Uniqueness of the partial travel time representation of a compact Riemannian manifold with strictly convex boundary

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

We consider a compact $n$-dimen-sional smooth manifold $M$ with smooth boundary $\partial M$, equipped with a smooth Riemannian metric $g$. For points $p,q\in M$ the Riemannian distance between them is denoted by $d(p,q)$. Then for $p\in M$ we define the boundary distance function $\hat{r}_{p}:\partial M\to{\bf R}$ given by $\hat{r}_{p}(z)=d(p,z)$. Let $\Gamma$ be a non-empty open subset of the boundary $\partial M$. We denote the restriction of the boundary distance function on this set as $r_{p}:=\hat{r}_{p}\big{|}_{\Gamma}$. The collection

are called the partial travel time data of $\Gamma\subset\partial M$. With these data we seek to recover the Riemannian manifold $(M,g)$ up to a Riemannian isometry. The following definition explains when two Riemannian manifolds have the same partial travel time data (1).

We use the notations $TM$ and $SM$ for the tangent and unit sphere bundles of $M$. Their respective fibers, for each point $p\in M$, are denoted by $T_{p}M$ and $S_{p}M$. In order to show that the data (1) determine $(M,g)$, up to an isometry or in other words that the Riemannian manifolds $(M_{1},g_{1})$ and $(M_{2},g_{2})$ of Definition 1 are Riemanian isometric, we need to place an additional geometric restriction. We assume that $(M,g)$ has a strictly convex boundary $\partial M$ which means that the shape operator $S\colon T\partial M\to T\partial M$ as a linear operator on each tangent space $T_{x}\partial M$ of the boundary $\partial M$ for a point $x\in\partial M$ is negative definite. This means that the second fundamental form

is strictly negative whenever $X,Y\in T_{x}\partial M$ agree, but do not vanish. It has been shown in [5, 3] that the strict convexity of the boundary implies the geodesic convexity of $(M,g)$. That is any pair of points $p,q\in M$ can be connected by a distance minimizing geodesic (not necessarily 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:

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

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

Let us construct an explicit example of a surface $M$ and a subset $\Gamma\subset\partial M$ so that our results fail with data recorded only on $\Gamma$ (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 by a $C^{1}$-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 $M_{1}$ and $M_{2}$ with respect to the line (red dotted line) that is normal to $\partial M$ at $x_{0}\in\partial M$ (blue dot). Then we choose a domain $\Gamma\subset\partial M_{1}$ (red arch) so that any minimizing curve joining a point on $\Gamma$ and a point in $M_{2}$ touches the boundary near $x_{0}$. The curve $P\subset M$ is any involute of the boundary, meaning that the distance from all points on $P$ to $x_{0}$ is the same. Because $d(z,p)=d(z,q)$ for any $z\in\Gamma$ and $p,q\in P$, from the point of view of our data (1), the set $P$ appears to collapse to a point.

Let $(M,g)$ be as in Theorem 2 and denote the Laplace-Beltrami operator of the metric $g$ by $\Delta_{g}$. Let $(p,t_{0})\in M^{int}\times{\bf R}$. In this paper we consider an inverse problem related to the wave equation

where the solution $u(x,t)$ is a spherical wave produced by an interior point source given by the delta function $\delta_{p}(x)\delta_{t_{0}}(t)$ of the space time $M\times{\bf R}$ at $(p,t_{0})$. We define the arrival time function $\mathcal{T}_{p,t_{0}}\colon\partial M\to{\bf R}$, of the wave $u$ by the formula

We recall that it was shown in [32, Proposition 3.1], by applying the results of [16, 18], that $\mathcal{T}_{p,t_{0}}(z)=d(p,z)+t_{0}.$ If the initial time $t_{0}$ is zero (or given), the knowledge of the arrival time function $\mathcal{T}_{p,t_{0}}$ on the open set $\Gamma\subset\partial M$ is equivalent to the corresponding travel time function $r_{p}(\cdot)$ as in (1).

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

was introduced for the first time in [26]. The reconstructions of the smooth atlas on the manifold and the metric tensor in these coordinates was originally considered in [23]. In contrast to the paper at hand, these earlier results do not need any extra assumption for the geometry, but have complete data in the sense that the measurement area $\Gamma$ is the whole boundary. Their counterpart of our Theorem 4, is [23, Lemma 2.15], which says that the boundary distance function $\hat{r}_{p}$ is smooth near its minimizers (the set of closest boundary points to the source point $p\in M$). This lemma is a key of their proof. However, the same technique 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 problems. For instance, it is a crucial step in proving uniqueness for Gel’fand’s inverse boundary spectral problem [23]. Gel’fand’s problem concerns the question whether the data

determine $(M,g)$ up to isometry, when $(\lambda_{j},\phi_{j})$ are the Dirichlet eigenvalues and the corresponding $L^{2}$-orthonormal eigenfunctions of the Laplace–Beltrami operator. Belishev and Kurylev provide an affirmative answer to this problem in [4].

In [24] the authors studied a question of approximating a Riemannian manifold under the assumption: For a finite set of receivers $R\subset\partial M$ one can measure the travel times $d(p,\cdot)|_{R}$ for finitely many $p\in P\subset M^{int}$ under the a priori assumption that $R\subset\partial M$ is $\varepsilon$-dense and that $\{d(p,\cdot)|_{R}:\>p\in P\}\subset\{d(p,\cdot)|_{R}:\>p\in M^{int}\}$ is also $\varepsilon$-dense. Thus $\{d(p,\cdot)|_{R}:\>p\in P\}$ is a finite measurement. The authors construct an approximate finite metric space $M_{\varepsilon}$ and show that the Gromov-Hausdorff distance of $M$ and $M_{\varepsilon}$ is proportional to some positive power of $\varepsilon$. 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 Riemannian manifold (a compact Riemannian manifold with strictly convex boundary where each pair of points is connected by the unique smoothly varying distance minimizing geodesic) by measuring the arrival times of wave fronts produced by several point sources, that go off at unknown times, and moreover, the signals from the different point sources are mixed together. To describe the similarity of two metric spaces ‘with the same boundary’ the authors defined a labeled Gromov-Hausdorff distance. This is an extension of the classical Gromov-Hausdorff distance which compares both the similarity of the metric spaces and the sameness of the boundaries — with a fixed model space for the boundary. In addition to reconstructing a discrete metric space approximation of $(M,g)$, the authors in [11] estimated the density of the point sources and established an explicit error bound for the reconstruction in the labeled Gromov-Hausdorff sense.

If we do not know the initial time $t_{0}$ in (3), but we can recover the arrival times $\mathcal{T}_{p,t_{0}}(z)$ for each $z\in\partial M$, then taking the difference of the arrival times one obtains a boundary distance difference function

for all $z_{1},z_{2}\in\partial M$, which is independent from the initial time $t_{0}$. In [32] it is shown that if $U\subset N$ is a compact subset of a closed Riemannian manifold $(N,g)$ with a non-empty interior, then distance difference data

determine $(N,g)$ up to an isometry. This result was generalized for complete Riemannian manifolds [22] and for compact Riemannian manifolds with boundary [14, 21]. These results require the full boundary measurement in the sense of $\Gamma=\partial M$, unlike Theorem 2 in the present paper.

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

for all $p\in M$ and $z_{1},z_{2}\in\partial M$. These functions give the lengths of the broken geodesics, that is, the union of the shortest geodesics connecting $z_{1}$ to $p$ and the shortest geodesics connecting $p$ to $z_{2}$. Also, the gradients of $D^{+}_{p}(z_{1},z_{2})$ with respect to $z_{1}$ and $z_{2}$ give the velocity 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 show that broken geodesic data determine the boundary distance data of any compact smooth manifold of dimension three and higher. Finally they use the results of [23, 26] to prove that the broken geodesic data determine the Riemannian manifold up to an isometry. A different variant of broken geodesic data was recently considered in [38].

The Riemannian wave operator is a globally hyperbolic linear partial differential operator of real principal type. Therefore, the Riemannian distance function and the propagation of a singularity initiated by a point source in space time are related to one another. We let $u$ be the solution of the Riemannian wave equation as in (3). In [17, 18] it is shown that the image, $\Lambda$, of the wave front set of $u$, under the musical isomorphism $T^{\ast}M\ni(x,\xi)\mapsto(x,g^{ij}(x)\xi_{i})\in TM$, coincides with the image of the tangent space  $T_{p}M$ at $p\in M^{int}$ under the geodesic flow of $g$. Thus $\Lambda\cap\partial(SM)$, 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 convex boundary, then generically the scattering data of point sources $(\partial M,R_{\partial M}(M))$ determine $(M,g)$ up to an isometry. Here, $R_{\partial M}(p)\in R_{\partial M}(M)$ for $p\in M$ stands for the collection of tangential components to the boundary of exit directions of geodesics from $p$ to $\partial M$.

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 a Riemannian manifold up to an isometry? For the full boundary data this problem was solved originally in [4] using the Boundary control method. Partial boundary data questions have been studied for instance in [31, 40]. Recently [29] extended these results for connection Laplacians. Lately also inverse problems related to non-linear hyperbolic equations have been studied extensively [28, 34, 51]. For a review of inverse boundary value problems for partial differential equations see [30, 50].

Maybe the most extensively studied geometric inverse problem formulated with the distance functions is the Boundary rigidity problem. This problem asks: Does the boundary distance 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 general Riemannian manifold the problem is false: Suppose the manifold contains a domain with very slow wave speed, such that all the geodesics starting and ending at the boundary avoid this domain. Then in this domain one can perturb the metric in such a way that the boundary distance function does not change. It was conjectured in [39] that for all simple Riemannian manifolds the answer is affirmative. In two dimensions this was verified in [44]. For higher dimensional cases the problem is still open, but different variations of it has been considered for instance in [7, 9, 48, 49].

In [13] the authors studied the recovery of a compact Finsler manifold from its boundary distance data. In contrast to earlier Riemannian results [23, 26] the data only determines the topological and smooth structures, but not the global geometry. However the Finsler function $F\colon TM\to[0,\infty)$ can be recovered in a closure of the set $G(M,F)\subset TM$, which consists of points $(p,v)\in TM$ such that the corresponding geodesic $\gamma_{p,v}$ is distance minimizing to the terminal boundary point. They also showed that if the set $TM\setminus G(M,F)$ is non-empty then any small perturbation of $F$ in this set leads to a Finsler metric whose boundary distance data agrees with 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 manifold is determined by this data. The same is not true if only the boundary distance function is known [20]. Thus a simple Finsler manifold is never boundary rigid. In [12] the main result of [13] was utilized to generalize the result of [27], about the broken geodesic data, on reversible Finsler manifolds, satisfying a convex foliation condition.

Although, simple Finsler manifolds are not boundary rigid there are results considering their rigidity questions for some special Finsler metrics. For instance it was shown in [41] that Randers metrics $F_{k}=G_{k}+\beta_{k}$ indexed with $k\in\{1,2\}$ with simple and boundary rigid Riemannian norm $G_{k}(x,v)=\sqrt{g_{ij}(x)v^{i}v^{j}}$ and closed one-form $\beta_{k}$, have the same boundary distance function if and only if $G_{1}=\Psi^{\ast}G_{2}$ for some boundary fixing diffeomorphism $\Psi\colon M\to M$ and $\beta_{1}-\beta_{2}=\mathrm{d}\phi$ for some smooth function $\phi$ vanishing on $\partial M$. It is worth mentioning that analogous results have been presented earlier on a Riemannian manifold in the presence of a magnetic field [2, 10].

In this section we consider a Riemannian manifold $(M,g)$ as in Theorem 4, and study the properties of several critical distance functions. We define the exit time function

where $\gamma_{p,v}$ is the geodesic of $(M,g)$ with the initial conditions $(p,v)\in SM$. Since the boundary of $M$ is strictly convex, ${\tau_{\text{exit}}}(p,v)$ is the first time when the geodesic $\gamma_{p,v}$ hits the boundary, and $(-{\tau_{\text{exit}}}(p,-v),{\tau_{\text{exit}}}(p,v))$ is the maximal interval where the geodesic is defined. We do not assume that ${\tau_{\text{exit}}}(p,v)<\infty$ for all $(p,v)\in SM$. That is, $(M,g)$ may have trapped geodesics. Here we denote by $J\subset SM$ the set of all non-trapped directions, that are those $(p,v)\in SM$ for which ${\tau_{\text{exit}}}(p,v)<\infty$. It is well known that on compact Riemannian manifolds with strictly convex boundary the set $J$ is open in $SM$, the exit time function ${\tau_{\text{exit}}}$ is continuous in $J$, and smooth on $J\setminus T\partial M$ (See for instance [47, Chapter 4]).

For any $p\in M$ we define a star shaped set

Thus $M_{p}$ is the largest subset of $T_{p}M$ where the exponential map of $p$

is defined. Since $\partial M$ 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 common point. This leads to the following definition of the cut distance function:

Thus the geodesic segment $\gamma_{p,v}\colon[0,t]\to M$ is a distance minimizing curve for any $t\in[0,{\tau_{\text{cut}}}(p,v)]$.

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

is known as the cut locus of the point $p\in N$ and each point in this set is called a cut point of $p$. Moreover, the cut locus of $p$ coincides with the closure of the set of those points $q\in N$ such that there is more than one distance minimizing geodesic from $p$ to $q$ (see for instance [25, Theorem 2.1.14]). It has been also shown in [45, Section 9.1] that $d(p,\cdot)$ is smooth in $N\setminus(\{p\}\cup{\text{cut}}_{N}(p))$ but not at any $q\in(\{p\}\cup{\text{cut}}_{N}(p))$. In order to understand the smoothness properties of the distance function on a Riemannian manifold $(M,g)$ with a strictly convex boundary, our aim is to define the set analogous to the one in (6) in this context.

If $N$ is a closed manifold and $(p,v)\in SN$ then by Klingenberg’s lemma [36, Proposition 10.32] either there is a second distance minimizing geodesic from $p$ to $\gamma_{p,v}({\tau_{\text{cut}}}(p,v))$ or these points are conjugate to each other along $\gamma_{p,v}$. In particular, the geodesic $\gamma_{p,v}$ is not a distance minimizer beyond the interval $[0,{\tau_{\text{cut}}}(p,v)]$. The following lemma extends this result in our case.

Since the manifold $M$ has a non-empty boundary $\partial M$ it holds that both the tangent bundle $TM$ and the unit sphere bundle $SM$ are manifolds with boundaries $\partial TM$ and $\partial SM$ respectively.

We equip $TM$ with the Sasaki metric $g_{S}$. Thus we can consider $TM$, and its submanifold $SM$, as Riemannian manifolds. In the following the convergence and other metric properties in $TM$ or $SM$ will be considered with respect to this metric.

As $M$ has a boundary, the definition of the cut time function ${\tau_{\text{cut}}}$, in the equation (5), has an issue. Namely if ${\tau_{\text{cut}}}(p,v)={\tau_{\text{exit}}}(p,v)$ for some $(p,v)\in SM$ we do not know a priori if the geodesic $\gamma_{p,v}$ just hits the boundary at $\gamma_{p,v}({\tau_{\text{cut}}}(p,v))$ or if it is possible to find an extension of $(M,g)$ such that $\gamma_{p,v}$ also extends as a distance minimizer.

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

where $d_{M}(\cdot,\cdot)$ and $d_{N}(\cdot,\cdot)$ are the distance functions of $M$ and $N$ respectively. The following proposition shows that while we stay close enough to $M$ we do not need to worry about these short cuts.

By Proposition 7 we may assume that $M$ is contained in the interior of some compact, Riemannian manifold $(\hat{M},g)$ with a smooth strictly convex boundary. Moreover the distance function of $\hat{M}$ restricts to the one of $M$. Thus for every $(p,v)\in S\hat{M}$ where $p$ is in $M$ we always have that

where $\hat{\tau_{\text{cut}}}$ and $\hat{\tau_{\text{exit}}}$ are the cut distance and the exit time functions of $(\hat{M},g)$ respectively. Motivated by this observation we define the cut locus of a point $p\in M$ as

The following result summarizes the basic properties of these sets.

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

Recall that we have assumed that $M$ is isometrically embedded in a closed Riemannian manifold $(N,g)$. Thus any geodesic starting from $M$ can be extended to the entire ${\bf R}$. Let $p\in N$. We define the conjugate distance function ${\tau_{\text{con}}}\colon S_{p}N\to{\bf R}\cup{\infty}$ by the formula:

As the infimum of the empty set is positive infinity we set ${\tau_{\text{con}}}(p,v)=\infty$ in the case when the geodesic $\gamma_{p,v}$ does not have any conjugate points to $p$. Since geodesics do not minimize the distance beyond the first conjugate point it holds that

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