arXiv:2201.01887v1 [math.DG] 6 Jan 2022

UNIQUENESS OF THE PARTIAL TRAVEL TIME REPRESENTATIONOF A COMPACT RIEMANNIAN MANIFOLD WITH STRICTLY

CONVEX BOUNDARY

ELLA PAVLECHKO AND TEEMU SAKSALA

Abstract. In this paper a compact Riemannian manifold with strictly convex boundaryis reconstructed from its partial travel time data. This data assumes that an open measure-ment region on the boundary is given, and that for every point in the manifold, the respectivedistance function to the points on the measurement region is known. This geometric in-verse problem has many connections to seismology, in particular to micro seismicity. Thereconstruction is based on embedding the manifold in a function space. This requires thedifferentiation of the distance functions. Therefore this paper also studies some global reg-ularity properties of the distance function on a compact Riemannian manifold with strictlyconvex boundary.

1. Introduction

This paper is devoted to an inverse problem for smooth compact Riemannian manifoldswith smooth boundaries. Suppose that there is a dense but unknown set of point sourcesgoing off in an unknown Riemannian manifold. For each of these point sources we measurethe travel time of the respective wave on some small known open subset of the boundary of themanifold. The point source can be natural (e.g. an earthquake as a source of seismic waves)or artificial (e.g. produced by focusing of waves or by a wave sent in scattering from a pointscatterer). We show that this partial travel time data determines the unknown Riemannianmanifold up to a Riemannian isometry under some geometric constraints. Namely, we assumethat the unknown manifold has a strictly convex boundary. We also provide an exampledemonstrating that the convexity is a necessary assumption. To prove our result, we embeda Riemannian manifold with boundary into a function space and use smooth boundarydistance functions to give a coordinate and the Riemannian structures.

The inverse problem introduced above can be rephrased as the following problem in seis-mology. Imagine that earthquakes occur at known times but unknown locations withinEarth’s interior and arrival times are measured on some small area on the surface. Aresuch partial travel time measurements sufficient to determine the possibly anisotropic elasticwave speed everywhere in the interior and pinpoint the locations of the earthquakes? Whileearthquake times are not known in practice, this is a fundamental mathematical problemthat underlies more elaborate geophysical scenarios.

Date: January 7, 2022.2020 Mathematics Subject Classification. 53C21, 53C24, 53C80, 86A22.Key words and phrases. Inverse problem, Riemannian geometry, Distance function, Geodesics.

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An elastic body — e.g. a planet — can be modeled as a manifold, where distance ismeasured in travel time: The distance between two points is the shortest time it takesfor a wave to go from one point to the other. If the material is isotropic or ellipticallyanisotropic, then this elastic geometry is Riemannian. However, this sets a very stringentassumption on the stiffness tensor describing the elastic system, and Riemannian geometryis therefore insufficient to describe the propagation of seismic waves in the Earth. If nostructural assumptions on the stiffness tensor apart from the physically necessary symmetryand positivity properties are made, this leads necessarily to modeling the planet by a Finslermanifold as was explained in [13].

An isotropically elastic medium carries pressure (P) and shear (S ) wave speeds that areconformally Euclidean metrics. Of these two the P -waves are faster [8]. In order to be trueto the isotropic elasticity we should measure both P - and S -wave arrivals. In this paper wesimplify this aspect of the problem by disregarding polarizations and considering only onetype of isotropic waves.

Acknowledgements. EP was funded by the AWM and NSF-DMS grant # 1953892 fortravel to the 2022 Joint Mathematics Meetings (JMM), where this paper is to be presented.

1.1. Main theorem and the geometric assumptions. We consider a compact n-dimen-sional smooth manifold M with smooth boundary ∂M , equipped with a smooth Riemannianmetric g. For points p, q ∈M the Riemannian distance between them is denoted by d(p, q).Then for p ∈ M we define the boundary distance function rp : ∂M → R given by rp(z) =d(p, z). Let Γ be a non-empty open subset of the boundary ∂M . We denote the restrictionof the boundary distance function on this set as rp := rp

∣∣Γ. The collection

(1) Γ and rp : Γ→ R : rp(z) = d(p, z),are called the partial travel time data of Γ ⊂ ∂M . With these data we seek to recover theRiemannian manifold (M, g) up to a Riemannian isometry. The following definition explainswhen two Riemannian manifolds have the same partial travel time data (1).

Definition 1. Let (M1, g1) and (M2, g2) be compact, connected and oriented Riemannianmanifolds of dimension n ∈ N, n ≥ 2 with smooth boundaries ∂M1 and ∂M2 and open non-empty regions Γi ⊂ ∂Mi respectively. We say that the partial travel time data of (M1, g1)and (M2, g2) coincide if there exists a diffeomorphism φ : Γ1 → Γ2 such that

(2) rp φ−1 : p ∈M1 = rq : q ∈M2.

We want to emphasize that the equality (2) is for the non-indexed sets of travel time func-tions. Thus, for any p ∈ M1 there exists a point q ∈ M2 such that rp(φ

−1(z)) = rq(z) forevery z ∈ Γ2. We do not know a priori where the point p ∈ M1 is or if there are severalpoints q ∈M2 that satisfy this equation.

We use the notations TM and SM for the tangent and unit sphere bundles of M . Theirrespective fibers, for each point p ∈M , are denoted by TpM and SpM . In order to show thatthe data (1) determine (M, g), up to an isometry or in the other words that the Riemannianmanifolds (M1, g1) and (M2, g2) of Definition 1 are Riemanian isometric, we need to place anadditional geometric restriction. We assume that (M, g) has a strictly convex boundary ∂M

BOUNDARY DISTANCE REPRESENTATION WITH PARTIAL DATA 3

which means that the shape operator S : T∂M → T∂M as a linear operator on each tangentspace Tx∂M of the boundary ∂M for a point x ∈ ∂M is negative definite. This means thatthe second fundamental form

Πx(X, Y ) = 〈S(x)X, Y 〉g,is strictly negative whenever X, Y ∈ Tx∂M agree, but do not vanish. It has been shownin [5, 3] that the strict convexity of the boundary implies the geodesic convexity of (M, g).That is any pair of points p, q ∈M can be connected by a distance minimizing geodesic (notnecessarily unique) which is contained in the interior M int of M modulo the terminal points.In particular any geodesic of M that hits the boundary exits immediately.

The main theorem of this paper is the following:

Theorem 2. Let (M1, g1) and (M2, g2) be compact, connected, and oriented Riemannianmanifolds of dimension n ∈ N, n ≥ 2 with smooth and strictly convex boundaries ∂M1 and∂M2 and open non-empty measurement regions Γi ⊂ ∂Mi respectively. If the travel timedata of (M1, g1) and (M2, g2) coincide, in the sense of Definition 1, then the Riemannianmanifolds (M1, g1) and (M2, g2) are Riemannian isometric.

Remark 3. Our assumptions in Theorem 2 do not prevent the existence of the conjugatepoints. Actually quite a lot of work in this paper is needed to handle their existence.

1.2. Outline of the proof of Theorem 2. The main tool of proving Theorem 2 is todifferentiate the travel time functions given in the equation (1). As these functions aredefined only on a small open subset of the boundary we need to develop some regularitytheorem for them. For this reason in Section 3 we study the regularity properties of thedistance function on Riemannian manifolds satisfying the geometric constraints of Theorem2. Section 3 has two main results. Theorem 4 is the aforementioned regularity result and thekey of the proof of Theorem 2. In order to prove Theorem 4 we need to study, for each pointin our manifold, the properties of its cut locus. This is the set past which the geodesics shotfrom the chosen point are not anymore distance minimizers. Theorem 12 collects the neededproperties of these sets. Up to the best of our knowledge the material presented in Section3 does not exist or is not easily accessible in the literature. Nevertheless, the correspondingresults for manifolds without boundaries are well known.

In Section 4 we apply Theorem 4 to reconstruct the Riemannian manifold from its partialtravel time data (1). This is done in five parts. Firstly we recover the geometry of themeasurement region. As the second step we recover the topological structure by embeddingthe unknown manifold into a function space. Then we determine the boundary. The fourthstep is to find local coordinates. Since our manifold has a boundary, we need differenttypes of local coordinates for the interior and boundary points. Lastly we reconstruct theRiemannian metric. All the steps in Section 4 are fully data driven. Finally, in Section 5we show that if two Riemannian manifolds, as in Theorem 2, have coinciding partial traveltime data, in the sense of the Definition 1, then they are isometric.

1.3. The convexity of the domain in Theorem 2 is necessary. Let us construct anexplicit example of a surface M and a subset Γ ⊂ ∂M so that our results fail with datarecorded only on Γ (this example was originally presented in [11]). We recall that every pair


of points on a smooth compact Riemannian manifold with boundary is always connected bya C1-smooth distance minimizing curve [1]. We choose our a manifold to be the horseshoe-shaped domain of Figure 1. We split the domain M into two pieces M1 and M2 with respectto the line (red dotted line) that is normal to ∂M at x0 ∈ ∂M (blue dot). Then we choose adomain Γ ⊂ ∂M1 (red arch) so that any minimizing curve joining a point on Γ and a pointin M2 touches the boundary near x0. The curve P ⊂ M is any involute of the boundary,meaning that the distance from all points on P to x0 is the same. Because d(z, p) = d(z, q)for any z ∈ Γ and p, q ∈ P , from the point of view of our data (1), the set P appears tocollapse to a point.

PM2

x0

Γ M1

Figure 1. A domain where partial data is insufficient.

2. Some geometric inverse problems arising from seismology

In the following subsections we review some seismologically relevant geometric inverseproblems on Riemannian and Finsler manifolds.

2.1. Related geometric inverse problems on Riemannian manifolds. Let (M, g) beas in Theorem 2 and denote the Laplace-Beltrami operator of the metric g by ∆g. Let(p, t0) ∈ M int × R. In this paper we consider an inverse problem related to the waveequation

(3)

(∂2t −∆g)u(x, t) = δp(x)δt0(t),

u(x, t) = 0, t < t0, x ∈M,

where the solution u(x, t) is a spherical wave produced by an interior point source given bythe delta function δp(x)δt0(t) of the space time M ×R at (p, t0). We define the arrival timefunction Tp,t0 : ∂M → R, of the wave u by the formula

Tp,t0(z) = supt ∈ R; the point (z, t) ∈ ∂M ×R has a neighborhood

U ⊂M ×R such that u(·, ·)∣∣U

= 0.


We recall that it was shown in [32, Proposition 3.1], by applying the results of [16, 18], thatTp,t0(z) = d(p, z) + t0. If the initial time t0 is zero, (or given) the knowledge of the arrivaltime function Tp,t0 on the open set Γ ⊂ ∂M is equivalent to the corresponding travel timefunction rp(·) as in (1).

The problem of determining the isometry type of a compact Riemannian manifold fromits boundary distance data

rp : ∂M → (0,∞); rp(z) = d(p, x), p ∈M int

was introduced for the first time in [26]. The reconstructions of the smooth atlas on themanifold and the metric tensor in these coordinates was originally considered in [23]. Incontrast to the paper at hand these earlier results do not need any extra assumption for thegeometry, but have a complete data in the sense that the measurement area Γ is the wholeboundary. Their counterpart of our Theorem 4, is [23, Lemma 2.15], which says that theboundary distance function rp is smooth near its minimizers (the set of closest boundarypoints to the source point p ∈ M). This lemma is a key of their proof. However, the sametechnique is not available to us as it requires the access to the whole boundary.

The problem of boundary distance data is related to many other geometric inverse prob-lems. For instance, it is a crucial step in proving uniqueness for Gel’fand’s inverse boundaryspectral problem [23]. Gel’fand’s problem concerns the question whether the data

(∂M, (λj, ∂νφj|∂M)∞j=1)

determine (M, g) up to isometry, when (λj, φj) are the Dirichlet eigenvalues and the cor-responding L2-orthonormal eigenfunctions of the Laplace–Beltrami operator. Belishev andKurylev provide an affirmative answer to this problem in [4].

In [24] the authors studied a question of approximating a Riemannian manifold under theassumption: For a finite set of receivers R ⊂ ∂M one can measure the travel times d(p, ·)|Rfor finitely many p ∈ P ⊂M int under the a priori assumption that R ⊂ ∂M is ε-dense andthat d(p, ·)|R : p ∈ P ⊂ d(p, ·)|R : p ∈M int is also ε-dense. Thus d(p, ·)|R : p ∈ P is afinite measurement. The authors construct an approximate finite metric space Mε and showthat the Gromov-Hausdorff distance of M and Mε is proportional to some positive power of ε.In [24] an independent travel time measurement is made for each interior source point in P ,whereas in [11] the authors studied the approximate reconstruction of a simple Riemannianmanifold (a compact Riemannian manifold with strictly convex boundary whose each pairof points is connected by the unique smoothly varying distance minimizing geodesic) bymeasuring the arrival times of wave fronts produced by several point sources, that go offat unknown times, and moreover, the signals from the different point sources are mixedtogether. To describe the similarity of two metric spaces ‘with the same boundary’, theauthors defined a labeled Gromov-Hausdorff distance. This is an extension of the classicalGromov-Hausdorff distance which compares both the similarity of the metric spaces and thesameness of the boundaries — with a fixed model space for the boundary. In addition toreconstructing a discrete metric space approximation of (M, g), the authors in [11] estimatedthe density of the point sources and established an explicit error bound for the reconstructionin the labeled Gromov-Hausdorff sense.


If we do not know the initial time t0 in (3), but we can recover the arrival times Tp,t0(z)for each z ∈ ∂M , then taking the difference of the arrival times one obtains a boundarydistance difference function

Dp(z1, z2) := d(p, z1)− d(p, z2)

for all z1, z2 ∈ ∂M , which is independent to the initial time t0. In [32] it is shown thatif U ⊂ N is a compact subset of a closed Riemannian manifold (N, g) with a non-emptyinterior, then distance difference data

((U, g|U), Dp : U × U → R | p ∈ N)determine (N, g) up to an isometry. This result was generalized for complete Riemannianmanifolds [22] and for compact Riemannian manifolds with boundary [14, 21]. These resultsrequire the full boundary measurement in the sense of Γ = ∂M , unlike Theorem 2 in thepresent paper.

If the sign in the definition of the distance difference functions is changed, we arrive atthe distance sum functions,

D+p (z1, z2) = d(z1, p) + d(z2, p)

for all p ∈ M and z1, z2 ∈ ∂M . These functions give the lengths of the broken geodesics,that is, the union of the shortest geodesics connecting z1 to p and the shortest geodesicsconnecting p to z2. Also, the gradients of D+

p (z1, z2) with respect to z1 and z2 give thevelocity vectors of these geodesics. The inverse problem of determining the manifold (M, g)from the broken geodesic data, consisting of the initial and the final points and directions,and the total length, of the broken geodesics, has been considered in [27]. The authors showthat broken geodesic data determine the boundary distance data of any compact smoothmanifold of dimension three and higher. Finally they use the results of [23, 26] to prove thatthe broken geodesic data determine the Riemannian manifold up to an isometry. A differentvariant of broken geodesic data was recently considered in [38].

The Riemannian wave operator is a globally hyperbolic linear partial differential operatorof real principal type. Therefore, the Riemannian distance function and the propagationof a singularity initiated by a point source in space time are related to one another. Welet u be the solution of the Riemannian wave equation as in (3). In [17, 18] it is shown thatthe image, Λ, of the wave front set of u, under the musical isomorphism T ∗M 3 (x, ξ) 7→(x, gij(x)ξi) ∈ TM , coincides with the image of the unit sphere SpM at p ∈ M int underthe geodesic flow of g. Thus Λ ∩ ∂(SM), where SM is the unit sphere bundle of (M, g),coincides with the exit directions of geodesics emitted from p. In [33] the authors show that if(M, g) is a compact smooth non-trapping Riemannian manifold with smooth strictly convexboundary, then generically the scattering data of point sources (∂M,R∂M(M)) determine(M, g) up to an isometry. Here, R∂M(p) ∈ R∂M(M) for p ∈ M stands for the collection oftangential components to boundary of exit directions of geodesics from p to ∂M .

A classical geometric inverse problem, that is closely related to the distance functions,asks: Does the Dirichlet-to-Neumann mapping of a Riemannian wave operator determine aRiemannian manifold up to an isometry? For the full boundary data this problem was solvedoriginally in [4] using the Boundary control method. Partial boundary data questions havebeen studied for instance in [31, 40]. Recently [29] extended these results for connection


Laplacians. Lately also inverse problems related to non-linear hyperbolic equations havebeen studied extensively [28, 34, 51]. For a review of inverse boundary value problems forpartial differential equations see [30, 50].

Maybe the most extensively studied geometric inverse problem formulated with the dis-tance functions is the Boundary rigidity problem. This problem asks: Does the boundarydistance function, that gives a distance between any two boundary points, determine (M, g)up to an isometry? In an affirmative case (M, g) is said to be boundary rigid. For a gen-eral Riemannian manifold the problem is false: Suppose the manifold contains a domainwith very slow wave speed, such that all the geodesics starting and ending at the boundaryavoid this domain. Then in this domain one can perturb the metric in such a way that theboundary distance function does not change. It was conjectured in [39] that for all simpleRiemannian manifolds the answer is affirmative. In two dimensions this was verified in [44].For higher dimensional case the problem is still open, but different variations of it has beenconsidered for instance in [7, 9, 48, 49].

2.2. Related geometric inverse problems on Finsler manifolds. In [13] the authorsstudied the recovery of a compact Finsler manifold from its boundary distance data. Incontrast to earlier Riemannian results [23, 26] the data only determines the topological andsmooth structures, but not the global geometry. The Finsler function F : TM → [0,∞)can be however recovered in a closure of the set G(M,F ) ⊂ TM , that consists of points(p, v) ∈ TM such that the corresponding geodesic γp,v is distance minimizing to the terminalboundary point. They also showed that if the set TM \G(M,F ) is non-empty then any smallperturbation of F in this set leads to a Finsler metric whose boundary distance data agreeswith the one of F . If G(M,F ) = TM , then the boundary distance data determines (M,F )up to a Finsler isometry. For instance the isometry class of any simple Finsler manifoldis determined by this data. The same is not true if only the boundary distance functionis known [20]. Thus a simple Finsler manifold is never boundary rigid. In [12] the mainresult of [13] was utilized to generalize the result of [27], about the broken geodesic data, onreversible Finsler manifolds, satisfying a convex foliation condition.

Although, simple Finsler manifolds are not boundary rigid there are results consideringtheir rigidity questions for some special Finsler metrics. For instance it was shown in [41]that Randers metrics Fk = Gk + βk indexed with k ∈ 1, 2 with simple and boundary rigid

Riemannian norm Gk(x, v) =√gij(x)vivj and closed one-form βk, have the same boundary

distance function if and only ifG1 = Ψ∗G2 for some boundary fixing diffeomorphism Ψ: M →M and β1−β2 = dφ for some smooth function φ vanishing on ∂M . It is worth of mentioningthat analogous results have been presented earlier on a Riemannian manifold in the presenceof a magnetic field [2, 10].

3. Distance functions on compact manifolds with strictly convex boundary

The aim of this section is to prove the following regularity result for the Riemanniandistance function.

Theorem 4. Let (M, g) be a smooth, compact, connected, and oriented Riemannian manifoldof dimension n ∈ N, n ≥ 2 with smooth and strictly convex boundary. For any p0 ∈M there


exists an open and dense set Wp0 ⊂ ∂M such that for every z0 ∈ Wp0 there are neighborhoodsUp0 ⊂ M of p0 and Vp0 ⊂ M of z0 such that the distance function d(·, ·) is smooth in theproduct set Up0 × Vp0.

This result is the key of the proof of Theorem 2.

3.1. Critical distances, extensions and the cut locus. In this section we consider aRiemannian manifold (M, g) as in Theorem 4, and study the properties of several criticaldistance functions. We define the exit time function

τexit : SM → R ∪ ∞, τexit(p, v) = supt > 0 : γp,v(t) ∈M int,

where γp,v is the geodesic of (M, g) with the initial conditions (p, v) ∈ SM . Since theboundary of M is strictly convex τexit(p, v) is the first time when the geodesic γp,v hitsthe boundary and (−τexit(p,−v), τexit(p, v)) is the maximal interval where the geodesic γp,v isdefined. We do not assume that τexit(p, v) <∞ for all (p, v) ∈ SM . That is, (M, g) may havetrapped geodesics. It is well known that the exit time function, on manifolds with strictlyconvex boundary, is continuous and its restriction on SM \ T∂M (for those (p, v) ∈ SM forwhich τexit(p, v) <∞) is smooth (See for instance [47, Chapter 4]).

For any p ∈M we define a star shaped set

(4) Mp :=

v ∈ TpM : v = 0, ‖v‖g ≤ τexit

(p,

v

‖v‖g

).

Thus Mp is the largest subset of TpM where the exponential map of p

expp : Mp →M, expp(v) = γp,v(1)

is defined. Since ∂M is strictly convex this map is onto, but it does not need to be one-to-one, since there can be several geodesics of the same length connecting p to some commonpoint. This leads to the following definition of the cut distance function:

(5) τcut : SM → R, τcut(p, v) = supt ∈ (0, τexit(p, v)] : d(p, γp,v(t)) = t.

Thus the geodesic segment γp,v : [0, t] → M is a distance minimizing curve for any t ∈[0, τcut(p, v)].

Traditionally on a closed Riemannian manifold (N, g) the set

(6) cutN(p) := γp,v(τcut(p, v)) ∈ N : v ∈ SpN

is known as cut locus of the point p ∈ N and each point in this set is called a cut point ofp. Moreover, the cut locus of p coincides with the closure of the set of those points q ∈ Nsuch that there is more than one distance minimizing geodesics from p to q (see for instance[25, Theorem 2.1.14]). It has been also shown in [45, Section 9.1] that d(p, ·) is smooth inN\(p∪cutN(p)) but not at any q ∈ (p∪cutN(p)). In order to understand the smoothnessproperties of the distance function on a Riemannian manifold (M, g) with a strictly convexboundary, our aim is to define the set analogous to the one in (6) in this context.

If N is a closed manifold and (p, v) ∈ SN then by the Klingenberg’s lemma [36, Proposition10.32] either there is a second distance minimizing geodesic from p to γp,v(τcut(p, v)) or thesepoints are conjugate to each other along γp,v. In particular, the geodesic γp,v is not a distance


minimizer beyond the interval [0, τcut(p, v)]. The following lemma extends this result in ourcase.

Lemma 5. Let Riemannian manifold (M, g) be as in Theorem 4 and (p, v) ∈ SM . If

τcut(p, v) < τexit(p, v)

then at least one of the following holds for q := γp,v(τcut(p, v)):

• There exists another distance minimizing geodesic from p to q.• q is the first conjugate point to p along γp,v.

Moreover, for any t0 ∈ (0, τcut(p, v)) the geodesic segment γp,v : [0, t0] → M has no conju-gate points and is the unique unit-speed distance minimizing curve between its endpoints.

Proof. Since the exit time function is continuous on SM and q is an interior point of M , theproof is identical to the proof of the analogous claim in [36, Proposition 10.32].

Lemma 6. Let Riemannian manifold (M, g) be as in Theorem 4. The cut distance functionτcut is continuous in SM .

Proof. Since the exit time function is continuous, the proof of this claim is almost identicalto the proof of the analogous claims in [36, Theorem 10.33] and [25, Lemma 2.1.5].

As M has a boundary, the definition of the cut time function τcut, in the equation (5), hasan issue. Namely if τcut(p, v) = τexit(p, v) for some (p, v) ∈ SM we don’t know a priori if thegeodesic γp,v just hits the boundary at γp,v(τcut(p, v)) or if it is possible to find an extensionof (M, g) such that also γp,v extends as a distance minimizer.

To address this question, from here onwards we assume that (M, g) has been isometricallyembedded in some closed Riemannian manifold (N, g). This can be done for instance byconstructing the double of the manifold M as explained in [35, Example 9.32] and extendingthe metric g smoothly across the boundary ∂M . The issue with this extension is that itmight create ‘short cuts’ in the sense that there can be a curve in N , connecting some pointsof M , which is shorter than any curve entirely contained in M . Therefore we always have

dM(p, q) ≥ dN(p, q), for all p, q ∈M,

where dM(·, ·) and dN(·, ·) are the distance functions of M and N respectively. The followingproposition shows that while we stay close enough to M we don’t need to worry about theseshort cuts.

Proposition 7. Let (N, g) be a smooth, connected, orientable and closed Riemannian man-ifold and M ⊂ N an open set whose boundary ∂M is a smooth strictly convex hyper-surfaceof (N, g). There exists an open subset M of N , that contains the closure of M, and whoseboundary is a smooth, strictly convex hyper-surface of N .

Moreover

(7) dM(p, q) = dM(p, q), for all p, q ∈ M.

Proof. Since ∂M is a smooth hyper-surface of N there exists a smooth function s : N → Rand a neighborhood U of ∂M such that

|s(x)| = dist(x, ∂M) := infdN(x, z) : z ∈ ∂M, and ‖grad s(x)‖g ≡ 1,


for every x ∈ U . Moreover for each x ∈ U there exists a unique z ∈ ∂M such thatdN(x, z) = |s(x)|. We choose the sign convention of s such that s(x) ≥ 0 for x ∈ U \M .Then on ∂M the gradient of the function s(·) agrees with the outward pointing unit normalvector field of ∂M . The existence of this function is explained for instance in [36, Example6.43].

By this construction, each p ∈ U can be written uniquely as

p = (z(p), s(p)) ∈ ∂M ×R,

where z(p) is the closest point of ∂M to p. Thus on U we write the Riemannian metric as afunction of (z, ε) ∈ ∂M ×R in the form ds2 + g(ε, z), where g(ε, z) is the first fundamentalform of the smooth hyper-surface Ω(ε) := s−1ε. By [36, Proposition 8.18] we can thenwrite the second fundamental form of Ω(ε) as a bi-linear form

Π(z,ε)(X, Y ) = −1

2

∂

∂εgαβ(ε, z)XαY β ∈ R

on TΩ(s). Thus the eigenvalues λ1(z, ε), . . . , λn−1(z, ε) of Π(z,ε) are continuous functions of(z, ε) [52, Appendix V, Section 4, Theorem 4A]. Since Ω(0) coincides with ∂M , which isstrictly convex, we have that λα(z, 0) < 0 for every α ∈ 1, . . . , n − 1. Thus there existsε0 > 0 so that

λα(z, ε) < 0, for every α ∈ 1, . . . , n− 1 and |ε| < ε0.

Therefore, for small enough ε < 0, we have that

M(ε) := s−1(−∞, ε) ⊂M ∪ U

is an open set of N that contains M , and whose boundary ∂M(ε) = Ω(ε) is a smooth strictly

convex hyper-surface of N . We choose ε ∈ (0, ε0) and set M = M(ε).

Let p, q ∈ M and choose a distance minimizing unit speed geodesic γ : [0, dM(p, q)] → Mthat connects these points. Now without loss of generality we may assume that γ(t) ∈ U forsome t ∈ [0, dM(p, q)]. If this is not true then the trace of γ is contained in M and we aredone.

Since U is open and γ(t) ∈ U we can choose an interval [a, b] ⊂ [0, dM(p, q)] such thatγ([a, b]) ⊂ U and define a smooth function

s : [a, b]→ R, s(t) := s(γ(t)).

Since p and q are in M we may without loss of generality assume that s(a), s(b) ≤ 0.We aim to verify that s is always non-positive. To establish this we show that the maximum

value m ∈ R of s is attained at the endpoints of the domain interval. So suppose thatm = s(t0) is attained in some interior point t0 ∈ (a, b). As t0 is a maximum point of s,laying in the interior of the domain interval, it must hold that ˙s(t0) = 0 and ¨s(t0) ≤ 0. Onthe other hand since γ is a geodesic, we have by Weingarten equation [36, Theorem 8.13 (c)]that(8)˙s(t0) = 〈grad s(γ(t0)), γ(t0)〉g, and ¨s(t0) = 〈Dt grad s(γ(t0)), γ(t0)〉g = −Πγ(t0)(γ(t0), γ(t0)).


Here Dt stands for the covariant differentiation along the curve γ. Therefore γ(t0) is tan-gential to the strictly convex hyper-surface Ω(m) which implies that Πγ(t0)(γ(t0), γ(t0)) < 0.

This in conjunction with (8) leads into a contradiction with ¨s(t0) ≤ 0. We have verified that

for all p, q ∈ M any distance minimizing geodesic in M , between these points, is containedin M . Therefore the equation (7) is true.

By Proposition 7 we may assume that M is contained in the interior of some compact,Riemannian manifold (M, g) with a smooth strictly convex boundary. Moreover the distance

function of M restricts to the one of M . Thus for every (p, v) ∈ SM where p is in M wealways have that

(9) τcut(p, v) ≤ ˆτcut(p, v), and τexit(p, v) < ˆτexit(p, v),

where ˆτcut and ˆτexit are the cut distance and the exit time functions of (M, g) respectively.Motivated by this observation we define the cut locus of a point p ∈M as

cut(p) :=γp,v(τcut(p, v)) ∈M : v ∈ SpM, τcut(p, v) = ˆτcut(p, v).(10)

The following result summaries the basic properties of these sets.

Proposition 8. Let Riemannian manifold (M, g) be as in Theorem 4. Let p ∈M .

• The cut locus cut(p) of the point p is a closed set of measure zero.• If q ∈ cut(p) and γ is a unit speed distance minimizing geodesic of M between p andq. Then at least one of the following holds:(1) There exists another distance minimizing geodesic from p to q.(2) q is the first conjugate point to p along γ.

Proof.

• The proof of the first claim is identical to the proof of [36, Theorem 10.34 (a)], thusomitted here.• If q ∈ cut(p) then by (9) there is v ∈ SpM such that

q = γp,v(τcut(p, v)), τcut(p, v) = ˆτcut(p, v) ≤ τexit(p, v) < ˆτexit(p, v),

Thus Lemma 5 and Proposition 7 yield the second claim.

The following result introduces an open and dense subset of M where the distance functionof an interior point is smooth.

Lemma 9. Let Riemannian manifold (M, g) be as in Theorem 4. Let p ∈M . The distancefunction d(p, ·) : M → R, is smooth precisely in the open and dense set M \ (p ∪ cut(p)).

Proof. The proof is identically to the proof of the analogous claim of [45, Chapter 5, Section9, Corollary 7] thus omitted here.

Proposition 10. Let Riemannian manifold (M, g) be as in Theorem 4. Let p ∈ M andq ∈ M \ (p ∪ cut(p)). There exist neighborhoods U ⊂ M of p and V ⊂ M of q such thatthe distance function d(·, ·) is smooth in the product set U × V .


Proof. Let (N, g) be a closed extension of (M, g) as in Proposition 7. We define a smoothmap

F : (x, v) ∈ TN 7→ (x, expx(v)) ∈ N ×N.Then the differential of this map can be written as

DF (x, v) =

[Id 0∗ D expx(v)

].

Since q is not in the cut locus of p there is a v0 ∈ TpN such that ‖v0‖g = dM(p, q), expp(v0) = qand D expp(v0) is not singular. Therefore det(DF (p, v0)) = det(D expp(v0)) does not vanish.

Thus the Inverse function theorem implies that there are neighborhoods W ⊂ TN of (p, v0)and W ⊂ N ×N of (p, q) such that the local inverse function of F ,

F−1 : W → W , F−1(x, y) = (x, exp−1x (y)),

is a diffeomorphism.Let (M, g) be the extension of (M, g) as in Proposition 7. Since q is not in the cut locus

of p we have and ‖F−1(p, q)‖g < ˆτcut

(p, v0‖v0‖g

). Thus by the continuity of the cut distance

function ˆτcut (·, ·) of (M, g) we can choose a neighborhood W1 ⊂ W of (p, q) such that

‖F−1(x, y)‖g < ˆτcut

(x,

F−1(x, y)

‖F−1(x, y)‖g

), for all (x, y) ∈ W1.

This gives dM(x, y) = ‖F−1(x, y)‖g, for all (x, y) ∈ W1.Finally we choose disjoint neighborhoods U ⊂M of p and V ⊂M of q such that U ×V is

contained in W1. Let (x, y) ∈ U × V then γ(t) := expx(tF−1(x, y)) for t ∈ [0, 1] is a geodesic

of M that connects x to y having the length of dM(x, y). Since both x and y are in M , weget from the proof of Proposition 7 that γ(t) is contained in M . This yields

(11) dM(x, y) = ‖F−1(x, y)‖g, for all (x, y) ∈ U × V.Since the sets U and V are disjoint we have that F−1 does not vanish in U × V . Henceequation (11) gives the smoothness of dM(·, ·) on U × V .

Recall that we have assumed that M is isometrically embedded in a closed Riemannianmanifold (N, g). Thus any geodesic starting from M can be extended to whole R. Letp ∈ N . We define the conjugate distance function τcon : SpN → R ∪∞ by the formula:

τcon(p, v) = inft > 0 : γp,v(t) is a conjugate point to p.As the infimum of the empty set is positive infinity we set τcon(p, v) = ∞ in the case whenthe geodesic γp,v does not have any conjugate points to p. Since geodesics do not minimizethe distance beyond the first conjugate point it holds that

τcut(p, v) ≤ τcon(p, v), if (p, v) ∈ SM.

The following result is well known, but we could not find it’s proof in the existing literature,so we provide one below.

Lemma 11. Let (N, g) be a closed Riemannian manifold and p ∈ N . The conjugate distancefunction is continuous on SpN .


Proof. Let vi ∈ SpN for i ∈ N converge to v.We set

C = τcon(p, v), B = lim infi→∞

τcon(p, vi), and A = lim supi→∞

τcon(p, vi).

It suffices to show that A ≤ C ≤ B.

We assume first that C = ∞. If A < ∞ we choose a sub-sequence vik of vi such thatτcon(p, vik) converges to A. Then det(D expp(τcon(p, vik)vik)) = 0, and the smoothness of theexponential map gives det(D expp(Av)) = 0 yielding that expp(Av) is conjugate to p alongγp,v. This implies that τcon(p, v) ≤ A, which is impossible. By the same argument we seethat B =∞.

Let C < ∞, and by the same limiting argument as above we get C ≤ B. Then we showthat A ≤ C. Choose a sub-sequence vik such that τcon(p, vik)→ A as k →∞. For a sake ofcontradiction we suppose that A > C. By the definition of the conjugate distance functionwe have that q = γp,v(C) is the first conjugate to p along γp,v. By [36, Theorem 10.26] forany ε ∈ (0, A − C) there exists a piecewise smooth vector field X on the geodesic segmentγp,v : [0, C + ε] → N , that vanishes on 0 and C + ε such that the index form of γp,v over Xis strictly negative. That is

(12) Iv(X,X) :=

∫ C+ε

0

〈DtX,DtX〉g + 〈R(γp,v, X)γp,v, X〉g dt < 0.

Here we used the notation Dt for the covariant differentiation along γp,v. The capital Rstands for the Riemannian curvature tensor.

We choose vectors E1, . . . , En of TpN that form a basis of TpN and extend them on γp,v(t)for t ∈ [0, C + ε] via the parallel transport. Since parallel transport is an isomorphism thevector fields E1(t), . . . , En(t) constitute a basis of Tγp,v(t)M . We write X(t) = Xj(t)Ej(t).Since X is piecewise smooth it holds that the component functions Xj(t) are piecewisesmooth. This lets us to ‘extend’ X on γp,vik by the formula

(13) Xk(t) = Xj(t)Ekj (t),

where the vector field Ekj (t) is the parallel transport of Ej along γp,vik . Thus Xk is a piecewise

smooth vector field on γp,vik that vanishes at t = 0 and t = C + ε.Since vik → v, as k →∞, it holds that

γp,vik (t)→ γp,v(t), and Ekj (t)→ Ej(t), uniformly in t ∈ [0, C + ε] as k →∞.

Therefore by (12), (13), and the continuity of the Levi-Civita connection and the Riemanniancurvature tensors we have

Ivik (Xk, Xk) < 0, for large enough k ∈ N.

By [36, Theorem 10.28] there exists sk ∈ (0, C+ε] so that γp,vik (0) and γp,vik (sk) are conjugate

points. Therefore we must have sk ≥ τcon(p, vik) and we arrive in a contradiction τcon(p, vik) <A. This ends the proof.


3.2. The Hausdorff dimension of the cut locus. We fix a point p ∈ M for this sub-section. By Lemma 9 we know that for any p ∈ M the distance function d(p, ·) is smoothin the open set M \ (cut(p) ∪ p). Moreover by Proposition 10 for each q in this open setthere are neighborhoods U of p and V of q such that the distance function is smooth in theproduct set U × V . As we are interested in the inverse problem where we study distancefunction d(p, ·) restricted on some open subset Γ of the boundary, we don’t know a priori ifthis function is smooth on Γ. In particular we don’t know the size of the set cut(p) ∩ ∂Myet. In this sub-section we show that the set ∂M \ cut(p), where d(p, ·) is smooth, is alwaysan open and dense subset of ∂M .

Proposition 8 yields that cut(p) can be written as a disjoint union of

• Conjugate cut points:

Q(p) := γp,v(t) ∈M : v ∈ SpM, t = τcut(p, v) = τcon(p, v) ⊂ cut(p),

that are those points q ∈ cut(p) such that there exits a distance minimizing geodesicfrom p to q along which these points are conjugate to each other. By Proposition 8and Lemma 11 the set Q(p) is closed in M .• Typical cut points: T (p) ⊂ (cut(p) \ Q(p)) that can be connected to p with exactly

two distance minimizing geodesics.• A-typical cut points: L(p) ⊂ (cut(p) \ Q(p)) that can be connected to p with more

than two distance minimizing geodesics. Thus an a-typical cut point is both non-conjugate and non-typical.

It was proven in [19] that the Haussdorff dimension of the cut locus on a closed Riemannianmanifold (N, g) is locally an integer that does not exceed dimN − 1. Moreover T (p) is asmooth hyper surface of N and the Hausdorff dimension of Q(p) ∪ L(p) does not exceeddimN − 2. In this paper we will extended these results for manifolds with strictly convexboundary. The main result of this section is as follows:

Theorem 12. Let (M, g) be a smooth, compact, connected, and oriented Riemannian man-ifold of dimension n ∈ N, n ≥ 2 with smooth and strictly convex boundary. If p ∈ Mthen

(1) The set T (p) of typical cut points is a smooth hyper-surface of M that is transverseto ∂M .

(2) The Hausdorff dimension of Q(p) ∪ L(p) does not exceed n− 2.(3) The Hausdorff dimension of cut(p) does not exceed n− 1.(4) The set ∂M \ cut(p) is open and dense in ∂M .

For the readers who want to learn more about Hausdorff measure and dimension we suggestto have look at [6, 37]. Some basic properties of the Hausdorff dimension dimH is collectedin the following lemma.

Lemma 13. Basic properties of the Hausdorff dimension are:

• If X is a metric space and A ⊂ X then dimH(A) ≤ dimH(X).• If X is a metric space and X is a countable cover of X then

dimH(X) = supA∈X

dimH(A).


• If X, Y are metric spaces and f : X → Y is a bi-Lipschitz map then dimH(A) =dimH(f(A)) for any A ⊂ X.• If U ⊂ M is open and M is a Riemannian manifold of dimension n ∈ N then

dimH(U) = n.

From here onwards we follow the steps of [19, 42] and develop machinery needed for theproof of Theorem 12. We recall that we have isometrically embedded M into the closedRiemannian manifold (N, g). The maximal subset Mp ⊂ TpM where the exponential mapexpp : Mp →M of (M, g) is well defined was given in (4) as

Mp =

X ∈ TpM : X = 0 or ‖X‖g ≤ τexit

(p,

X

‖X‖g

).

Thus the exponential function expp : TpN → N of N agrees with that of M in Mp.Let v0 ∈ Mp be such that the exponential map expp is not singular at v0. The Inverse

function theorem yields that there are neighborhoods U ⊂ TpN of v0 and V ⊂ N of x0 =

expp(v0) ∈M such that expp : U → V is a diffeomorphism. We want to emphasize that evenwhen v0 ∈Mp the set U ⊂ TpN does not need to be contained in Mp. If we equate TpN andTv0(TpN) we note that the formula

Y (x) = D expp

∣∣∣∣exp−1

p (x)

exp−1p (x)

defines a smooth vector field on V that satisfies the following properties

(14) Y (x) = γp,exp−1p (x)(1), and ‖Y (x)‖g = ‖ exp−1

p (x)‖g.

The vector field Y is called a distance vector field related to p and U . Let x ∈ V andX ∈ TxN . It holds by the similar proof as [42, Lemma 2.2.] that

(15) X‖Y (x)‖g =〈X, Y (x)〉g‖Y (x)‖g

.

In what follows we will always consider cut(p) as defined for (M, g) in equation (10). Letq ∈ cut(p) \ Q(p) and λ = d(p, q). By Proposition 8 it holds that there are at least twoM -distance minimizing geodesics from p to q. Thus the set

Ep,q := exp−1p q ∩ SλM, where SλM = w ∈ TpM : ‖w‖g = λ,

contains at least two points.By the compactness of SλM and the continuity of the exit time function we get from the

assumption that q /∈ Q(p) that the set Ep,q is finite. We write

Ep,q = wi : i ∈ 1, . . . , kp(q),where kp(q) ∈ N is the number of distance minimizing geodesics from p to q. Since the setQ(p) is closed in M , the complement cut(p) \ Q(p) is relatively open in cut(p), and thereexists a open neighborhood W ⊂ M of q such that Q(p) ∩W = ∅. Thus by the previousdiscussion for any x ∈ cut(p) ∩ W there are only kp(x) ∈ N many distance minimizinggeodesics connecting p to x. The following lemma is an adaptation of the analogous resultgiven in [42].


Lemma 14. Let (M, g) be a Riemannian manifold as in Theorem 12. Let p ∈ M andq ∈ cut(p) \ Q(p). Let the closed manifold (N, g) be as in Proposition 7. Then there is aneighborhood V of q in M such that

(16) kp(x) ≤ kp(q), for every x ∈ cut(p) ∩ V.

Proof. Since the set Ep,q is finite we can choose disjoint neighborhoods Ui ⊂ TpN for each

wi ∈ Ep,q, so that for each i ∈ 1, . . . , kp(q) the map expp : Ui → V is a diffeomorphism on

some open set V ⊂ N that contains q. We want to show that there is a neighborhood V ⊂Mof q such that for every x ∈ V and for any M -distance minimizing unit speed geodesic γfrom p to x there is i ∈ 1, . . . , kp(q) such that

γ(t) = expp

(tX

‖X‖g

), for some X ∈Mp ∩ Ui.

Clearly this implies the inequality (16).If the former is not true then there exist a sequence qk ∈ M that converges to q and

Xk ∈Mp so that for each k ∈ N we have

• expp(Xk) = qk• ‖Xk‖g = dM(p, qk) ≤ τexit(p,

Xk

‖Xk‖g)

• expp

(t Xk

‖Xk‖g

)for t ∈ [0, dM(p, qk)] is a unit speed distance minimizing geodesic from

p to qk.• Xk /∈ U1 ∪ . . . ∪ Ukp(q).

These imply thatlimk→∞‖Xk‖g = lim

k→∞dM(p, qk) = dM(p, q).

Moreover, the sequence Xk ∈ TpM is contained in some compact subset K of TpM . Afterpassing to a sub-sequence we may assume Xk → X ∈ TpM and the continuity of the exittime function gives ‖X‖g ≤ τexit(p,

X‖X‖g ). Thus X ∈ Mk and by the continuity of expp we

get expp(X) = q, and ‖X‖g = dM(p, q).

Therefore t 7→ expp

(t X‖X‖g

)is a M -distance minimizing geodesic from p to q and X

must coincide with wi for some i ∈ 1, . . . , kp(q). Therefore Xk ∈ Ui for large enough

k ∈ N. This contradicts the choice of Xk and possible after choosing a smaller V we can setV = M ∩ V .

Suppose now that q ∈ T (p) is a typical cut point, and V ⊂ M is a neighborhood of q asin Lemma 14. Then by (16) it holds that

(17) cut(p) ∩ V ⊂ T (p),

and Ep,q = w1, w2 ⊂ TpN are the directions that give the two distance minimizing geodesicsexpp(twi), t ∈ [0, 1] from p to q. Let U1, U2 ⊂ TpN be the neighborhoods of w1 and w2 and

V ⊂ N a neighborhood of q as in the proof of Lemma 14. Finally we consider the distancevector fields Y1, Y2 related to p and U1 and U2. Since these vector fields do not vanish on Vthe function

ρ : V → R, ρ(x) = ‖Y1(x)‖g − ‖Y2(x)‖g,


is smooth. The following result is an adaptation of [42, Propositions 2.3 & 2.4].

Lemma 15. Let Riemannian manifold (M, g) be as in Theorem 12 and p ∈ M . Let q ∈T (p) ⊂M and define the closed manifold N as in Proposition 7. Let the neighborhood V ⊂ Nof q and function ρ : V → R be as above. Then possibly after choosing a small enough V wehave

(18) ρ−10 ∩M = V ∩ cut(p).

Moreover, the set ρ−10 is a smooth hyper-surface of N whose tangent bundle is given bythe orthogonal complement of the vector field Y1 − Y2.

Proof. We prove first the equation (18).

• Let x ∈ ρ−10∩M . By the proof of Lemma 14 we can assume that aM -distance min-imizing speed geodesic from p to x is given by expp (tX1) , t ∈ [0, 1], for some X1 ∈Mp ∩ U1. Also x = expp(X2) for some X2 ∈ U2, but we don’t know a priori ifexpp(tX2) ∈ M for all t ∈ [0, 1] or equivalently if X2 ∈ Mp. However, by the defini-tion of the distance vector fields and the assumption x ∈ ρ−10 we have that

(19) dM(p, x) = ‖X1‖g = ‖Y1(x)‖g = ‖Y2(x)‖g = ‖X2‖g.

Let M be as in Proposition 7. Thus we can assume that V ⊂ M . Since q ∈ T (p)

there is w2 ∈ U2 so that expp(w2) = q and expp(tw2) ∈ M ⊂ M, for every t ∈ [0, 1].

Since q is an interior point of M we can again choose choose smaller V so thatexpp(tX) ∈ M, for every t ∈ [0, 1] and X ∈ Ui, for i ∈ 1, 2. Since x ∈ M and

expp(tX2) is a geodesic of M that connects p to x having the length of ‖X2‖g, theequation (19) and Proposition 7 imply that expp(tX2) ∈ M for every t ∈ [0, 1].

Therefore equation (19) gives x ∈ V ∩ cut(p).• Let x ∈ V ∩cut(p) ⊂ T (p). Thus there are exactly two distance minimizing geodesics

of M from p to x. Since x ∈ V , it holds by the proof of Lemma 14 that one of thesegeodesics has the initial velocity in U1 and the other in U2. Therefore ρ(x) is zero bythe definition of the distance vector fields.

Then we prove that the set ρ−10 is a smooth hyper-surface whose tangent bundle isorthogonal to the vector field Y1 − Y2. By (15) we get

(20) Xρ(x) =〈X, Y1(x)〉g‖Y1(x)‖g

− 〈X, Y2(x)〉g‖Y2(x)‖g

=〈X, Y1(x)− Y2(x)〉g

‖Y1(x)‖g, for every x ∈ ρ−10.

Moreover the vector field Y1−Y2 does not vanish on V , since the geodesics related to thesetwo vector fields are different. This implies that the differential of the map ρ does not vanishin V . Thus the set ρ−10 is a smooth hyper-surface of N , and by (20) its tangent bundleis given by those vectors that are orthogonal to Y1 − Y2.

Now we consider the set of conjugate cut points Q(p). First we define a function

δ : SpN → 0, 1, . . . , n− 1, δ(v) is the dimension of the kernel of D expp at τcon(p, v)v

If τcut(p, v) =∞ we set λ(v) = 0.


Lemma 16. Let (N, g) be a closed Riemannian manifold. Let p ∈ N and v0 ∈ SpN be suchthat δ(v0) = 1. There exists a neighborhood U ⊂ SpN of v0 such that the δ(·) is the constantfunction one in U .

Proof. We define the function

i : [0,∞)× SpN → N, i(t, v) is the index of the index form of γp,v on the interval [0, t].

By this we mean that i(v, t) ∈ N is the dimension of the vector space V (t, v) of piecevisesmooth vector fields X, Y along the geodesic γp,v that vanishes at 0, t and for which thebi-linear symmetric form

It,v(X, Y ) :=

∫ t

0

〈DsX,DsY 〉g + 〈R(γp,v, X)γp,v, Y 〉g ds,

is negative semi-definite. By the proof of the Morse index theorem (see [15, Chapter 11]) itholds that

i(t, v) =

0, t < τcon(p, v)δ(v), t ∈ [τcon(p, v), ε(v)),

where ε(v) > 0 depends on v ∈ SpN . Moreover, no γp,v(t), for t ∈ (τcon(p, v), ε(v)) isconjugate to p along γp,v.

We choose t ∈ (τcon(p, v0), ε(v0)). By the proof of the Morse index theorem, the indexform It,v0 can only be negative semi-definite on a finite dimensional vector space V−(0, t),which is independent to the geodesics that are close to γp,v0 . Since γp,v0(t) is not conjugateto p along γp,v0 it follows from [15, Chapter 11, Corollary 2.4] that It,v0(Y, Y ) > 0 for vectorfields Y in a sub-space Wt,v0 ⊂ V−(0, t), which by the Finite-dimensional spectral theoremsatisfies V−(0, t) = Wt,v0⊕Vt,v0 . Moreover, the linear operator Lt,v0 : V−(0, t)→ V−(0, t), cor-responding to the bi-linear form It,v0 , has i(t, v0) negative and dimV−(0, t)− i(t, v0) positiveeigenvalues. Since these eigenvalues depend continuously of the initial direction v ∈ SpN ,that are near v0, we can find a neighborhood U ⊂ SpN of v0 such that the linear operatorLt,v, related to the bi-linear form It,v, has i(t, v0) negative and dimV−(0, t)− i(t, v0) positiveeigenvalues for every v ∈ U . Furthermore, the continuity of the first conjugate time function,that was established in Lemma 11, gives τcon(p, v) < t for v ∈ U . By the proof of the Morseindex theorem the function i(t, v) is increasing in t. We have in particular that

1 ≤ δ(v) = i(τcon(p, v), v) ≤ i(t, v) = i(t, v0) = δ(v0) = 1, for every v ∈ U.

This ends the proof.

Lemma 17. Let (N, g) be a closed Riemannian manifold, p ∈ N and suppose that δ isconstant in some open set U ⊂ SpM . Then τcon(p, ·) is smooth in U .

Proof. If δ is zero in U then τcon(p, ·) is infinite and we are done. So we suppose that δ equalsto k ∈ 1, . . . , n− 1 in U and get τcon(p, v) <∞ for every v ∈ U .

Let ξ1, . . . , ξn−1 be a base of Tv0SpM and use the formula

Jv0,β(t) = D expp

∣∣∣∣tv0

tξβ, for β ∈ 1, . . . , n− 1,


from [36, Proposition 10.10], to define (n − 1)-Jacobi fields Jv0,1(t), . . . , Jv0,n−1(t) along thegeodesic γp,v0 . They span the vector space of all Jacobi fields along γv0(t) that vanishes att = 0 and are normal to γv0(t). As δ(v0) = k we may assume that Jv0,β(τcon(p, v0)) = 0 forβ ∈ 1, . . . , k, implying DtJv0,β(τcon(p, v0)) 6= 0 and Jv0,α(τcon(p, v0)) 6= 0 for β ∈ 1, . . . , kand α ∈ k + 1, . . . , n− 1. Moreover, the vectors

(21) DtJv0,1(τcon(p, v0)), . . . , DtJv0,k(τcon(p, v0)), Jv0,k+1(τcon(p, v0)), . . . Jv0,n−1(τcon(p, v0))

are linearly independent due to [25, Proposition 2.5.8 (ii)] and the properties of the geodesicflow on SN presented in [43, Lemma 1.40].

Since Jacobi fields are solutions of the second order ODE they depend smoothly on thecoefficients of the respective equation. In particular after choosing smaller U if necessarywe can construct family of Jacobi fields Jv,1(t), . . . , Jv,n−1(t) along the geodesic γp,v(t) thatdepend smoothly on v ∈ U , and span the vector space of all Jacobi fields along γp,v(t) thatvanishes at t = 0 and are normal to γp,v(t). Therefore the function

f : U × [0, τcon(p, v0) + 1]→ R, f(v, t) = det(Jv,1(t), . . . , Jv,n−1(t)),

is smooth and vanishes at (v, t) if and only if γp,v(t) is conjugate to p.We choose a parallel frame E1(t), . . . , En−1(t) along γp,v0(t) that is orthogonal to γp,v0(t).

With respect to this frame we write

Jv0,β(t) = jαβ (t)Eα(t), for α, β ∈ 1, . . . , n− 1,

for some some smooth functions jαβ (t). From here we get

∂j

∂tjf(v0, t0) =

∂j

∂tj

(∑σ

sign(σ)jσ(1)1 (t)j

σ(2)2 (t) · · · jσ(n−1)

n−1 (t)

)∣∣∣∣t=τcon(p,v0)

= 0,

for every j ∈ 0, . . . , k − 1. Above σ is a permutation of the set 1, . . . , n − 1. Moreoverthe covariant derivative of Jv0,β along γp,v0 writes as DtJv0,β(t) =

(ddtjαβ (t)

)Eα(t).

If A is the square matrix whose column vectors are given in the formula (21) we have

∂k

∂tkf(v0, τcon(p, v0)) = k! det(A) 6= 0.

Since δ(·) is constant k in the set U we have that ∂k−1

∂tk−1f(v, τcon(p, v)) = 0 for every v ∈ U .Therefore the Implicit function theorem gives that the conjugate distance function is smoothin some neighborhood V ⊂ U of v0. Since v0 ∈ U was chosen arbitrarily the claim follows.

Let v0 ∈ SpN be such that τcon(p, v0) < ∞. Then by lemmas 6 and 11 the functioneq(v) = expp(τcon(p, v)v) ∈ N is well defined and continuous on some neighborhood U ⊂ Spnof v0. Moreover, we have that

(22) Q(p) = eq(v) ∈M : τcut(p, v) = τcon(p, v).

The following result is an adaptation of [19, Lemma 2].

Proposition 18. Let Riemannian manifold (M, g) be as in Theorem 12 and p ∈ M . TheHausdorff dimension of Q(p) does not exceed n− 2.


Proof. By (22) we can write the conjugate cut locus Q(p) as a disjoint union of the sets

A1 = eq(v) ∈M : τcut(p, v) = τcon(p, v), δ(v) = 1

and

A2 = eq(v) ∈M : τcut(p, v) = τcon(p, v), δ(v) ≥ 2.To prove the claim of this proposition it suffices to show that

(23) A1 ⊂ eq(v) ∈ N : dim (Deq(TvSpM)) ≤ n− 2,

since clearly we have that

A2 ⊂ expp(w) ∈ N : w ∈ TpN, dim(D expp(Tw(TpN))

)≤ n− 2,

and therefore by the generalization of the classical Sard’s theorem [46] the Hausdorff dimen-sion of Q(p) = A1 ∪ A2 is at most n− 2.

We choose v0 ∈ SpN such that eq(v0) ∈ A1. By the properties of the Jacobi fields normal toγp,v0 , we can identify the kernel D expp(τcon(p, v0)v0) with some vector sub-space of Tv0SpM.

Since dimTv0SpM = n− 1 we can verify the inclusion (23) if we show that

(24) ker D expp(τcon(p, v0)v0) ⊂ ker Deq(v0).

Since δ(v0) = 1 we get by Lemmas 16 and 17 that there exists a neighborhood U ⊂ SpN ofv0 where the conjugate distance τcon(p, ·) and the map eq(v) = expp(τcon(p, v)v) are smooth.Let ξ ∈ Tv0SpM be in the kernel of the differential of the exponential map. Then by thechain and Leibniz rules we get

Deq(v0)ξ = γp,v0(τcon(p, v0))Dτcon(p, v0)ξ.

Therefore Deq(v0)ξ = 0 if and only if Dτcon(p, v0)ξ = 0. So we suppose that Deq(v0)ξ 6= 0.By the Rank theorem [35, Theorem 4.12] we get that the subset of TpN , near τcon(p, v0)v0,

where D expp vanishes is diffeomorphic to a smooth sub-bundle of TU ⊂ TSpN . Then weuse the existence of the ODE theorem to choose a smooth curve v(·) : (−1, 1) → U ⊂ SpNsuch that v(0) = v0, v(0) = ξ and v(t) ∈ ker D expp(τcon(p, v(t))v(t)) for every t ∈ (−1, 1).

Thus c(t) := eq(v(t)) is a smooth curve in M that satisfies

(25) c(t) = γp,v(t)(τcon(p, v(t)))d

dt(τcon(p, v(t))).

Since ddt

(τcon(p, v(t0))) = Dτcon(p, v0)ξ we can assume that ddt

(τcon(p, v(t))) > 0 on someinterval (−ε, ε) for 0 < ε < 1. Thus by equation (25) and the fundamental theorem ofcalculus we get that the length of c(t) on [−ε, 0] is τcon(p, v0)− τcon(p, v(−ε)). From here bythe assumption τcon(p, v0) = τcut(p, v0) and the triangle inequality we get

τcon(p, v(−ε)) ≥dM(p, eq(v(−ε))) ≥ dM(p, eq(v0))− L(c) ≥ τcon(p, v(−ε)),

and the inequality above must hold as an equality. Therefore

dM(p, eq(v(−ε))) + L(c) = dM(p, eq(v0)),


and the curve c(·) : [−ε, 0] → M is part of some distance minimizing geodesic γ of M fromp to eq(v(0)) that contains eq(v(−ε)). Thus we have after some reparametrization t = t(s)that

γ(s) = eq(v(t(s))) = c(t(s))

for every t(s) ∈ (−ε, 0). By (25) we get that γ is a parallel to γp,v(t) for every t ∈ (−ε, 0).This is not possible unless the geodesics γp,v(t) are all the same for every t ∈ (−ε, 0). Hencev(t) and c(t) are constant curves. This leads into a contradiction. The inclusion (24) isconfirmed and the proof is complete.

We are ready to present the proof of Theorem 12.

Proof of Theorem 12. Let p ∈M . In this proof we combine the observations made earlier inthis section. The proofs of the four sub-claims are given below.

(1) By Lemma 15 we know that T (p) is a smooth hyper-surface of M whose tangent spaceis normal to the vector field ν(q) = Y1(q) − Y2(q) for q ∈ T (p). Since Y1(q) 6= Y2(q)and ‖Y1(q)‖g = ‖Y2(q)‖ we get from Cauchy-Schwarz inequality that

〈Y1(q), ν(q)〉 > 0 and 〈Y2(q), ν(q)〉〉 < 0.

Thus Y1(q) and Y2(q) hit T (p) from different sides. If q ∈ T (p) ∩ ∂M and thesesurfaces are tangential to each other at q we arrive in a contradiction: Since ν(q)is normal to both T (T (p)) and T∂M we can without loss of generality assume thatY2(q) is inward pointing at q. This is not possible since the geodesic related to Y2(q),that connects p to q, is contained in M . Thus by equation (14) Y2(q) is also outwardpointing which is not possible.

(2) By Proposition 18 we know that the Hausdorff dimension of the conjugate cut locusQ(p) does not exceed n − 2. If we can prove the same for the set L(p) of a-typicalcut points the claim (2) follows from Lemma 13.

Recall that L(p) ⊂ (cut(p) \Q(p)) is the set of points in M that can be connectedto p with more than two distance minimizing geodesics of M . Let q ∈ L(p) and definekp(q) ∈ N to be the number of distance minimizing geodesics from p to q. Then wechoose vectors w1, . . . , wkp(q) ∈Mp and their respective neighborhoods Ui ∈ TpN suchthat for each i ∈ 1, . . . , kp(q)

expp(wi) = q, and expp : Ui → V

is a diffeomorphism on some open set V ⊂ N . Let Yi for i ∈ 1, . . . , kp(q) be thedistance vector fields related to the Ui and p. Then we define a collection of smoothfunctions

ρij : V → R, ρij(x) = ‖Yi(x)‖g − ‖Yj(x)‖g, i, j ∈ 1, . . . , kp(q).

By the proof of Lemma 15 it holds that the sets Kij := ρ−1ij 0, for i < j are smooth

hyper-surfaces of N that contain q. Also by [42, Proposition 2.6] it holds that thesets

Ki,j,k := Kik ∩Kjk, for i < j < k


are smooth submanifolds of N of co-dimension two. Next we set K(q) :=⋃i<j<kKi,j,k

and claim that

(26) L(p) ∩ V = K(q) ∩M.

Since the sets Ki,j,k are smooth sub-manifolds of dimension n − 2 their Hausdorffdimension is also n− 2. Thus the equation (26) and Lemma 13 imply that Hausdorffdimension of L(p) does not exceed n− 2.

Finally we verify the equation (26). If x ∈ L(p)∩ V it holds there are at least threedistance minimizing geodesics of M connecting p to x. Thus there are 1 ≤ i < j <k ≤ kp(q) so that

‖Yi(x)‖g = ‖Yj(x)‖g = ‖Yk(x)‖g = dM(p, x),

which yields

ρik(x) = ρjk(x) = 0, and x ∈ Ki,j,k ⊂ K(q).

If x ∈ K(q) ∩M then x ∈ Ki,j,k ∩M for some i < j < k. Thus by the proof ofLemma 15 it holds that there are at least three distance minimizing geodesics of Mconnecting p to x. Therefore x ∈ L(p) ∩ V .

(3) Since we can write the cut locus of p as a disjoint union cut(p) = T (p)∪L(p)∪Q(p)the parts (1) and (2) in conjunction with Lemma 13 yield the claim of part (3).

(4) Since ∂M is a smooth hyper-surface of a n-dimensional Riemannian manifold M wehave by part (1) that T (p) ∩ ∂M is a smooth sub-manifold of dimension n− 2, thusit has the Hausdorff-dimension n− 2. Also by part (3) we know that the Hausdorffdimension of L(p)∩Q(p) does not exceed n− 2. We have proven that the Hausdorffdimension of the closed set cut(p)∩∂M does not exceed n−2. Since the boundary ofM has the Hausdorff-dimension n− 1 it follows that ∂M \ cut(p) is open and densein ∂M . The density claim follows from the observation that by Lemma 13 the setcut(p)∩ ∂M cannot contain any open subsets of ∂M as their Hausdorff dimension isn− 1.

We are ready to prove Theorem 4.

Proof of Theorem 4. The proof follows from Proposition 10 and Theorem 12.

4. Reconstruction of the manifold

4.1. Geometry of the measurement region. In this section we consider only one Rie-mannian manifold (M, g) that satisfies the assumptions of Theorem 4 and whose partialtravel time data (1) is known. Let ν(z) be the outward pointing unit normal vector field atz ∈ ∂M . The inward pointing bundle at the boundary is the set

∂inTM = (z, v) ∈ TM | z ∈ ∂M, 〈v, ν(z)〉g < 0.

We restrict our attention to the vectors that are inward pointing and of unit length: ∂inSM =(z, v) ∈ ∂inTM : ‖v‖g = 1. We emphasize that this set or its restriction on the open


measurement region Γ ⊂ ∂M is not a priori given by the data (1). Our first task is torecover a diffeomorphic copy of this set. We consider the orthogonal projection

(27) h : ∂inSM → T∂M, h(z, v) = v − 〈v, ν(z)〉gν(z),

and denote the set, that contains the image of h, as P (∂M) = (z, w) ∈ T∂M : ‖w‖g < 1.It is straightforward to show that the map h is a diffeomorphism onto P (∂M).

For the rest of this section we will be considering the vectors in P (∂M), and with a slightabuse of notation, each vector (z, v) ∈ P (∂M) represents an inward-pointing unit vector atz. In the next lemma we show that the data (1) determines the restriction of P (∂M) on Γ.

Lemma 19. Let Riemannian manifold (M, g) be as in Theorem 4. The first fundamentalform g|Γ of Γ and the set

P (Γ) = (z, v) ∈ T∂M : z ∈ Γ, ‖v‖g < 1can be recovered from data (1).

Proof. Let (z, v) ∈ TΓ. We choose a smooth curve c : (−1, 1) → Γ for which c(0) = z,and c(0) = v. Since the boundary ∂M of M is strictly convex the inverse function of theexponential map expz is smooth and well defined near z on M . In addition, we have that

rz(c(t)) = d(z, c(t)) = ‖ exp−1z (c(t))‖g.

We set c(t) = exp−1z (c(t))) ∈ TzM. As the differential of the exponential map at the origin is

an identity operator we get c(0) = 0, and ˙c(0) = v ∈ TzM. From here the continuity of thenorm yields

(28) limt→0

rz(c(t))

|t|= lim

t→0

∥∥∥∥ c(0)− c(t)t

∥∥∥∥g

=∥∥ ˙c(0)

∥∥g

= ‖v‖g.

By the data (1) and the choice of the path c(t) ∈ Γ we know the left hand side of equation(28). Therefore we have recovered the length of an arbitrary vector (z, v) ∈ TΓ. Moreover,the set P (Γ) is recovered.

Since we know the unit sphere v ∈ Tz∂M : ‖v‖g = 1 for each z ∈ Γ the reconstructionof the first fundamental form of Γ can be carried out as explained in the next lemma.

Lemma 20. Let (X, g) be a finite dimensional inner product space. Let a > 0 and S(a) :=v ∈ X : ‖v‖g = a. Then any open subset U of S(a) determines the inner product g on X.

Proof. This proof is the same as the one in [23, Lemma 3.33] and thus omitted here.

Let p0 ∈ M . By Theorem 4 we can find a boundary point z0 ∈ Γ and neighborhoods Up0and Vp0 for p0 and z0 respectively such that the distance function d(·, ·) is smooth in theproduct set Up0 × Vp0 . For each z ∈ Γ ∩ Vp0 we let γz be the unique distance minimizingunit speed geodesic from p0 to z. If we decompose the velocity of the geodesic γz at rp0(z)into it’s tangential and normal components to the boundary, then the tangential componentcoincides with the boundary gradient of the travel time function rp0 at z. For this vector fieldwe use the notation grad∂M rp0(z) ∈ Pz(Γ). Furthermore, by Lemma 19 we have recoveredthe metric tensor of the measurement domain Γ ⊂ ∂M . Thus we can compute grad∂M rp0(z)whenever the respective travel time function rp0 is differentiable on Γ.


4.2. Topological reconstruction. We first show that the data (1) separates the points inthe manifold M .

Lemma 21. Let (M, g) be as in Theorem 4. Let Γ ⊂ ∂M be open and p1, p2 ∈ M be suchthat rp1(z) = rp2(z) for all z ∈ Γ, then p1 = p2.

Proof. First we choose open and dense subsets W1,W2 ⊂ ∂M for the points p1 and p2 aswe have for the point p0 in Theorem 4. Then we choose any point z0 ∈ Wp1 ∩ Wp2 ∩ Γ,neighborhoods Up1 of p1, Up2 of p2 and Vp1 , Vp2 of z0 as we have for p0 in Theorem 4. Thusthe distance function d(·, ·) is smooth in the product sets Up1 × V and Up2 × V , whereV = Vp1 ∩ Vp2 is an open neighborhood of z0. Moreover for each (p, z) ∈ Upi × V, i ∈ 1, 2there exists a unique distance minimizing geodesic of M connecting p to z.

If γi is the distance minimizing geodesic from pi to z0 for i = 1, 2 then by the discussionpreceding this lemma we have that grad∂M rpi(z0) represents the tangential component of γiat rpi(z0). Since rp1 = rp2 the tangential components of γ1 and γ2 are the same. Since thevelocity vectors of γi at rpi(z0) have the unit length, also they must coincide. We get

z0 = γ1(rp1(z0)) = γ2(rp2(z0)) and γ1(rp1(z0)) = γ2(rp2(z0)).

Thus the geodesics γ1 and γ2 agree and we have p1 = p2.

We are now ready to reconstruct the topological structure of (M, g) from the partial traveltime data (1). Let B(Γ) be the collection of all bounded functions f : Γ→ R and ‖ · ‖∞ thesupremum norm of B(Γ). Thus (B(Γ), ‖ · ‖∞) is a Banach space. Since (M, g) is a compactRiemannian manifold each travel time function rp, p ∈M is bounded by the diameter of M ,which is finite. Thus

rp = d(p, ·) : Γ→ [0,∞)| p ∈M ⊂ B(Γ),

and the map

(29) R : (M, g)→ (B(Γ), ‖ · ‖∞), R(p) = rp

is well defined.

Proposition 22. Let Riemannian manifold (M, g) be as in Theorem 4. The map R as in(29) is a topological embedding.

Proof. By Lemma 21, we know that the map R is injective, and by the triangle inequality weget that it is also continuous. Let K be a closed set in M . Since M is a compact Hausdorffspace the set K is compact. Since the image of a compact set under a continuous mapping iscompact, it follows that R(K) is closed. This makes R a closed map and thus a topologicalembedding.

4.3. Boundary Determination. We recall that the data (1) only gives us the subset Γ ofthe boundary, and we do not know yet if the travel time function r ∈ R(M) is related to aninterior or a boundary point of M . In this subsection we will use the data (1) to determinethe boundary of the unknown manifold M as a point set. However, due to Proposition 22we may assume without loss of generality that the topology of M is known. Also the setP (Γ), as in Lemma 19, is known to us.


Let (z, v) ∈ P (Γ), and define the set,

σ(z, v) = p ∈M | rp : Γ→ R is smooth near z and grad∂M rp(z) = −v ∪ z,(30)

where grad∂M rp(z) is the boundary gradient of rp at z ∈ Γ. In the next lemma we generalizethe result [32, Lemma 2.9] and relate σ(z, v) to the the maximal distance minimizing segmentof the geodesic γz,v.

Lemma 23. Let (z, v) ∈ P (Γ) then σ(z, v) = γz,v([0, τcut(z, v)]).

Proof. Let t ∈ [0, τcut(z, v)) and define y := γz,v(t). By Lemma 5, γz,v : [0, t] → M is theunique distance minimizing curve from z to y. Then Lemma 9 implies that ry(·) is smoothat z and grad∂M ry(z) = −v. So y is in σ(z, v) and the inclusion γz,v([0, τcut(z, v))) ⊆ σ(z, v)

is true. This gives γz,v([0, τcut(z, v)]) ⊆ σ(z, v).Let p ∈ σ(z, v), then rp(z) is smooth in a neighborhood of z and grad∂M rp(z) = −v. Thus

γz,v is the unique distance minimizing geodesic connecting z to p. Since the geodesic γz,v isnot distance minimizing beyond the interval [0, τcut(z, v)] we have p ∈ γz,v([0, τcut(z, v)]) and

therefore σ(z, v) ⊂ γz,v([0, τcut(z, v)]).

We set

(31) Tz,v := supp∈σ(z,v)

rp(z) = supp∈σ(z,v)

d(p, z), for (z, v) ∈ P (Γ).

Notice that this number is determined entirely by the data (1), as opposed to τcut(z, v) whichrequires our knowledge of when the geodesics were distance minimizing. By the followingcorollary, whose proof is evident, these two numbers are the same.

Corollary 24. For any (z, v) ∈ P (Γ) we have that Tz,v = τcut(z, v).

We will use the sets σ(z, v), for (z, v) ∈ P (Γ) to determine the boundary ∂M of M . Sincethe topology of M is known by Proposition 22, we can determine the topology of these σ setsfrom the data. The next lemma shows if σ(z, v) is closed then γz,v(Tz,v) is on the boundaryof M .

Lemma 25. Let (z, v) ∈ P (Γ). If σ(z, v) is closed then Tz,v = τexit(z, v).

Proof. By the definition of Tz,v we must have Tz,v ≤ τexit(z, v).Suppose that Tz,v < τexit(z, v). From Corollary 24 then we also know

(32) τcut(z, v) = Tz,v < τexit(z, v)

Let p = γz,v(Tz,v), and by Lemma 23 it holds that p ∈ σ(z, v) = σ(z, v). By Lemma 5 thereeither exists a second distance minimizing geodesic from from z to p or p is conjugate to zalong γz,v. Clearly in the first case rp is not differentiable at z. If the second case is valid,and since p ∈M int, we get by a similar proof as in [25, Theorem 2.1.12] that the exponentialmap expz is not a local injection at Tz,vv ∈ TzM . This implies that there is a sequence ofpoints pi ∈ M int, that converge to p and can be connected to z by at least two distanceminimizing geodesics. Thus rpi is not differentiable at z for any i ∈ N implying that neitheris rp. We arrive in a contradiction, meaning that the inequality (32) cannot occur and wemust have Tz,v = τexit(z, v).


Lemma 26. Let p0 ∈ ∂M and z0 ∈ Γ, Up0, and Vp0 be as in Theorem 4. For every p ∈ Up0we denote η(p) = − grad∂M rp(z0). There exists a neighborhood U ′p0 ⊆ Up0 of p0 such that forall p ∈ U ′p0 we have that p is in the closed set σ(z0, η(p)).

Proof. Define v0 = η(p0) and t0 = τexit(z0, v0), then p0 = expz0(t0v0). Since z0 was chosen tobe a point outside the cut locus of p0, these points are not conjugate to each other along thegeodesic γz0,v0 connecting them. Therefore the differential D expz0 of the exponential map isinvertible at t0v0 ∈ Tz0M . From here the claim follows from the Inverse function theoremfor expz0 near t0v0, the continuity of the exit time function and the inequality

rp(z0) = ‖ exp−1z0

(p)‖q ≤ τexit(z0, η(p)), for p ∈ Up0 .We omit the further details.

Corollary 27. Let p0 ∈ ∂M , z0 ∈ Γ and U ′p0 be as in Lemma 26. If we denote η(p) =− grad∂M rp(z0) then Tz0,η(p) is smooth for all p ∈ U ′p0.

Proof. Since the exit time function is smooth on those (z, v) ∈ ∂inSM that satisfy τexit(z, v) <∞ we only need to show that Tz,η(p) = τexit(z0, η(p)). This equation follows from Lemmas 25and 26.

We are now ready to determine the boundary of M from the data (1).

Proposition 28. Let (M, g) be as in Theorem 4 and p0 ∈M . Then p0 ∈ ∂M if and only ifthere exists (z, v) ∈ P (Γ) such that p0 ∈ σ(z, v) and rp0(z) = Tz,v.

Proof. If p0 ∈ ∂M then we get from Lemma 26 that there exists (z0, v) ∈ P (Γ) such thatp0 is in the closed set σ(z0, v). By Lemma 25 we have Tz0,v = τexit(z0, v). Firstly the strictconvexity of ∂M implies that each geodesic has at most two boundary points. Secondlysince p0 6= z0 are both boundary points contained in σ(z0, v), which is a trace of a distanceminimising geodesic, it follows that Tz0,v = rp0(z0).

To show the reverse direction, let there be a pair (z, v) ∈ P (Γ) where p0 ∈ σ(z, v) andTz,v = rp0(z). Thus γz,v([0, rp0(z)]) ⊆ σ(z, v), and it follows from Lemma 23 and Corollary24 that σ(z, v) is closed. By Lemma 25, the closedness of σ(z, v) implies Tz,v = τexit(z, v).Thus, rp0(z) = τexit(z, v), making p0 ∈ ∂M .

4.4. Local Coordinates. By Proposition 28 we have reconstructed the boundary ∂M of thesmooth manifold M . In this section we use the partial travel time data (1) to construct twolocal coordinate systems for p0 ∈M . Since M has a boundary, we need different coordinatessystems based on whether p0 ∈M int or p0 ∈ ∂M .

Proposition 29. Let (M, g) be as in Theorem 4. Let p0 ∈ M int, and choose z0 ∈ Γ, Up0,and Vp0 as in Theorem 4. Let the map α : Up0 → Pz0(Γ)×R be defined as

(33) α(p) = (− grad∂M rp(z0), rp(z0)).

This map is a diffeomorphism onto it’s image α(Up0) ⊂ Pz0(Γ)×R.

Proof. Since the distance function d(·, ·) is smooth in Up0×Vp0 also the function α is smoothon Up0 . By a direct computation we see that the inverse function of α, is given as,

α−1(v, t) = expz0(th−1(v)

), for (v, t) ∈ Pz0(Γ)×R.


where h : ∂inSz0M → Pz0(Γ), is the orthogonal projection given in (27). By the smoothnessof h−1 and the exponential map, it follows that α−1 is smooth. Thus, α is a diffeomorphismonto it’s image, which is open in Pz0(Γ)×R.

In particular, the function α, in (33), gives a local coordinate system near the interiorpoint p0. In order to define a coordinate system for a point at the boundary we will adjustthe last coordinate function of α to be a boundary defining function.

Proposition 30. Let (M, g) be as in Theorem 4. Let p0 ∈ ∂M and choose z0 ∈ Γ, and U ′p0as in Lemma 26. Let η(p) := − grad∂M rp(z0) and βz0 : U ′p0 → Pz0(Γ)×R be defined as

(34) β(p) = (η(p), Tz0,η(p) − rp(z0)).

This map is a diffeomorphism onto it’s image β(U ′p0) ⊂ Pz0(Γ)×R.

Proof. Since the distance function d(·, ·) is smooth in U ′p0 × Vp0 and p0 ∈ ∂M we have byCorollary 27, that the map Tz0,η(p) is smooth for all p ∈ U ′p0 . Thus β is smooth in U ′p0 . Againby a direct computation we get that the inverse function of β is given as

β−1(v, t) = expz0((Tz0,v − t)h−1(v)

)for (v, t) ∈ Pz0(Γ)×R.

By the local invertibility of the exponential map expz0 at rp0(z0)h−1(η(p0)) ∈ Tz0M and theequation rp(z0) = ‖ exp−1

z0(p)‖g for p ∈ U ′p0 , the set η(U ′p0) ⊂ Pz0(Γ) is open and the function

v 7→ Tz0,v, in this set is smooth, making β−1 smooth. Thus, β is a diffeomorphism onto it’simage, which is open in Pz0(Γ)×R.

Finally by Proposition 25 we get that Tz0,η(p) − rp(z0) = 0 if and only if p ∈ U ′p0 ∩ ∂M .Thus this function defines the boundary.

Combining the results of Propositions 29 and 30, we know that for p0 ∈ M , either thefunction α as in (33) or the function β as in (34), gives a smooth local coordinate system.Moreover these maps can be recovered fully from the data (1). As these two types ofcoordinate charts cover M the smooth structure on M is then the same as the maximalsmooth atlas determined by these coordinate charts [36, Proposition 1.17].

4.5. Reconstruction of the Riemannian Metric. So far we recovered both the topolog-ical and smooth structures of the Riemannian manifold (M, g) from the data (1). In thissection we recover the Riemannian metric g. We recall that by Lemma 19 we know the firstfundamental form of Γ.

In order to recover the metric we consider the distance function

d(p, z) = rp(z), for (p, z) ∈M × Γ,

which we have recovered by Proposition 22. Let p0 ∈ M . By Theorem 4 we can choosez0 ∈ Γ and neighborhoods Up0 and Vp0 for p0 and z0 respectively such that the distancefunction d(p, z) for (p, z) ∈ Up0 × Vp0 is smooth. Thus the map

(35) Hp0 : Vp0 ∩ Γ→ T ∗p0M, Hp0(z) = Dd(p0, z)

is well defined and smooth. Here D stands for the differential of the distance function d(p, z)with respect to the p variable in the open set Up0 ⊂ M and T ∗p0M is the cotangent space atp0. As we have recovered the smooth structure of M we can find Hp0 .


For z ∈ Vp0 . The gradient gradpd(p, z) for p ∈ Up0 is the velocity of the distance minimizingunit speed geodesic from z to p (see for instance [36, theorems 6.31, 6.32]). In particular themap

Hp0 : (Vp0 ∩ Γ) 3 z → gradpd(p0, z) ∈ Sp0M

is well defined and satisfies Hp0(z) = Hp0(z)], where ] : T ∗p0M → Tp0M is the musical isomor-

phism, raising the indices, given in any local coordinates near p0 as (ξ])i = gij(p0)ξj. Notethat the inverse of ] is given by [ : Tp0M → T ∗p0M , that lowers the indices. Although we

know the map Hp0 , we do not know its sister map Hp0 .

Lemma 31. Let p0 ∈ M . Let z0 ∈ Γ, Up0 and Vp0 be as in Theorem 4. Then the image ofthe map Hp0, as in (35), contains an open subset of the unit co-sphere

S∗p0M := ξ ∈ T ∗p0M : ‖ξ‖g−1 = 1.

Proof. Let v ∈ Tp0M such that expp0(v) = z0. Since [ : Tp0M → T ∗p0M is a linear isomor-phism that preserves the inner product, the claim holds due to the local invertibility ofthe exponential map expp near v, the equality d(p0, z) = ‖ exp−1

p0(z)‖g, which is true for all

z ∈ Vp0 , and the continuity of the exit time function near v‖v‖g ∈ Sp0M . We omit the further

details.

Finally Lemma 20 in conjunction with the previous lemma lets us recover the inversemetric g−1(p0) and thus the metric gp0 . This is formalized in the proposition below.

Proposition 32. Let (M, g) be as in Theorem 4 and p0 ∈M . The data (1) determines themetric tensor g near p0 in the local coordinates given in Propositions 29 and 30.

Proof. Let z0 ∈ Γ, Up0 and Vp0 be as in Theorem 4. By Proposition 28 we can tell whetherp0 is an interior or a boundary point. Based on this we choose local coordinates of p0 asin Proposition 29 or as in Proposition 30. Then we consider the function Hp0 given in theequation (35). By Lemma 31 we know that image of the function Hp0 contains an opensubset of S∗p0M .

From here, by applying Lemma 20 we determine the inverse metric gij(p0) in the afore-mentioned coordinates. Finally taking the inverse of gij(p0) determines gij(p0). As thisprocedure can be done for any point p ∈M , which is close enough to p0, we have recoveredthe metric g near p0 in the appropriate local coordinates.

5. The proof of Theorem 2

Let Riemannian manifolds (M1, g1) and (M2, g2) be as in Theorem 2. We recall thatthe partial travel time data of these manifolds coincide in the sense of Definition 1. Let(B(Γi), ‖ · ‖∞), for i ∈ 1, 2, be the Banach space of bounded real valued functions on Γi.We set a mapping

(36) F : B(Γ1)→ B(Γ2), F (f) = f φ−1,

where φ is the diffeomorphism from Γ1 to Γ2. By the triangle inequality we have that F is ametric isometry whose inverse mapping is given by F−1(h) = h φ. Taking Ri : (Mi, gi) →


(B(Γi), ‖ · ‖∞), as in the equation (29), we have by the equation (2) in Definition 1 that

F (R1(M1)) = R2(M2).

Therefore we get from Proposition 22 that the map

(37) Ψ : (M1, g1)R1−→ (B(Γ1), ‖ · ‖∞)

F−→ (B(Γ2), ‖ · ‖∞)R−1

2−−→ (M2, g2),

is a well defined homeomorphism, that satisfies the equation

(38) d2(Ψ(x), φ(z)) = F (d1(x, ·))(φ(z)) = d1(x, z), for all (x, z) ∈M1 × Γ1.

Here di(·, ·) is the distance function of (Mi, gi). The goal of this section is to show that Ψ is aRiemannian isometry. In the following lemma we show first that φ preserves the Riemannianstructure of the measurement regions.

Lemma 33. Let Riemannian manifolds (M1, g1) and (M2, g2) be as in Theorem 2. ThenΨ|Γ1 = φ and φ : (Γ1, g1)→ (Γ2, g2) is a Riemannian isometry.

Proof. Let z1 be in Γ1. From the equation (38) we get d2(Ψ(z1), φ(z1)) = 0. Thus Ψ(z1) =φ(z1) and we have verified the first claim Ψ|Γ1 = φ. It follows from the proof of Lemma19 and equation (38) that ‖Dφv‖g2 = ‖v‖g1 , for all v ∈ TΓ1. Then the polarization identityimplies that the differential Dφ of φ also preserves the first fundamental forms:

〈Dφv1,Dφv2〉g2 = 〈v1, v2〉g1 , for all v1, v2 ∈ TΓ1,

making φ a Riemannian isometry.

In particular we get from this lemma that Dφ(P (Γ1)) = P (Γ2). Next we show that themapping Ψ maps the boundary of M1 onto the the boundary of M2. In light of Proposition28 we need to understand how this map carries over the sets σ(z, v), as in (30). The followinglemma gives an answer to this question.

Lemma 34. Let Riemannian manifolds (M1, g1) and (M2, g2) be as in Theorem 2. If (z0, v) ∈P (Γ1) then Ψ(σ(z0, v)) = σ(φ(z0),Dφv).

Proof. Clearly we have that Ψ(z0) = φ(z0) ∈ σ(φ(z0),Dφv). So suppose that p ∈ σ(z0, v) \z0. Moreover by Lemma 23 we get that z0 is not in the cut-locus of p. Thus by Proposition10 we can choose a neighborhood V ⊂ Γ1 of z0 such that rp : V → R is smooth. By (38)we have rΨ(p)(φ(z)) = rp(z), for every z ∈ V . Thus rΨ(p) is smooth on φ(V ) ⊂ Γ2, and sinceφ : Γ1 → Γ2 is a Riemannian isometry we have that

〈Dφ(grad∂M1rp),Dφy〉g2 = 〈grad∂M2

rΨ(p),Dφy〉g2 ,for all y ∈ TV. Here D stands for the differential, grad∂M1

for the boundary gradient of Γ1

and grad∂M2for that of Γ2. Since φ is a diffeomorphism this equation gives Dφ(grad∂M1

rp) =grad∂M2

rΨ(p) in V . In particular

grad∂M2rΨ(p)(φ(z0)) = Dφ

(grad∂M1

rp(z0))

= −Dφv,

implying Ψ(σ(z0, v)) ⊂ σ(φ(z0),Dφv). On the other hand after reversing the roles of M1 andM2 we can use the same proof to show σ(z0, v) ⊃ Ψ−1(σ(φ(z0), dφv)), implyingΨ(σ(z0, v)) =σ(φ(z0),Dφv). This ends the proof.


Lemma 35. Let Riemannian manifolds (M1, g1) and (M2, g2) be as in Theorem 2. ThenΨ(∂M1) = ∂M2. Moreover, Ψ(M int

1 ) = M int2 .

Proof. Let p ∈ ∂M1. Due to Proposition 28 there is a (z, v) ∈ P (Γ1) such that p is inthe closed set σ(z, v) and rp1(z) = Tz,v. Thus Lemma 34 gives Ψ(σ(z, v)) = σ(φ(z),Dφv),and since Ψ is a homeomorphism, also the set σ(φ(z),Dφv) is closed and contains Ψ(p).Furthermore, by equation (38) we have that rΨ(q)(φ(z)) = rq(z), for all q ∈ σ(z, v). Therefore

Tφ(z),Dφv = Tz,v = rp(z) = rΨ(p)(φ(z)).

From here Proposition 28 implies that Ψ(p) is in ∂M2. Thus Ψ(∂M1) ⊂ ∂M2 and by usingthe same argument for Ψ−1 it follows that Ψ(∂M1) = ∂M2. Since M int

1 and ∂M1 are disjointand Ψ is a bijection we also have that Ψ(M int

1 ) = M int2 .

Lemma 36. Let Riemannian manifolds (M1, g1) and (M2, g2) be as in Theorem 2. Themapping Ψ : M1 →M2, given in formula (37), is a diffeomorphism.

Proof. Let p0 ∈ M1, and choose Wp0 ⊂ ∂M1 as in Theorem 4. Since φ : Γ1 → Γ2 is adiffeomorphism, the set φ(Wp0∩Γ1) is open and dense in Γ2. Then for Ψ(p0) ∈M2 we chooseWΨ(p0) ⊂ ∂M2 as in Theorem 4 and consider the non-empty open set WΨ(p0)∩φ(Wp0 ∩Γ1) ⊂Γ2. We pick z0 ∈ Wp0 ∩ Γ1 such that φ(z0) ∈ WΨ(p0) ∩ φ(Wp0 ∩ Γ1).

Let neighborhoods Up0 ⊂M1 of p0 and Vp0 ⊂M1 of z0 be such that the distance functiond1(·, ·) is smooth in the product set Up0 × Vp0 . We also choose neighborhoods UΨ(p0) ⊂ M2

of Ψ(p0) and VΨ(p0) ⊂ M2 of φ(z0) = Ψ(z0) to be such that the distance function d2(·, ·) issmooth in the product set UΨ(p0) × VΨ(p0). Since Ψ: M1 →M2 is a homeomorphism we maychoose these four sets in such a way that they satisfy

Ψ(Up0) = UΨ(p0), and Ψ(Vp0) = VΨ(p0).

By Lemma 35 we know that Ψ(p0) ∈ M int2 if and only if p0 ∈ M int

1 and Ψ(p0) ∈ ∂M2 if andonly if p0 ∈ ∂M1. Next we consider the interior and boundary cases separately.

Suppose first that p0 is an interior point of M1. The functions

Up0 3 p 7→ α1(p) = (− grad∂M1rp(z0), rp(z0)) ∈ Pz0(Γ1)×R,

andUΨ(p0) 3 q 7→ α2(q) = (− grad∂M2

rq(φ(z0)), rq(φ(z0))) ∈ Pφ(z0)(Γ2)×R,

as in Proposition 29, are smooth local coordinate maps of M1 and M2 respectively. Moreover,by the computations done in the proof of Lemma 34 we get for every p ∈ Up0 that

rp(z0) = rΨ(p)(φ(z0)), and Dφ(z0) grad∂M1rp(z0) = grad∂M2

rΨ(p)(φ(z0)).

Therefore for any (v, t) ∈ α1(Up0) we have that

(α2 Ψ α−11 )(v, t) = (Dφ(z0)v, t).

Thus we have proven that the map α2 Ψ α−11 : α1(Up0)→ α2(UΨ(p0)) is smooth.

Then we let p0 be a boundary point of M1. Let η1(p) := − grad∂M1rp(z0) for p ∈ Up0

and choose U ′p0 ⊂ Up0 as in Lemma 26 to be such that the set σ(z0, η1(p)) is closed and thefunction p 7→ Tz0,η1(p) is smooth for every p ∈ U ′p0 . Let U ′Ψ(p0) := Ψ(U ′p0) ⊂ UΨ(p0) and denote

η2(q) := − grad∂M2rq(φ(z0)) for q ∈ U ′Ψ(p0). Since we have that Dφ(z0)η1(p) = η2(Ψ(p)) it


holds by Lemma 34 that the set σ(φ(z0), η2(Ψ(p))) = Ψ(σ(z0, η1(p))), is closed, for everyp ∈ U ′p0 , and thus the function U ′Ψ(p0) 3 q → Tφ(z0),η2(q) is smooth by Corollary 24. Moreover,

we have Tz0,η1(p) = Tφ(z0),η2(Ψ(p)) for every p ∈ U ′p0 .Then we consider local coordinate maps

U ′p0 3 p 7→ β1(p) = (η1(p), Tz0,η1(p) − rp(z0)) ∈ Pz0(Γ1)×R,

of M1 and

U ′Ψ(p0) 3 q 7→ β2(q) = (η2(q), Tφ(z0),η2(q) − rq(φ(z0))) ∈ Pφ(z0)(Γ2)×R,

of M2, as in Proposition 30. By the discussion above we have for any (v, t) ∈ β1(U ′p0) that

(β2 Ψ β−11 )(v, t) = (Dφ(z0)v, t),

which implies that the map (β2 Ψ β−11 ) : β1(U ′p0)→ β2(U ′Ψ(p0)) is smooth.

By combining these two cases we have proved that for every p0 ∈M a local representationof the map Ψ is smooth, making Ψ: M1 → M2 smooth. Finally by an analogous argumentfor Ψ−1 we can show that also this map is smooth. Thus Ψ: M1 →M2 is a diffeomorphismas claimed.

We are ready to present the proof of our main inverse problem:

Proof of Theorem 2. By Lemma 36 we know that the map Ψ: M1 →M2 is a diffeomorphism.We define a metric tensor g2 on M1 as the pull back of the metric g2 with respect to map Ψ.Thus it suffices to consider a smooth manifold M = M1 with an open measurement regionΓ = Γ1 ⊂ ∂M and two Riemannian metrics g1 and g2. Moreover ∂M is strictly convex withrespect to both of these metrics.

Let d2(·, ·) be the distance function of g2. We note that due to equation (38) we have

d1(p, z) = d2(p, z), for all (p, z) ∈ M × Γ. By Lemma 33 we get that g1(p) = g2(p) for allp ∈ Γ. Let p0 ∈M . Thus the map Hp0 given by (35) is the same for both metrics. From hereLemma 31 and Proposition 32 imply that g1(p0) = g2(p0). Therefore map Ψ is a Riemannianisometry as claimed.

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Department of Mathematics, North Carolina State University, Raleigh, NC 27695, USA([email protected])

Department of Mathematics, North Carolina State University, Raleigh, NC 27695, USA([email protected])

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