ∈ V the set R E ∂M (z) ∩ U ⊂ S q N has a cardinality of 1.
Therefore, P q , coincides with Θ q in V. We will show that if (60) 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 q N that converges to η and η j = η for any j ∈ N. Choose t j < 0 such that exp q (t j η j ) = p. Due to (13) 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 j η j ) = exp q (tη).
Since (q, η) ∈ K(p) it must hold that t = t p . As p = exp q (t j η j ) for every j ∈ N and t j η j → tη in T q N , we end up with a contradiction to the assumption (q, η) ∈ K G (p).
Thus we have proven the existence of such sets V and U for which (60) is valid. Using data (21) we can verify for a given r and the neighborhoods U of η and R E ∂M (V ) of r whether the property (60) is valid. We will denote by U 1 ⊂ S q N a neighborhood of η and V q a neighborhood of p that satisfy (60). The next step is to prove that for any r = R E ∂M1 (p) ∈ R E ∂M1 (M 1 ) and (q, η) ∈ K G (p) the equation (59) is valid. Choose neighborhoods U 1 ⊂ S q N of η and V q ⊂ M 1 of p for which (60) is valid. By (46) 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 (22) DΦ(U 1 ) ⊂ S φ(q) N 2 is a neighborhood of DΦη. Therefore (60) 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)) ∩ DΦ(U 1 ) and the equation (59) follows.
Finally we assume that p ∈ ∂M . We notice that by Lemma (2.2) 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 .

4.2.
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 q → S q N exists for some neighborhood V q of p. Let v ∈ T q N and define a smooth map Here is a schematic picture about the map Θ q, q,v evaluated at point p ∈ M , where the point p ∈ M is the blue dot and V q ∩ V q is the blue ellipse. The higher red&blue dot is q and the lower is q. The blue vector is the given direction v ∈ T q N .
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 → R n is a coordinate map near p ∈ N if and only if detDf (p) = 0. Here Df (p) is the Jacobian matrix of f at p.  Let (N, g) and M be as in the Theorem 1.1. If r ∈ R E ∂M (M ) is given and p ∈ M is the unique point for which R E ∂M (p) = r, then there exist (q, η), ( q, η) ∈ K G (p) and v ∈ T q N such that Θ q, q,v is a smooth coordinate mapping from a neighborhood V q, q,v of p.
Moreover, for given r = R E ∂M (p) ∈ R E ∂M (M ) and (q, η), ( q, η) ∈ K G (p) the data More precisely, let (N i , g i ) and M i be as in Theorem 1.1 such that (15) and (23) hold. Then for any p ∈ M 1 and (q, η), ( q, η) ∈ K G (p) and v ∈ T q N the following holds: Let ξ ∈ S q N be outward pointing and s ∈ R. Then Proof. Choose (q, η) ∈ K G (p). By Lemma 4.2 the set π(K G (p)) is open and, there exists ( q, η) ∈ K G (p), such that By Lemma 4.4 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 p N such that ξ and ξ are not parallel and (γ p,ξ (t),γ p,ξ (t)) = (q, η) and (γ p, ξ ( t),γ p, ξ ( t)) = ( q, η). Choose linearly independent vectors w 1 , . . . , w n−1 ∈ S q N that are all perpendicular to η. Then vectors are linearly independent and perpendicular to ξ. Therefore, the vectors ξ 1 , . . . , ξ n−1 , ξ span T p N . Let v ∈ T q N be such that v, Θ q (p) g = 0 and v, DΘ q (p)ξ g = 0. Notice that by Lemma 4.3 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 where V is the Jacobian matrix of Θ q at p with respect to (n − 1) first coordinates, a, 0 ∈ R n−1 and c = ∂ ∂xn v, Θ q (x) g | x=p . Then by the choice of basis ξ 1 , . . . , ξ n−1 , ξ and Lemma 4.3 it holds that V is invertible. Moreover and we conclude that detDΘ q, q,v (p) = (±1)c detV = 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 (21) determines v ∈ T q N , a neighborhood R E ∂M (V q, q,v ) of r and the map . By Lemma 4.2, the data (21) determines the set K G (p) and it is not empty. Choose any (q, η) ∈ K G (p). By equation (25) we can choose ( q, η) ∈ K G (p) such that (63) is valid. By Lemmas 2.8 and 4.4 we can construct the neighborhoods V q , V q of p and the map Θ q, q,v : V q, q → S q N × R, where V q, q := V q ∩ V q , for any v ∈ T q N . Suppose now on that the points (q, η) and ( q, η) are given. By (64) and (65) 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 q N \ ((DΘ q (p)ξ) ⊥ ∪ Θ q (p) ⊥ ) that is open and dense.
Lastly we will introduce a method to test, if v ∈ T q N is admissible, that is, detDΘ q, q,v (p) = 0. By the Propostion 3.5 the map R E ∂M : M → R E ∂M (M ) is a homeomorphism. Thus we can determine the set H w,w is smooth and det(DH w,w (Θ q, q,w (p)) = 0)}. Thus A q × A q ⊂ C 2 q and the data (21), 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 Write C q for the projection of C 2 q into T q N 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 q N such that If v is admissible, then for every w ∈ A q holds (v, w) ∈ C 2 q . Therefore, {v}×A q ⊂ C 2 q and (67) follows. Suppose then that (67) 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 q N \A q . Then it follows that ({v} × (A ∩ A q )) ∩ C 2 q = ∅. But this contradicts (66) and it must hold that v ∈ A q , which means that v is admissible.
If v, w ∈ T q N, 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 (21). Let U v,w ⊂ U v,w be an open neighborhood of Θ q, q,w (p) such that detDH 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 q N × R is a smooth coordinate map, can be defined for instance by formula V q, q,v := Θ −1 q, q,v (U v,w ).
Lastly we notice that B φ( q) = DΦ(B q ). Therefore, we conclude that for given (q, η), ( q, η) ∈ K G (p) for which (63) is valid the equation (65) holds for v ∈ T q N 1 if and only if the same holds for DΦv ∈ T φ( q) N 2 . Therefore, the equation (62) follows from (61).

Boundary coordinates.
Next we construct the coordinates near ∂M . Let p ∈ ∂M . Let I be the set of self-intersection directions of p (see (40)). By Lemma 4.1 the set S p N \ I is open and dense in S p N . As p ∈ ∂M the data (21) determines the set I, since it holds that 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 (21). 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. Lemma 4.6. 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 is well defined and moreover, there exists T p ∈ (0, T ) and > 0 such that the restriction Proof. Since W (z) ∈ (∂ + SM ) int for every z ∈ ∂M and ∂M is compact, the Lemma 2.5 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 ) n−1 j=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 In the next Lemma we show that the data (21) determines a continuous map Π E W that coincides with Π W near p. Later we will use the map Π E W to construct (n − 1)-coordinates for points near p.  Let (N, g) and M be as in the Theorem 1.1. For given r ∈ R E ∂M (∂M ) where p ∈ ∂M is the unique point for which R E ∂M (p) = r and let W be a nonvanishing, non-tangential inward pointing smooth vector field such that W (p) / ∈ I, the data (21) 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 q N is well defined. Moreover,γ q,−Θq(p) (τ exit (q, −Θ q (p))) ∦ W (p) (70) and a map Π E q,W : V q,W → ∂M ∩ V q,W , is determined by (21).
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 1.1 and suppose that (15) and (23) hold. Let W be a smooth non-vanishing, non-tangential, inward pointing vector field on ∂M 1 such that W (p) / ∈ I. If ∈ V q,W and for every z ∈ V q,W the property (59) holds. are given. Thenγ and Proof. We start by showing how property (70) can be verified. Let R E ∂M (U ) ⊂ R E ∂M (M ) be a neighborhood of r = R E ∂M (p). For every r ∈ (R E ∂M (U )∩R E ∂M (∂M )), r = r we define the number N (r ) to be the cardinality of the set A(r, r ) := {ξ ∈ (S z N ) ∩ r; Σ(z, ξ) ⊂ R E ∂M (U ), (z, ξ) ∈ K G (p)}, here z = (R E ∂M ) −1 (r ). By Lemma 2.4 there exists a neighborhood R E ∂M (U ) ⊂ R E ∂M (U ) of r such that N (r ) = 1 for every r ∈ (R E ∂M (U ) ∩ R E ∂M (∂M )), r = r. Since the sets A(r, r ) are determined by r and (21) we can find the set R E ∂M (U ). We denote by U ⊂ U the unique neighborhoods of p that are related to R E ∂M (U ) and R E ∂M (U ). We write z = (R E ∂M ) −1 (r ) ∈ U , for given r ∈ R E ∂M (U ). Therefore, by Lemma 4.4 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 z N is smooth and well defined. Moreover there exists a unique (p, v z ) ∈ S p N so that γ p,vz (τ exit (p, v z )) = z andγ p,vz (τ exit (p, v z )) = Θ z (p) ∈ A(r, r ) and Σ(z, Θ z (p)) = Σ(p, v z ). Then v z =γ z,−Θz(p) (τ exit (z, −Θ z (p))) and thus the data (21) determines, if v z and W (p) are parallel. It follows from Lemma 4.6 that there exists q ∈ U such that v q is not parallel to W (p). Therefore we can use data (21) to determine a point q that satisfies the property (70). Let (q, η) ∈ A(r, R E ∂M (q)). Write V q ⊂ U for a neighborhood of p, q / ∈ V q such that the map Θ q : V q → S q N is well defined.
Next we will construct the map Π E W , see (71). 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 selfintersecting point on geodesic γ p,−W . We will prove that there exists 0 < T ≤ T and > 0 such that for any t ∈ [0, T ] and z ∈ B ∂M (p, ) 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 ⊂ R such that for every j ∈ N, we have T j > t j ≥ 0, T j ≤ τ exit (p j , −W ), γ pj ,−W (t j ) = γ pj ,−W (T j ) and t j → 0 as j → ∞.
By Lemma 2.5 we may assume that T j → T 0 ∈ R + as j → ∞. Then Notice that T 0 > 0 since otherwice geodesic loops γ pj ,−W ([t j , T j ]) would be short and this would contradict Lemma 2.4. Therefore due to Corollary 2.3 it must hold that T 0 = τ exit (p, −W ). But this contradicts the assumption that p is not a selfintersecting point of γ p,−W .
Then for a point z ∈ O(p, T (2) , (2) ) there exists a unique w ∈ (∂M ∩O(p, T (2) , (2) )) such that R E ∂M (z) ∈ C(w, T, (2) , (2) ). Next we will show how we can determine a set V q,W ⊂ V q that is similar to O(p, T (2) , (2) ) just using the data (21). Let > 0 be so small that B ∂M (p, ) ⊂ V q . For every w ∈ B ∂M (p, ) we write C(w) for the maximal connected subset of Σ(w, W ) that contains R E ∂M (w) and satisfies (P1) The sets C(w) are homeomorphic to [0, 1). (P2) For all w, z ∈ B ∂M (p, ), w = z the sets C(w) and C(z) do not intersect.
neighborhood V q,W of p as in Lemma 4.7 and v ∈ T q N 1 we have that (73) holds. If s ∈ R, w ∈ ∂M 1 and z ∈ V q then Q q,v (z) = (s, w) if and only if Q φ(q),DΦv (Ψ(z)) = (s, φ(w)), and Here is a schematic picture about the map ( Q q,v , Π E q,W ) evaluated at a point x ∈ M that is close to ∂M , where the point x ∈ M is the blue dot. The right hand side red&blue dot is Π E q,W (x) and the left hand side red&blue dot is q. The blue arrow is the given vector v ∈ T q N .
Proof. Recall that by Lemma 4.3 and (70) it holds that W (p) is not contained in ker(DΘ q ). Therefore there exists v ∈ T q N such that v, DΘ q W (p) g = 0.
Choose the local coordinates at p with the map E W (as in Lemma 4.6). Then with respect to these coordinates we have 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 (78), Lemma 4.6 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 → R vanishes only in V q,v,W ∩ ∂M . Therefore we have proven that ( Q q,v , Π E q,W ) : V q,v,W → R n is a smooth boundary coordinate map at p.
Notice that for given r ∈ R E ∂M (∂M ), W , (q, η), ∈ K G (p) and v ∈ T q N the data (21) 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 4.5.
The equations (76) and (77) can be proven by a similar argument as in the proof of Lemma 4.5.
Let A be such an index set that collections (V α q, q,v , Θ α q, q,v ) α∈A and (V β q,v,W , ( Q q,v , Π E q,W ) β ) β∈A , that are as in Lemmas 4.5 and 4.8, form a smooth atlas of (M , g).
We define an atlas for R E ∂M (M ) using the following charts  Proof. Since maps R E ∂Mi , i ∈ {1, 2} are diffeomorphisms it suffices to prove that DΦ : Let p ∈ M 1 . By Lemma 4.5 there exist (q, η), ( q, η) ∈ K G (p), v ∈ T q M 1 and a neighborhood V = V q, q,v of p such that the map Θ q, q,v : V → (S q M 1 × R) is a smooth coordinate map. Then it follows from (61) and (62) that Θ φ(q), φ(q),DΦv : Ψ(V ) → (S q M 2 × (R \ {0})) is a smooth coordinate map. Therefore the local representation of DΦ, with respect to the smooth structures as in (79), on R E ∂M (V ) is given by Since Φ : K 1 → K 2 (see (20)) is a diffeomorphism it holds that DΦ is diffeomorphic on R E ∂M1 (V ). Let p ∈ ∂M 1 . By Lemma 4.8 there exist (q, η) ∈ K G (p), v ∈ T q M 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 → R × ∂M 1 defines smooth local boundary coondinates. Moreover by (76) and (77) the map ( Q φ(q),DΦv , Π E φ(q),DΦW ) : Ψ(U ) → R×∂M 2 defines smooth local boundary coordinates near φ(p). Therefore the local representation of DΦ on R E ∂M1 (U ), with respect to the smooth structures as in (79), is given by (R × ∂M 1 ) (s, z) → (s, φ(z)) ∈ R × ∂M 2 .
Since φ : ∂M 1 → ∂M 2 is a diffeomorphims it holds that DΦ on R E ∂M1 (U ) is diffeomorphic to its image.
Thus we have proved that Ψ is a local diffeomorphism. By Theorem 3.7 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.
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 5.1 g and g coincide on the boundary ∂M . Next we will show that they also have the same Taylor expansion at the boundary.
Proposition 5.2. Suppose that N, M, g and g are as in Theorem 1.1 such that data (6) and (15) is valid with φ = id.
Thus by (44) it holds that L g = L g =: L, where L g is the scattering relation of metric tensor g.
It is shown in [23] Proposition 2.1 that the scattering relation L g with Lemma 5.1 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 p M 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∈∂M V p .
We call the pair (L| V , τ exit | V ) the local lens data. In Theorem 2.1. of [16] it is shown that the local lens data implies (82). Lemma 5.1 and Proposition 5.2 imply that we can smoothly extend g onto N such that g = g in N \ M . Next we will show that the geodesics of g and g are the same.
Lemma 5.3. Metrics g and g are geodesically equivalent, i.e., for every geodesic γ : R → N of metric g, there exists a reparametrisation α : R → R such that the curve γ • α is a geodesic of metric g, and vice versa.
Proof. 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 (44) 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 thatα > 0, γ(t) = (γ • α)(t).
See Subsection 2.0.5 of [15] for more details. Now we are ready to prove the main result of this section. The proof is similar to the one given in [15] which works for more general settings. The key ingredient of the proof is the Theorem 1 of [27].
Proposition 5.4. The metrics g and g coincide in M .
Proof. Define a smooth mapping I 0 : T N → R as note that the function f (p) := det(g)| p det( g)| p is coordinate invariant, since in any smooth local coordinates (U, (x j ) n j=1 ) the function f coincides with the function p → det( g ik (p)g jk (p)) in U where ( g ij ) n i,j=1 is the inverse matrix of ( g ij ) n i,j=1 and the (1, 1)-tensor field g ik g jk ∂ ∂x i ⊗ dx j can be considered as a linear automorphism on T U .
Let γ g be a geodesic of metric g. Define a smooth path β in T N as β(t) = (γ g (t),γ g (t)). Then β is an integral curve of the geodesic flow of metric g. Theorem 1 of [27] 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 z N I 0 (z, v) = g z (v, v) = g z (v, v).
Proof of 1.1. The proof of Theorem 1.1 follows from Theorems 3.7, 4.10 and Proposition 5.4.
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 Met(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 5.4 the following set contains precisely one element that is (G, G), for every α ∈ N 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.