Boundary rigidity for Riemannian manifolds

Selcuk Journal of

Applied Mathematics

Vol. 4, No. 2, pp. 5–12, 2003

Boundary rigidity for Riemannian manifolds

Department of Mathematics, Devrek Arts and Sciences Faculty, Karaelmas Uni-versity, Incivez 67100, Zonguldak, Turkey;

e-mail: amirov@karaelmas.edu.tr

Received: October 12, 2003

Summary. In the work question of the uniqueness of the solution of the problem of restoring the Riemannian metric by the distances between the pairs of the points of boundary of the region are inves-tigated. The uniqueness of solution of the problem, up to the diffeo-morphism identical on the boundary of the region, for the sufficiently wide class of the metrics is proven.

Key words: Hodograph, semigeodesic coordinates, inverse kine-matic problem, integral geometry problem, special kinetic equation

2000 Mathematics Subject Classification: 53C65

In this paper the problem of reconstructing a Riemannian metric

g = (gij(x)) in the bounded domain D of space Rn, (n > 1) with the boundary S of class C3 from the distances between the boundary points of the domain in a metric, and connected with it problems are investigated. Here questions connected with the uniqueness of a solution to the problem in question will be discussed. A domain D is called convex with respect to a metric g, if any two points x₀, x∈ ¯D,

can be joint by unique geodesic of this metric all points of which with the exception, of possible ends, belong to domain D.

For the points x₀, x∈ S let us denote through Hg(x, x0) the dis-tance between the points x₀, x in the metric g. Function Hg(x, x0) determined on the set S× S is called the hodograph of the metric g. Problem 1. Determine a metric g in the region D if the hodo-graph Hg(x, x0) is known.

Problem 1 arose in geophysics in connection with the study of distribution of the velocities of propagation of elastic waves inside terrestrial globe and its linearization has, in particular, applications in tomography.

It is easy to show nonuniqueness of a solution of the problem 1. Indeed, let ϕ be the diffeomorphism of region D to itself from class

C1 identical on S. It transforms each simple metric g₁ again into a simple metric g₂ = ϕ∗g₁, in the sense that for any vectors ξ, η∈ TxD,

ξ, η(2)x =ϕ_∗ξ, ϕ_∗η1_ϕ(x) equality holds, where ϕ_∗- the differential of the map ϕ,., .i_x- scalar product on TxD is determined by the metric

gi, i = 1, 2, D is convex with respect to a metric gi. These two metrics have different families of geodesic, but the same hodograph.

The questions naturally arise:

1) are there other types of the nonuniqueness of solution of prob-lem 1 ?

2) when is a metric determined by its hodograph up to isometry, identical on ?

3) for what classes of metrics hodograph determines a metric uniquely?

Let us clarify the formulation of problem 1 as follows:

Problem 2. Let g1, g2 be two metrics which are convex in D. Does the existence of a diffeomorphism ϕ : D → D follow from the equality H₁(x, x₀) = H₂(x, x₀), such that ϕ|S = 1, and g2 = ϕ∗g1, where Hk(x, x0) the hodograph of metric gk, k = 1, 2 and equality

ϕ|S = 1 it means that the mapping ϕ is identical on S?

Positive answer to the question formulated in problem 2, is ob-tained only for few class of metrics (see [1-11]). Below (in theorems 1, 2, 3) it is assumed that the domain D is convex with respect to a metric gk = (gij(k)(x)) ∈ C4( ¯D), k = 1, 2. Furthermore it is assumed that a metrics gk are known on the set Dε/D and coin-cide on Dε/D, where Dε is the neighborhood of set D, ε > 0, i.e

Dε = {x ∈ Rn/d(x, D) < ε}, d(x, D)- the Euclidean distance be-tween the point x∈ Rn and the set D, d(x, D) = inf|x − y|, y ∈ D. Let us note that the last condition is not, generally speaking, re-striction on metrics gk in D if they hodographs coincide. Indeed, it is proved in [ 10 ] that, if H₁ and H₂ coincide, then in the suitable coordinates g₁ and g₂ will coincide in the space C2(S). Consequently, it is possible to continue g₂ from the boundary S ∈ C3 to Dε/D by equality g₂ = g₁. Then the metrics g₁ and ¯g₂ will be from C2(Dε), have the same hodograph and coincide on Dε/D, where ¯g2 = g2when

Theorem 1. Let H1 = H2, then

a) there exists a diffeomorphism ϕ : D → D, that ϕ|S = 1 and

g₂= ϕ∗g₁, ϕ∈ C3(D),

b) if gk

1i = δ1i, where i = 1, 2, . . . , n; k = 1, 2; δ1i - Kronecker’s

symbols, then metrics g₁ and g₂ coincide in D.

Let V (x, t) be a solution of the equation

(1) Vtt−

n



i,j=1

aij(x)Vx_ix_j = 0

satisfying the conditions

(2) V, Vt|t=0= 0,

∂V

∂N|∂Dx(0,T )= f (γ, t),

where f continuous on Sx(0, T ), N = N (γ) is outer normal to Sx(0, T ),

γ ∈ S, t ∈ (0, T ), T ≥ T₀, T₀ -the diameter of the region D in the metric g, (aij(x)) inverse to (gij(x)) matrix.

Let us assume that on C(Sx(0, T )) the operator of reaction is known: Raf = V (γ, t), where a = (aij(x)).

Problem 3. It is necessary to determine vector function (aij(x))n₁ from equation (1) in the region D, if the operator Ra is known. Theorem 2. Let

1) the domain D is convex with respect to a metric (g_ijk(x)), where (g_ijk(x)) is the inverse of (a(k)_ij (x)) matrix

2) the operators of reaction corresponding to matrix- functions ak= (a(k)ij (x)), k = 1, 2 coincide, i.e Ra1 = Ra₂. Then

a) there exists a diffeomorphism ϕ : D → D from class C3(D),

such that ϕ|S = 1 and a2 = ϕ∗a1 .

b) if a(k)₁₁ = 1, a(k)_1i = 0 , k = 1, 2; i = 2, 3, . . . , n then a(1)_ij (x) =

a(2)_ij (x) in the region D, i, j = 2, 3, . . . , n.

Problem 1 is connected the following so-called inverse kinematic Problem 4. It is necessary to find a vector function a = (aij(x))n₁ in D if, τ (x, x₀) is known for points x, x₀ ∈ S and satisfies in D the equation (3) n  i,j=1 aij(x)τx_iτx_j = 1,

It is not difficult to see that (3) is the characteristic equation of equation (1), when the corresponding characteristic surface is repre-sented in the form t = τ (x, x₀), where τ (x, x₀)(see [ 1 ]) is the distance between the points x and x₀ in the metric g.

Theorem 3. Let the condition 1) of the theorem 2 be satisfied. Then

if H₁ = H₂ then

a) there exists a diffeomorphism ϕ : D → D from class C3(D),

such that ϕ|S = 1 and a2 = ϕ∗a1 .

b) under the conditions a(k)₁₁ = 1, a(k)_1i = 0 , k = 1, 2; i = 2, 3, . . . , n

matrix - functions a₁ and a₂ coincide in D.

Assertion b) of theorem 3 is the key result of this paper and there is an improvement of the theorem 1 of work [ 5], namely, in this paper the assertion b) is proved without restriction to a metric g of the form (x∈ ¯D, ξ∈ Rn) −1 2 ∂ ∂x₁ n  i,j=2 gijξiξj ≥ α0|ξ|2, α0 > 0,ξ = (ξ2, ξ3, . . . , ξn). Let us give the outline of the proof of assertion b) of theorem 3. For this let us introduce the following notations relating to the metric

gk= (gij(k)(x)): Γk(x, x0) - ray connecting the points x0∈ ∂D, x ∈ ¯D, a τk(x, x0) - the distance between these points; γ2(x, ζ) - the ray of metric g₂ starting from the point x ∈ D in the direction ζ , where (i = 1, 2, . . . , n) ζ = (ζ1, ζ2, . . . , ζn), ζi= n  j=1 a(2)_ij (x)pj₀, pi₀ = (τ₁(x, x₀) + τ₂(x, x₀))x_i.

Let us assume that under the conditions a(k)₁₁ = 1, a(k)_1i = 0 ,

i = 2, 3, . . . , n problem 4 has two solutions ak = (a(k)ij ), k = 1, 2, with the same data, i.e. H₁ = H₂ Then in equation (3) if we first take aij = a(2)_ij , and than aij = a(1)_ij , and subtracting the resulting equations from each other and transforming the obtained equation correspondingly, for the functions d(x, x₀) = τ₂(x, x₀)− τ₁(x, x₀) and

bij = a(2)_ij (x)− a(1)_ij (x) we have (4) n  i,j=1 a(2)_ij pi₀dx_j + n  i,j=2 bijτ1x_iτ1x_j = 0.

Let us note that, since a(k)_1j = 0, j = 2, 3, . . . , n, a(k)₁₁ = 1, we have b_1j = 0, j = 1, 2, . . . , n. It is easy to see that the expression

n

i,j=1a(2)ij pi0dxj is a derivative of d(x, x0) along γ2(x, ζ). Integrating

equality (4) along the ray γ₂(x, ζ) and taking into account that, for the points x, x₀ ∈ S, d(x, x₀) = 0, we will obtain

(5) 0≡ d(x, x₀) =  γ₂(x,ζ) n  i,j=2 bijτ_1zi(z, x₀)τ_1zj(z, x₀)dt.

Recalling, that ( see [1] )

v_i(1)(z, x₀)≡ dz (i) (1) dt = n  j=1 a(1)_ij (z)τ_1zj(z, x₀), i = 1, 2, . . . , n, from (5) we have (6) d(x, x₀) =  γ2(x,ζ) n  k,l=2 cklv_k(1)(z, x0)v(1)_l (z, x0)dt, where ckl = n i,j=2bijg(1)_ik g_jl(1), x, x₀ ∈ S.

Equation (6) is a problem of integral geometry for the matrix-function (ckl) but instead of it we will investigate the inverse prob-lem for the special kinetic equation connected with it. In order to formulate it, we need some notations. Let rays Γ₂(x, x₀), Γ₁(x, x₀) leave from the point x∈ D at angles ξ, f(ξ) correspondingly. In view of the condition on a metric gk(x), k = 1, 2 for each fixed x ∈ D functions ξ = ξ(x, x₀), η = f (ξ) ∈ C3(S₁(2)), are invertible and

x₀ = x₀(ξ)∈ C3(S₁(2)), ξ = f−1(η) where S₁(2), is the sphere of radius 1 of the metric g₂(x) at point x.

Furthermore Jacobian|∂f_∂ξ(ξ)| > 0 and x₀ ∈ S. Using these facts it is possible to prove that for each fixed x∈ D the equation

(7) F (ξ)≡ ξ + f(ξ) = ζ

is invertible ξ = F−1(ζ), where ξ ∈ S₁(2), ζ = F (ξ) ∈ Sv(2), Sv(2) - the sphere of a radius v(ξ) = (n_i,j=1a(2)_ij (x)pi₀pj₀)1/2 of the metric g₂(x),

F−1(ζ) in the domain of definition. Moreover using the condition that metric gk(x) is written in semigeodesic coordinates, the existence of a function I(ζ) = (I2(ζ), . . . , In(ζ))∈ C3 such, that

(8) v_i(1)(x, x₀) = Ii(ζ), i = 2, 3, . . . , n

with

where Rn₀ = Rn/{0} is Rn without the origin of coordinates, can be proven. Here and subsequently through ξ(ζ) is denoted the vector ξ∈ Rn (ζ∈ Rn) without the first component. In this case if function

f (ξ) is continued from the set S(2)₁ by the formula f (lξ) = lf (ξ) , (l > 0, ξ ∈ S₁(2) ) then functions F (ξ), F−1(ζ), I(ζ) will be also

homogenous functions.

Let us introduce a function u(x, ζ) according to the formula

u(x, ζ) =  γ₂(x,ζ) n  i,j=2 cijIi(˙z)Ij(˙z)dt,

where ˙z = (dz_dt2,dz_dt3, . . . ,dz_dtn). Differentiating u(x, ζ) at point x in the direction ζ we have the following kinetic equation

(10) n  i=1 ζiux_i+ n  i,j,k=1 Γ_ijkζiζju_ζk = n  i,j=2 cijIi(ζ)Ij(ζ),

where Γ_ijk - Christoffel symbols of the metric g₂. It follows from equal-ity (6) and (8) that for any x∈ S we have u(x, ζ) = 0.

Thus for the determination u(x, ζ) and a matrix- function (cij), we have the inverse problem (10), u(x, ζ) = 0. Modifying work technique [5] from relations (10), u(x, ζ) = 0 using the condition (9), it is proven, that cij = 0, i.e. a(1)ij = a(2)ij , i, j = 1, 2, . . . , n. Assertion b) of theorem 3 is proved.

Outline of the proof of theorem 1. Assertion b) of theorem 1 fol-lows from assertion b) of theorem 3. Let us prove now assertion a) of theorem 1. Let problem 2 have two solutions, with the same data i.e.,H₁ = H₂. In the region D for the metric gk, k = 1, 2, we in-troduce semigeodesic coordinates as follows. Let us select any point

V₀∈ Dε/ ¯D and let us consider the geodesics outgoing from it. The so-called geodesic hyperspheres orthogonally intersect these geodesics. We take the ends of the segments of a constant length sk= r on the geodesics, outgoing from V₀. These ends form the hypersurface, which is called the geodesic hypersphere of a radius r with center in V₀of the metric gk. Let us examine a certain region on the hypersphere (which lies outside ¯D) with the parameters u1_k, u2_k, . . . , un_k−1. We will carry the geodesics to the same parameters, connecting center hyperspheres

V₀ with the points of region D. We will characterize position of an arbitrary point L on the geodesic with arc length sk = V0L. Then it is obvious that in view of the condition on the metric gk the vari-ables u1_k, u2_k, . . . , un_k−1, sk form the semigeodesic coordinate system in the region D (see [12]). We will denote subsequently coordinates

u1_k, u2_k, . . . , un_k−1, sk through xk₁, xk₂, . . . , xkn, or x = (xk₁, xk₂, . . . , xkn). Let Dk be the domain of semigeodesic coordinates of the metric gk we introduced in the region D, k = 1, 2. It is easy to establish one-to-one correspondence between the regions D₁ and D₂ as follows: To each point x(1) ∈ D₁ one assigns a point x(2) ∈ D₂ so that both points become images of the same point in the region D in the semi-geodesic coordinates with respect to corresponding metrics g₁ and

g₂. Then it is unnecessary to construct the independent coordinate system in each domain D₁, D₂. Namely, it is possible to transform a coordinate system in the region D₂ into the region D₁ as follows: To each point x(1)in the region D₁, the same coordinates xiare assigned which are already a coordinates in the region D₂of the corresponding point x(2).Thus, for the corresponding points x(1) and x(2) we have

x(1)_i = x(2)_i , i = 1, 2, . . . , n. In this case boundary of the region D₁ is mapped into boundary of the region D₂. Nevertheless, generally speaking, metrics in both spaces remain different. This means that, from analytical point of view there is one region ˜D in which we have

two different metrics ˜g₁, ˜g₂. Moreover both metrics are written down in the semigeodesic coordinates in ˜D. It is known that (see [12 ]) ˜gk is written down in the semigeodesic coordinates in ˜D if and only if

˜

gk_1i = δ_1i, where i = 1, 2, . . . , n; k = 1, 2; δ_1i - Kronecker’s symbols. By the condition of theorem 1 on the metrics ˜g₁, ˜g₂ in the region ˜D.

they have the same hodograph, therefore, by assertion b) of theorem 3 they coincide, i.e. ˜g₁ = ˜g₂.

Let us build diffeomorphism ϕ according to equalities x(1)_i = x(2)_i ,

i = 1, 2, . . . , n, as follows: let, us assign to a point x1 ∈ D with the

semigeodesic coordinates (u1₁, u2₁, . . . , un₁−1, s₁) in the metric g₁ the point x2 ∈ D with the semigeodesic coordinates (u1₂, u2₂, . . . , un₂−1, s₂) in the metric g₂ if the equalities ui₂ = ui₁, 1, 2, . . . , n, s₂ = s₁ hold. It is not difficult to see that ϕ|S = 1. Actually, for the point x ∈ S the rays Γ₁(x, V₀), Γ₂(x, V₀) outside of the region D coincide, since outside D we have g₁ = g₂, where Γk(x, V0) - geodesic of the metric

gk connecting points x ∈ D and V0, k = 1, 2. Here, in construction of semigeodesic coordinates for the metrics g₁ and g₂ we take the same geodesic hypersphere with the center at the point V₀ which lies outside D. Then by the uniqueness of the ray Γk(x, V0) and from the definition of the coordinates u1₁, u2₁, . . . , un₁−1, the first (n− 1)

components of the semigeodesic coordinates of the point x∈ S in the metrics g₁and g₂coincide, i.e ui₂= ui₁, 1, 2, . . . , n−1. But the equality of last components (s₂ = s₁) follows from the equality H₁ = H₂ and from the fact that rays Γ₁(x, V₀) and Γ₂(x, V₀) outside of D coincide. Consequently, we have:

1) for x∈ S, ϕ(x) = x,

2) the regions D₁ and D₂ are coincide, i.e. D₁= D₂ = ˜D.

Then taking into account 2), the convexity of domain D with re-spect to gk, k = 1, 2, and determination of ϕ we will obtain, that ϕ transforms D to itself. According to the theorem about continuous-differentiability dependence of the solution to the Cauchy problem (determining the ray of the metric g_(k)∈ C(4)(D), k = 1, 2) from the initial data and to condition S ∈ C3 and also from the determination of the ϕ we have, that ϕ∈ C3(D). Equality ˜g₁ = ˜g₂, the

determina-tion of mappings ϕ and ϕ∗ give us, g₂ = ϕ∗g₁. This proves Theorem

1.

Assertion a) of Theorem 3 follows from the proof of assertion a) of Theorem 1. Theorem 2 is proven using the properties of the fun-damental solution of equation (1) and the assertion of theorem 1.

For the number of the inverse problems of the theory of scatter-ing and spectral analysis of those connected with problem 2, from theorem 1 it is possible to obtain the corresponding results.

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