BOUNDARY DETERMINATION OF A RIEMANNIAN METRIC BY THE LOCALIZED BOUNDARY DISTANCE FUNCTION

GUNTHER UHLMANN AND JENN-NAN WANG

Abstract. In this paper we give a reconstruction procedure for the Taylor series of a Riemannian metric on the boundary in boundary normal coordinates from the localized boundary distance function.

1. Introduction

Let (M, g) be a Riemannian manifold with smooth boundary Γ := ∂M. Define the boundary distance function

d_{g}(x, y) = dist(x, y) for x, y ∈ Γ

which is the geodesic distance between boundary points. An interesting inverse problem is whether one can determine g from its associated boundary distance function d_{g}. It is easy to see that uniqueness is not possible. Indeed, let ψ : M → M be a diffeomorphism that leaves Γ invariant, i.e. ψ|_{Γ} = Id, then d_{ψ^{∗}g} = d_{g}. Therefore, the inverse problem one would like to address is whether this is the only obstruction to uniqueness. This problem arose in geophysics in attempting to determine the inner structure of the Earth. The boundary distance function measures the travel times of seismic waves going through the Earth and the metric is the index of refraction [1], [5]. This problem is also called the boundary rigidity problem in Riemannian geometry [2], [3], [9]. The boundary rigidity problem has been extensively studied in the last three decades. The reader is referred to the article [12] for a recent survey.

In this paper we would like to look at this rigidity problem from the boundary perspective. That is, we want to investigate what information of g on Γ can be determined from d_{g}. As indicated above, unique determination of g|_{Γ} is not possible. Nevertheless, in this article we are able to show that one can reconstruct the Taylor series of the metric g on Γ in suitable coordinates from the localized boundary distance function. Precisely, we prove that for any x ∈ Γ all derivatives of g at x in boundary normal coordinates can be determined from the knowledge of d_{g}(x, y) for all y ∈ Γ sufficiently close to x. More importantly, we will give explicit formulas for all the normal derivatives. To state the main theorem, we first introduce the notion of convexity. We say that the boundary Γ is locally convex at p ∈ Γ if there exists an open neighborhood O of p such that for any two points x, y ∈ O ∩ Γ, x ≠ y, there is a unique geodesic joining x and y and all inner points of this geodesic lie entirely in O. Also, M is said to be extendible near p if we can extend M to another smooth manifold M̃ ⊃ M across O ∩ Γ.

This work was done when the authors were participating in the Inverse Problems Program at MSRI in the fall 2001. The authors would like to thank MSRI for providing a very stimulating research environment and partial financial support, through the NSF grant DMS-9810361. The second author was supported by NSF and a John Simon Guggenheim fellowship.

Main Theorem Let (M, g) be a Riemannian manifold of dimension n with smooth boundary Γ. Let p ∈ Γ and Γ be locally convex there. Assume that g is smooth up to Γ near p and also M is extendible near p. Let {x^{1}, ⋯, x^{n}} be the boundary normal coordinates near p. Then we can reconstruct ∂_{x}^{γ}g(p) for all |γ| ≥ 0 from the knowledge of d_{g}(q, r) for all q, r ∈ Γ sufficiently near p.

Previous results on the boundary determination of the Taylor series of g from d_{g} have been given in [9] (up to order 2), [10] (infinite order but n = 2) and recently in [7] (general case). But, none of these papers gave reconstruction formulas. The main tool of this paper is an identity which relates the metric g to the lengths of geodesics. This identity is similar to the one derived by Stefanov and Uhlmann in [11]. It should be noted that the identity derived in [11] relies on the hypotheses that two unknown metrics coincide with the Euclidean one up to certain order near the boundary and have the same boundary distance function. Since we are dealing with the reconstruction problem here, we have only one unknown metric with no boundary information available. To utilize Stefanov and Uhlmann’s arguments, we therefore need to choose an appropriate reference metric to go with the unknown metric (see Section 2). With the key identity at hand, we are able to get the boundary information of the metric by differentiating the identity and letting the boundary distance go to zero (see Section 3).

For the readers’ convenience, we now outline the reconstruction procedure. It should be noted that we are working in boundary normal coordinates near any boundary point, say p (∈ Γ), throughout the whole reconstruction.

Step 1.: Using Michel’s arguments [9], we can determine g(p) in the tangential

directions. Step 2.: Differentiating the key identity (2.5), where the reference metric g_{0} is

given in (3.5), leads to a new formula whose one side is solely determined by d_{g}. We construct the first normal derivatives g^{−1} at p by taking d_{g} → 0 in the new formula. Therefore, we can recover the first normal derivative of g at p.

Step 3.: Inductively, by further differentiating the identity (2.5) and repeating the arguments in Step 2, we can determine all normal derivatives of g at p.

We remark that the result here is the pointwise boundary reconstruction of the metric by the knowledge of the boundary distance function. This problem is somehow related to the inverse problem of determining the metric by boundary measurements from the Dirichlet-to-Neumann map associated with the Laplace-Beltrami operator Δ_{g}. It is well-known that the Dirichlet-to-Neumann map is a pseudo-differential operator of order one. By computing its full symbol (and this is requires knowledge only of the local Dirichlet-to-Neumann map), one can determine the boundary value of the metric up to any order in boundary normal coordinates [8]. For another approach see [6].

2. Key identity

In this section we will derive an identity which plays a key role in the proof of the main theorem. Let H_{g} denote a Hamiltonian related to g and H_{g}(x, ξ) = (1/2)(∑_{i,j=1}^{n} g^{ij}ξ_{i}ξ_{j} − 1) in local coordinates, where (g^{ij}) = (g_{ij})^{−1}. Here we are interested in integral curves associated with the Hamiltonian vector field induced by H_{g}. The projection of these integral curves are geodesics. Let x^{(0)} ∈ Γ and {O, x} be a local chart near x^{(0)}. Assume that Γ is locally convex at x^{(0)} and the Riemannian manifold (M, g) satisfies the assumptions near x^{(0)} as in the main theorem. Let ξ^{(0)} ∈ T_{x^{(0)}}^{∗}M satisfy

ν(x^{(0)}) · g^{−1}(x^{(0)})ξ^{(0)} < 0 (2.1)

and

ξ^{(0)} · g^{−1}(x^{(0)})ξ^{(0)} = 1, (2.2)

where ν(x^{(0)}) is the unit outer normal (co)vector (relative to g^{−1}) to Γ at x^{(0)}. Considered in this local coordinates {x^{1}, ⋯, x^{n}} near x^{(0)}, let x_{g}(s, x^{(0)}, ξ^{(0)}) and ξ_{g}(s, x^{(0)}, ξ^{(0)}) solve the Hamiltonian system

\begin{cases}\frac{dx^{m}}{ds} = ∑_{j=1}^{n} g^{mj}ξ_{j}, & \frac{dξ_{m}}{ds} = −\frac{1}{2} ∑_{i,j=1}^{n} \frac{∂g^{ij}}{∂x^{m}} ξ_{i}ξ_{j} \\ x|_{s=0} = x^{(0)}, & ξ|_{s=0} = ξ^{(0)} m = 1, ⋯, n. \end{cases} (2.3)

Note that x_{g}(s) is the geodesic with initial condition (x^{(0)}, g^{−1}(x^{(0)})ξ^{(0)}) and s in (2.3) is the arc length parameter. Denote X^{(0)} = (x^{(0)}, ξ^{(0)}) and t := t(X^{(0)}) the length of the geodesic issued from X^{(0)} with endpoint on O ∩ Γ. Here we choose ξ^{(0)} appropriately such that all inner points of x_{g}(s) lie entirely in O. This requirement is obviously true if ξ^{(0)} is taken so that t is sufficiently small.

Now let us fix an X^{(0)} satisfying (2.1) and (2.2). Assume that g_{0} is a Riemannian metric in M such that (2.1) holds at X^{(0)} with g being replaced by g_{0}. Notice that we do not require that X^{(0)} satisfy (2.2) with respect to g_{0}. In the coordinates chart {O, x}, we let (x_{g_{0}}(s, x^{(0)}, ξ^{(0)}), ξ_{g_{0}}(s, x^{(0)}, ξ^{(0)})) be the solution to the Hamiltonian system (2.3) with respect to g_{0} having the initial X^{(0)}. It is readily seen that x_{g_{0}}(s, x^{(0)}, ξ^{(0)}) ⊂ O for 0 < s ≤ t provided that t is small. Denote X = (x, ξ). The solutions to (2.3) related to g and g_{0} can be written as X_{g}(s, X^{(0)}) = X_{g}(s, x^{(0)}, ξ^{(0)}) and X_{g_{0}}(x, X^{(0)}) = X_{g_{0}}(s, x^{(0)}, ξ^{(0)}), respectively. Let F (s) = X_{g}(t − s, X_{g_{0}}(s, X^{(0)})), then

\begin{aligned}∫_{0}^{t} F′(s)ds & = F (t) − F (0) \\ & = X_{g}(0, X_{g_{0}}(t, X^{(0)})) − X_{g}(t, X_{g_{0}}(0, X^{(0)})) \\ & = X_{g_{0}}(t, X^{(0)}) − X_{g}(t, X^{(0)}). \end{aligned} (2.4)

It should be noted that x component of F may not be in Ω. In order to make sense of F, we can extend g smoothly to M̃. Thus, M̃ becomes a Riemannian manifold carrying a metric which is a smooth extension of g. Nevertheless, the integral in (2.4) is independent of the extension of g. It is shown in [11] that

F′(s) = \frac{∂X_{g}}{∂X^{(0)}} (t − s, X_{g_{0}}(s, X^{(0)}))(V_{g_{0}} − V_{g})(X_{g_{0}}(s, X^{(0)})),

where V_{g_{0}} = (∂H_{g_{0}}/∂ξ, −∂H_{g_{0}}/∂x) and V_{g} is defined similarly. In conclusion, we obtain that

∫_{0}^{t} \frac{∂X_{g}}{∂X^{(0)}} (t − s, X_{g_{0}}(s, X^{(0)}))(V_{g_{0}} − V_{g})(X_{g_{0}}(s, X^{(0)}))ds = X_{g_{0}}(t, X^{(0)}) − X_{g}(t, X^{(0)}). (2.5)

Before leaving this section, we want to remark that the right-hand side of (2.5) is solely determined by d_{g} near x^{(0)}. This property will be verified in the following section.

3. Proof of Main Theorem

Assume that (M, g) satisfies the assumptions of Main Theorem near x^{(0)} = 0 ∈ Γ. Let Õ be an open neighborhood of 0 in M̃ and the metric g has been extended smoothly in Õ, still denoted by g. We now introduce the boundary exponential map exp_{Γ}(s, p) = exp_{p}(sμ(p)) near 0, where p ∈ Õ ∩ Γ and μ(p) ∈ T_{p}M̃ is the unit inner normal to Γ with respect to g. It is clear that exp_{Γ} is a diffeomorphism if Õ is small. By virtue of this map, we can introduce coordinates, still denoted by {x^{1}, ⋯, x^{n−1}, x^{n}}, in Õ such that Õ is mapped onto an open neighborhood U of 0 and the boundary Γ is determined by x^{n} = 0 and x^{n} > 0 in Õ ∩ M. Moreover, in this coordinates system, the metric g is given by

g = (\begin{matrix}& & & 0 \\ & g_{αβ} & & ⋮ \\ & & & 0 \\ 0 & ⋯ & 0 & 1 \end{matrix}) and g^{−1} = (\begin{matrix}& & & 0 \\ & g^{αβ} & & ⋮ \\ & & & 0 \\ 0 & ⋯ & 0 & 1 \end{matrix}) .

Hereafter the indices α and β run from 1 and n − 1.

First, we want to determine g(0). Clearly, in boundary normal coordinates, it is enough to determine the information of g(v, v)|_{x=0} for any tangent vector v. We use here the following argument of Michel’s paper [9]. Denote e_{α} the α-th standard basis of ℝ^{n}. Let us define a set of vectors V = {v_{αβ}, α ≤ β}, where v_{αα} = e_{α} and v_{αβ} = e_{α} + e_{β} for α < β. Let c : [0, ε) → U be a curve on {x^{n} = 0} with c(0) = 0 and c′(0) = v, where v is an element of V. Here we choose ε small enough so that all geodesics joining 0 and c(τ) for 0 < τ ≤ ε lie entirely in U^{+} = {x ∈ U : x_{n} > 0} except two endpoints. It is easy to see that

lim_{τ→0} \frac{d_{g}(0, c(τ))}{τ} = ∥v∥_{g} = g(v, v)^{1/2}.

Now by repeating the arguments for all v ∈ V, we can determine g_{αβ}(0) for all α, β and hence g(0). Clearly, using the same method, we can find g(x′, 0) for |x′| < δ with δ sufficiently small, where x′ = (x^{1}, ⋯, x^{n−1}).

Having found g on Γ_{δ} := {(x′, 0) | |x′| < δ} for small δ, we can determine the righthand side of (2.5) by knowing d_{g}(p, g) for all p, g ∈ Γ_{δ}. This can be seen, using the notation of the previous section, by observing that d_{g} is the generating function of the canonical relation obtained by projecting the set {(X^{(0)}, X_{g}(t, X^{(0)})} onto T^{∗}Γ×T^{∗}Γ. This set is called the scattering relation in [4]. A more differential geometric way to see this is via the formula derived in [9]

γ′(t(p, q)) = i_{∗}(∇′_{q}t(p, q)) − \sqrt{1 − ∥∇′_{q}t(p, q)∥_{i^{∗}g}^{2}} μ(q), (3.1)

where γ is the geodesic issued from p and parametrized by the arc length, i : Γ_{δ} → U is the inclusion map and ∇′ is the gradient operator on the boundary x^{n} = 0. Let ξ^{(0)}(p, q) and ξ_{g}(t(p, q), ξ^{(0)}(p, q)) =: ξ_{g}(p, q) be the initial and final covectors related to the geodesic connecting p, q ∈ Γ_{δ}. Reinterpreting (3.1) in the covector setting, we can see that for the geodesic joining p and q, p ≠ q, ξ^{(0)}(p, q) and ξ_{g}(p, q) satisfy

ξ^{(0)}(p, q) = g(p)i_{∗}(∇′_{p}t(p, q)) − \sqrt{1 − ∥∇′_{p}t(p, q)∥_{i^{∗}g}^{2}} g(p)μ(p) (3.2)

and

ξ_{g}(p, q) = g(q)i_{∗}(∇′_{q}t(p, q)) − \sqrt{1 − ∥∇′_{q}t(p, q)∥_{i^{∗}g}^{2}} g(q)μ(q). (3.3)

Next we would like to determine all derivatives of g at 0. Since g^{−1}g = I, it suffices to determine all derivatives of g^{−1} at 0. Let v ∈ V and define c̃(τ), 0 ≤ τ ≤ ε̃, be a curve on {x^{n} = 0} with c̃(0) = 0 and c̃′(0) = g^{−1}(0)η, where η = v(v · g^{−1}(0)v)^{−1/2}. As before, we choose ε̃ sufficiently small so that all geodesics joining 0 and c̃(τ) for 0 < τ ≤ ε̃ lie entirely in U^{+} except two endpoints. Now we are at the position to choose our reference Riemannian metric g_{0} in M. The goal here is to choose a g_{0} such that ν(0) · g_{0}^{−1}(0)η < 0, i.e

∑_{j=1}^{n} g_{0}^{nj}(0)v_{j} > 0 (3.4)

for all v = (v_{1}, ⋯, v_{n}) ∈ V. One possible choice is that g_{0}^{−1} is of the form

g_{0}^{−1} = (\begin{matrix}& & & λ_{1} \\ & I_{n−1} & & ⋮ \\ & & & λ_{n−1} \\ λ_{1} & ⋯ & λ_{n−1} & 1 \end{matrix}) , (3.5)

where I_{n−1} is the identity matrix of size n − 1 and λ_{α} > 0. Since det(g_{0}^{−1}) = 1 − ∑_{α=1}^{n−1} λ_{α}^{2}, we can choose λ_{α} sufficiently small for all α to guarantee the positivedefiniteness of g_{0}^{−1}. Here we want to point out that η does not satisfy the incoming direction (2.1) with respect to g, but it satisfies (2.1) in terms of g_{0}. In fact, we can see that ν(0) · g^{−1}(0)η = 0. With this choice of g_{0}, we obtain that the solution to the Hamiltonian system (2.3) with respect to g_{0} can be written explicitly as

(x_{g_{0}}(s, 0, ξ^{(0)}), ξ_{g_{0}}(s, 0, ξ^{(0)})) = (sg_{0}^{−1}ξ^{(0)}, ξ^{(0)}),

where the initial (0, ξ^{(0)}) satisfies (2.1) in terms of g_{0}. Note that the curve x_{g_{0}}(s, 0, ξ^{(0)}) lies entirely in U^{+} for all small s.

Now consider the geodesic (relative to g) connecting 0 and c̃(τ) for 0 < τ < ε̃. In view of the formulas (3.2) and (3.3), it is readily seen that given c̃(τ) we can determine ξ^{(0)} = ξ^{(0)}(τ) and X_{g}(t(τ), X^{(0)}(τ)) = X_{g}(τ) from the boundary distance function d_{g}(0, c̃(τ)) = t(τ). Notice that if t(τ) is sufficiently small (i.e. τ is small), then ξ^{(0)}(τ) is close to η and (0, ξ^{(0)}(τ)) satisfies the incoming condition (2.1) related to g_{0}. Furthermore, we can see that ξ^{(0)}(τ) → η as τ → 0. Expressing every variable in the identity (2.5) in terms of τ, we have that

\begin{matrix}∫_{0}^{t(τ)} \frac{∂X_{g}}{∂X^{(0)}} (t(τ) − s, X_{g_{0}}(s, X^{(0)}(τ)))(V_{g_{0}} − V_{g})(X_{g_{0}}(s, X^{(0)}(τ)))ds \\ = X_{g_{0}}(τ) − X_{g}(τ) = (t(τ)g_{0}^{−1}ξ^{(0)}(τ) − x_{g}(τ), ξ^{(0)}(τ) − ξ_{g}(τ)). \end{matrix} (3.6)

Differentiating both sides of (3.6) in τ yields

\begin{matrix}\frac{∂X_{g}}{∂X^{(0)}} (0, X_{g_{0}}(t(τ), X^{(0)}(τ)))(V_{g_{0}} − V_{g})(X_{g_{0}}(t(τ), X^{(0)}(τ)))t′(τ) \\ + ∫_{0}^{t(τ)} \frac{d}{dτ} \{\frac{∂X_{g}}{∂X^{(0)}} (t(τ) − s, X_{g_{0}}(s, X^{(0)}(τ)))(V_{g_{0}} − V_{g})(X_{g_{0}}(s, X^{(0)}(τ)))\}ds \\ = (t′(τ)g_{0}^{−1}ξ^{(0)}(τ) + t(τ)g_{0}^{−1}ξ^{(0)}′(τ) − x′_{g}(τ), ξ^{(0)}′(τ) − ξ′_{g}(τ)). \end{matrix} (3.7)

Taking τ → 0 in (3.7), we obtain that

\begin{matrix}\frac{∂X_{g}}{∂X^{(0)}} (0, X_{g_{0}}(0, X^{(0)}(0)))(V_{g_{0}} − V_{g})(X_{g_{0}}(0, X^{(0)}(0)))t′(0) \\ = I_{2n×2n} · (V_{g_{0}} − V_{g})(0, η)t′(0) \\ = (t′(0)g_{0}^{−1}η − x′_{g}(0), ξ^{(0)}′(0) − ξ′_{g}(0)), \end{matrix} (3.8)

where I_{2n×2n} is the identity matrix of size 2n. It has been shown previously that t′(0) = ∥c̃′(0)∥_{g} = 1. Writing out the formula (3.8), we can see that

g^{−1}(0)η = g_{0}^{−1}(0)η − t′(0)g_{0}^{−1}η − x′_{g}(0) (3.9)

and

\frac{1}{2}η · ∂_{x}g^{−1}(0)η = ξ^{(0)}′(0) − ξ′_{g}(0). (3.10)

It should be noted that we do not use (3.9) as the reconstruction formula for g(0) since we need to choose the curve c̃ before proceeding the arguments. The curve c̃ has already used the information g(0). It follows from (3.10) that

v · ∂_{x}g^{−1}(0)v = 2(v · g^{−1}(0)v)(ξ^{(0)}′(0) − ξ′_{g}(0)). (3.11)

By repeating the above arguments for each one element of V, we can derive (3.11) for all v ∈ V. In turn we are able to determine ∂_{x}g^{−1}(0). Using the same method, we can find ∂_{x}g^{−1}(x′, 0) for (x′, 0) near 0. Thus, ∂_{x^{α}}^{k}∂_{x^{n}}g^{−1}(0) is also determined for all positive integer k.

To continue the proof, we differentiate (3.7) in τ and set τ = 0. To end, we get that

\begin{matrix}I_{2n×2n} ∑_{j=1}^{n} ∂_{x^{j}}(V_{g_{0}} − V_{g})(g_{0}^{−1}η)_{j}(t′(0))^{2} \\ = (t′′(0)g_{0}^{−1}η + 2t′(0)g_{0}^{−1}ξ^{(0)}′(0) − x′′_{g}(0), ξ^{(0)}′′(0) − ξ′′_{g}(0)) \\ +Ψ, \end{matrix} (3.12)

where (g_{0}^{−1}η)_{j} is the j-th component of g_{0}^{−1}η and Ψ is a 2n vector which consists of terms containing only g^{−1}(0) and ∂_{x}g^{−1}(0). Therefore, Ψ is a known vector-valued function. It is easily observed that only the last n components of ∑_{j=1}^{n} ∂_{x^{j}}(V_{g_{0}} − V_{g})(g_{0}^{−1}η)_{j} contain the second derivatives of g^{−1} at 0. Moreover, since we have found ∂_{x^{α}}∂_{x^{n}}g^{−1}(0), the only term yet to be determined is ∂_{x^{n}}^{2}g^{−1}(0). Singling out the last component of (3.12), we have

\frac{1}{2}η · [∑_{j=1}^{n} ∂_{x^{j}}∂_{x^{n}}g^{−1}(0)(g_{0}^{−1}η)_{j}]η = (ξ^{(0)}′′(0) − ξ′′_{g}(0))_{n} + (Ψ)_{2n}

from which we get that

\begin{matrix}\{v · ∂_{x^{n}}^{2}g^{−1}(0)v\}(g_{0}^{−1}v)_{n} \\ = 2(v · g^{−1}(0)v)^{3/2}\{η · [∑_{α=1}^{n−1} ∂_{x^{α}}∂_{x^{n}}g^{−1}(0)(g_{0}^{−1}η)_{α}]η + (ξ^{(0)}′′(0) − ξ′′_{g}(0))_{n} + (Ψ)_{2n}\}. \end{matrix} (3.13)

Since (g_{0}^{−1}v)_{n} is not zero (see (3.4)), we can determine v·∂_{x^{n}}^{2}g^{−1}(0)v from (3.13). Once again, repeating the arguments for all v ∈ V and noting that (g_{0}^{−1}v)_{n} is never zero for any v ∈ V, we can determine v · ∂_{x^{n}}^{2}g^{−1}(0)v for all v ∈ V and hence ∂_{x_{n}}^{2}g^{−1}(0). Using the same procedure, we can determine ∂_{x^{n}}^{2}g^{−1}(x′, 0) for |x′| < δ with δ sufficiently small. In turn we can find ∂_{x′}^{α′}∂_{x^{n}}^{2}g^{−1}(0) for any multi-index α′.

Inductively, assume that we have determined ∂_{x′}^{α′}∂_{x^{n}}^{l}g^{−1}(0) with 0 < l < ℓ and arbitrary α′. Now by differentiating (3.6) ℓ times in τ and setting τ = 0, we single out the term containing ∂_{x^{n}}^{ℓ}g^{−1}(0) and find that

{v · ∂_{x^{n}}^{ℓ}g^{−1}(0)v}(g_{0}^{−1}v)_{n}^{ℓ−1} = R, (3.14)

where R is a known value which is determined by the induction assumption. Deriving (3.14) for each v ∈ V and noting that (g_{0}^{−1}v)_{n} is never zero, we can determine v · ∂_{x^{n}}^{ℓ}g^{−1}(0)v for all v ∈ V and therefore ∂_{x^{n}}^{ℓ}g^{−1}(0).

References

[1] Creager, K.C., Anisotropy of the inner core from differential travel times of the phases PKP and PKIPK, Nature, 356,(1992), 309-314.

[2] Croke, C.B., Rigidity and the distance between boundary points, J. Differential Geom., 33(1991), 445-464.

[3] Gromov, M., Filling Riemannian manifolds, J. Differential Geom., 18(1983), 1-148.

[4] Guillemin, V., Sojourn times and asymptotic properties of the scattering matrix. Proceedings of the Oji Seminar on Algebraic Analysis and the RIMS Symposium on Algebraic Analysis (Kyoto Univ., Kyoto, 1976), Publ. Res. Inst. Math. Sci., 12(1976/77), supplement, 69–88.

[5] Herglotz, G., Über die elastizität der erde bei berücksichtigung ihrer variablen dichte, Zeitschr. für Math. Phys., 52(1905), 275-299.

[6] Kang, H. and Yun, K., Boundary determination of conductivities and Riemannian metrics via local Dirichlet-to-Neumann operator, preprint.

[7] Lassas, M., Sharafutdinov, V., Uhlmann, G., Semiglobal boundary rigidity for Riemannian metrics, preprint.

[8] Lee, J. and Uhlmann, G., Determining anisotropic real-analytic conductivities by boundary measurements, Comm. Pure Appl. Math., 42(1989), 1097-1112.

[9] Michel, R., Sur la rigidité imposee par la longuer des géodésiques, Invent. Math., 65(1981), 71-84.

[10] Michel, R., Restriction de la distance géodésique á un arc et rigidité, Bull. Soc. math. France, 122(1994), 435-442.

[11] Stefanov, P. and Uhlmann, G., Rigidity for metrics with the same lengths of geodesics, Math. Research Letters, 5(1998), 83-96.

[12] Uhlmann, G,, Travel time tomography, J. Korean Math. Soc., 38(2001),711-722.

Gunther Uhlmann, Department of Mathematics, University of Washington, Seattle, WA 98195-4350, USA

  E-mail address: gunther@math.washington.edu

Jenn-Nan Wang, Department of Mathematics, National Taiwan University, Taipei 106, Taiwan

  E-mail address: jnwang@math.ntu.edu.tw