Determining Riemannian Manifolds From Nonlinear Wave Observations at a Single Point
aa r X i v : . [ m a t h . A P ] F e b DETERMINING RIEMANNIAN MANIFOLDS FROM NONLINEARWAVE OBSERVATIONS AT A SINGLE POINT
LEO TZOU
Abstract.
We show that on an a-priori unknown Riemannian manifold (
M, g ), mea-suring the source-to-solution map for the semilinear wave equation at a single pointdetermines the topological, differential, and geometric structure. Introduction
The work of [KLU14, KLU18, KLOU14] developed the new idea of exploiting non-linearity to solve inverse problems involving nonlinear hyperbolic equations. Roughlyspeaking, these results reconstruct the topological, differential, and conformal structureof a Lorentzian manifold from the source-to-solution data of some nonlinear hyperbolicequation measured in an open space-time cylinder of a timelike path. Other works involv-ing the use of this idea to solve inverse problems for nonlinear hyperbolic equations can befound in [UW20, UW18, UZ19, LUW18, LUW17, dHUW19, dHUW20, FO19, OSSU20]Formally, the source-to-solution data considered in [KLU14, KLU18, KLOU14] (andthe works which followed) are over-determined. As such it is natural to ask if we candeduce the same results with less data. This paper provides a “proof of concept” thatwhen the Lorentzian metric is a direct product of R with a time-independent Riemannianmanifold, data is only needed at a single spacial point. The ultimate goal is of courseto recover the conformal structure of a general Lorentzian manifold using the singleobserver data for other semilinear hyperbolic equations.Beyond just mathematical curiosity, there are practical reasons to consider single de-tector inverse problems. For example, in measuring potential seismic activities on Mars,we have at most one reliable detector, the InSight Mars Lander. Even on earth, some seis-mological procedures such as the Tremorscope (http://seismo.berkeley.edu/research/tremorscope.html)require expensive preliminary drilling to install the detector, severely restricting theiravailability. On a cosmological scale, the size of our planet is comparable to neither itsdistance to the nearest galaxy (Andromeda) nor the diameter of that galaxy so observa-tions made on earth is effectively done by a single observer.Consider two Riemannian manifolds ( M , g ) and ( M , g ), and the correspondingLorentzian manifold ( M j , − dt + g j ) where M j = R × M j . Let p j ∈ M j and considersmall open neighborhoods U j ⊂ M j containing p j , and a diffeomorphism identifying Date : Feb 3, 2021.2000
Mathematics Subject Classification.
Key words and phrases. wave front propagation, nonlinear wave interaction, geodesics, Riemanniangeometry, inverse problems. U ≡ U and p ≡ p . We will denote the common point to be and the commonneighborhood to be U .For a Riemannian manifold ( M, g ), denote by ✷ g = − ∂ t + ∆ g the corresponding waveoperator. The choice of nonlinearity we make here is cubic, and this is motivated by thecubic nonlinear interaction in Yang-Mills equations (see [CMOP19] for this motivation), ✷ g u + u = f in M (1.1) u = 0 , t < − . This Cauchy problem has a unique solution (see e.g. [KLU18] or [CMOP19]) if we assume f is compactly supported in the cylinder ( − , ∞ ) × U and has small C (( − , ∞ ) × U )norm.The data we wish to collect is the source-to-solution map L M,g, T +1 f := u ( t, ) | t ∈ (0 , T +1) , f ∈ C (( − , T + 1) × U ) (1.2)for a fixed T >>
1. Note that while we are allowed to place sources in a cylindricalneighborhood of the timelike segment { ( t, ) ∈ M | t ∈ ( − , T + 1) } , we are onlycollecting data on the time-like curve (0 , T + 1) × . We denote by B g j ( x j , r j ) theRiemannian open ball of ( M j , g j ) centered at x j ∈ M j with radius r j . Our result is thefollowing: Theorem 1.1.
Let ( M , g ) and ( M , g ) be three-dimensional Riemannian manifolds.If their the source-to-solution map coincide L M ,g , T +1 = L M ,g , T +1 , then the Rie-mannian balls ( B g ( , T ) , g ) and ( B g ( , T ) , g ) are isometric. Note that a-priori the two Riemannian balls need not to have the same topologicalstructure. To the best of the author’s knowledge, this is the first inverse problems resultwhere measurements are done at a single point.
Remark 1.2.
The intervals [ − , T + 1] in (1.1) and (0 , T + 1) in the definition ofsource-to-solution can be replaced by [ − δ, T + δ ] and (0 , T + δ ) for any fixed δ > δ > g | U = g | U := g | U (1.3)since deducing this fact from the source-to-solution map is trivial. We may assumewithout loss of generality that U = B g ( ; δ ) (1.4)where δ <
12 min (inj( B g ( , T ) , g ) , inj( B g ( , T ) , g )) (1.5)is chosen small enough for U to be convex.Our idea is partly based on the nonlinear interaction approach pioneered by [KLU18].We motivate our idea by considering the special case when g and g are both Riemannianmetrics on R and assume the absence of conjugate points. Let y ∈ R and x ∈ U . ETERMINING RIEMANNIAN MANIFOLDS FROM MEASUREMENTS AT A SINGLE POINT 3
At time t = 0 we create a point source at which produces a wave whose singularitypropagates along the unit speed g -geodesic ray [ , y ] g joining and y (see [MU79]).At another time t (which may be greater or less than 0), we send another singularityalong the unit speed g -geodesic ray [ x , y ] g so that the two rays collide at y at time t = R := d g ( , y ). Arrange for an auxiliary singularity to go along unit speed g -geodesic ray [ x , y ] g from x ∈ U to y so that all three ray collide at the same time.When arranged judiciously (see Section 3), the nonlinear interaction given by (1.1) willresult in a singularity coming back from y to along the g -geodesic ray [ , y ] g . Thesingle observer sitting at will then see this singularity at time t = 2 R . We then deduce,using the fact that source-to-solution maps are equal, that the same collisions must haveoccurred for the g metric at time R at some point y which is distance R away from .This allows us to conclude that d g ( y , x ) = d g ( y , x ) = R − t for all x ∈ U (see Prop4.7 and its corollary). This, together with a simple Riemannian argument (Prop. 4.12),implies that the Riemannian balls ( B g ( , T ) , g ) and ( B g ( , T ) , g ) are isometric.The difficulty in carrying out this program is that sources do not produce waves whichpropagate along a single light ray but rather traveling planes. The interaction of threetraveling plane waves generically produce a spacelike curve as an artificial source (seeSect 3.2) and light emanating from such a curve can produce focal points even if weassume the absence of conjugate points. A single observer along a timelike path maynot be able to “see” such singularity. See Remark 2.13.To overcome this issue we place small spherical “mirrors” near . When the locationand timing of these mirrors are chosen judiciously, further nonlinear interaction betweenthe returning light wave and these mirrors will reflect a visible conormal wave back to (see Sect 3.3) which is visible to the lone observer, allowing us to circumvent thedifficulty of focal points.We note that it may be possible to avoid focal points by replacing the cubic nonlinear-ity in (1.1) by a quadratic nonlinearity and work with artificial point sources resultingfrom a four wave interaction. In addition to its relationship to Yang-Mills equations,we have chosen to consider the cubic nonlinearity because it motivates us to develop arobust method which has the potential to be applied to a broader range of problems.For example, this approach of using mirrors can potentially allow us to recover the “ear-liest light observation set” in the general Lorentzian setting (see [KLU18]) from artificialsources to a narrow cylinder in space-time by only looking at singularity along a timelikecurve contained in the cylinder. Acknowledgement
The author wishes to thank Marco Mazzucchelli, Alan Greenleaf, Suresh Eswarathasan,Mikko Salo, and Lauri Oksanen for the useful discussions. The author is supported byARC DP190103302 and ARC DP190103451.2.
Preliminaries
Basic Definitions.
Let (
M, g ) be a Riemannian manifold. Associated to thisRiemannian structure is a stationary Lorentzian metric − dt + g on the space-time M = { ( t, x ) | t ∈ R , x ∈ M } . LEO TZOU
The time orientation on M is given by ∂ t . It is sometimes convenient to have a Rie-mannian structure on M and the obvious choice is G := dt + g .Throughout this article, covectors in M are denoted by ξ = τ dt + ξ ′ , where τ ∈ R and ξ ′ ∈ T ∗ M . We denote by Char g ⊂ T ∗ M the subspace of lightlike covectors, whichare those covectors that are multiple of ∓ dt + ξ ′ , with ξ ′ ∈ S ∗ M . Here, the covectors − dt + ξ ′ and + dt + ξ ′ are future and past pointing respectively. Notice that Char g is thecharacteristic set of the homogeneous principal symbol of ✷ g .If Λ , Λ ⊂ T ∗ M we use the notationΛ + Λ := { ( t, x, ξ ) | ( t, x ) ∈ π (Λ ) ∩ π (Λ ) , ξ = ξ + ξ , with ( t, x, ξ j ) ∈ Λ j } . When Λ ∩ T ∗ ( t,x ) M and Λ ∩ T ∗ ( t,x ) M are linearly independent for all ( t, x ) ∈ π (Λ ) ∩ π (Λ ),we will emphasize this fact with ⊕ in place of just +.A smooth curve s ( t ( s ) , γ ( s )) ∈ M is causal when its tangent vectors are timelike(i.e. − ˙ t ( s ) + k ˙ γ ( s ) k g <
0) or lightlike (i.e. − ˙ t ( s ) + k ˙ γ ( s ) k g = 0). The curve is future-directed if ˙ t ( s ) >
0, and past-directed if ˙ t ( s ) <
0. The future and past causal cones at( t , x ) ∈ M are defined respectively by I + g ( t , x ) = { ( t, x ) | there exists a future-directed causal curve from ( t , x ) to ( t, x ) } ,I − g ( t , x ) = { ( t, x ) | there exists a past-directed causal curve from ( t , x ) to ( t, x ) } . Geodesics and Conjugate Points.
Let H denote the Hamiltonian vector fieldon T ∗ M associated to the principal symbol of ✷ g , that is, to the Hamiltonian function( t, x, τ dt + ξ ′ ) (cid:0) k ξ ′ k g − τ (cid:1) . (2.1)The flow of H preserves Char g , and its orbits on Char g are precisely the lifts of thelightlike geodesics of M . We define the future oriented flow on Char g by e + ( s, t, x, ξ ) = e s + ( t, x, ξ ) := ( e sH ( t, x, ξ ) , if ξ is future pointing, e − sH ( t, x, ξ ) , if ξ is past pointing (2.2)for s ≥
0. If ( t, x, ξ ) ∈ Char g we denote its future oriented flowout byFLO + g ( t, x, ξ ) := { e s + ( t, x, ξ ) | s ≥ } . Since the Lorentzian metric is a direct sum of − dt with a Riemannian metric g , theprojection to M of the lightlike geodesics in M are Riemannian geodesics. If ( t, x, ξ ) ∈ Char g and ξ = − dt + ξ ′ with ξ ′ ∈ S ∗ x M , then π ◦ e s + ( t, x, ξ ) = ( t + s, γ g ( s, ξ ′ )) (2.3)where γ g ( s, ξ ′ ) is the unit speed geodesic in M with initial condition ( x, ξ ′ ). We saythat a Riemannian geodesic segment is length minimizing when it is the shortest curvejoining its endpoints; we further say that it is unique length minimizing if every othercurve joining the same endpoints is strictly longer. Lemma 2.1.
Assume that, for some ( y , η ′ ) ∈ S ∗ M and R > , the geodesic segment γ g ([0 , R ] , η ′ ) is unique length minimizing and does not contain conjugate points. Then,there exist δ > and an open neighborhood U ′ ⊂ S ∗ M of ( y , η ′ ) such that, for each y ∈ π ( U ′ ) , the map (0 , R + δ ] × ( S ∗ y M ∩ U ′ ) → M, ( t, η ′ ) exp y ( tη ′ ) (2.4) ETERMINING RIEMANNIAN MANIFOLDS FROM MEASUREMENTS AT A SINGLE POINT 5 is a diffeomorphism onto its image. Up to reducing both U ′ and δ , for any ( y, η ′ ) ∈ U ′ and t ∈ (0 , R + δ ) , the geodesics segment γ g ([0 , t ] , η ′ ) is unique length minimizing; infact, there exists c > such that, for all ( y, η ′ ) ∈ U ′ , t ∈ ( R − δ, R + δ ) , ˆ η ′ ∈ S ∗ M \ { η ′ } ,and ˆ t > , if exp y ( tη ′ ) = exp y (ˆ t ˆ η ′ ) then ˆ t > t + c .Proof. The fact that (2.4) is a diffeomorphism onto its image is well known due to thefact that the endpoints of γ ([0 , R ] , η ′ ) are not conjugate along the geodesic segment. Asfor the second part of the statement, assume by contradiction that there exist sequences S ∗ M ∋ ( y j , η ′ j ) → ( y , η ′ ), t j → t ∈ (0 , R ], ˆ η ′ j ∈ S ∗ y j M \ { η ′ j } , and ˆ t j ≤ t j + j − suchthat γ g (ˆ t j , ˆ η ′ j ) = γ g ( t j , η ′ j ) . (2.5)Since the map (2.4) is a diffeomorphism onto its image, we must have that ˆ η ′ j / ∈ U ′ .Therefore, up to extracting a subsequence, ˆ η ′ j → ˆ η ′ ∈ S ∗ y M \ U ′ and ˆ t j → ˆ t . In particularˆ η ′ = η ′ . This implies γ g (ˆ t , ˆ η ′ ) = γ g ( t , η ′ ), contradicting the fact that γ g ([0 , t ] , η ′ ) isunique length minimizing. (cid:3) We recall that a Lagrangian submanifold Λ ⊂ T ∗ M is conic when λξ ∈ Λ for all λ > ξ ∈ Λ. The following Lemmas are classical, and we will not provide a proof.
Lemma 2.2.
Let K ⊂ M be a spacelike curve then the intersection N ∗ K ∩ Char g ⊂ T ∗ M is a three-dimensional submanifold. (cid:3) Lemma 2.3.
Let Λ ⊂ T ∗ M be a conic Lagrangian submanifold such that the intersection Λ ∩ Char g is a smooth three-dimensional submanifold nowhere tangent to H . Then itsflowout FLO + g (Λ ∩ Char g ) is Lagrangian. (cid:3) Let ξ ∈ T ∗ ( t ,x ) M . For any h > B h ( ξ ) := (cid:8) ξ ∈ T ∗ ( t ,x ) M (cid:12)(cid:12) k ξ − ξ k G < h (cid:9) Denote by ccl B h ( ξ ) its cone closure. We recall that a hypersurface S ⊂ M is calledlightlike when the Lorentzian metric − dt + g restricts to a degenerate bilinear form on T S , or equivalently its conormal bundle consists of lightlike covectors.
Lemma 2.4.
Let ξ ′ ∈ S ∗ x M and t > be such that the end points of the geodesic γ ([0 , t ] , ξ ′ ) are not conjugate. We set ξ := − dt + ξ ′ ∈ T ∗ (0 ,x ) M ∩ Char g , and considerfor h > small enough the Lagrangian submanifolds Λ ± := FLO + g (ccl B h ( ± ξ ) ∩ Char g ) . Then, for any sufficiently small open neighborhood V ⊂ M of ( t , γ ( t , ξ ′ )) , there existsa lightlike codimension-one hypersurface S ⊂ V such that (Λ + ∪ Λ − ) ∩ T ∗ V = N ∗ S. Proof.
Let U ⊂ S ∗ x M be a small open neighborhood of ξ ′ , and ǫ > ι ± : ( t − ǫ, t + ǫ ) × U → Char g ⊂ T ∗ M ,ι ± ( t, ξ ′ ) = e + ( t, , x , ∓ dt ± ξ ′ ) = ( t, γ ( t, ξ ′ ) , ∓ dt ± ˙ γ ( t, ξ ′ ) ♭ ) , LEO TZOU where ˙ γ ( t, ξ ′ ) := ∂ t γ ( t, ξ ′ ) and ˙ γ ( t, ξ ′ ) ♭ := g ( ˙ γ ( t, ξ ′ ) , · ). Since the end points of thegeodesic γ ([0 , t ] , ξ ′ ) are not conjugate, π ◦ ι is a diffeomorphism onto its image, where π : T M → M is the projection onto the factor M of the base M = R × M . If V ⊂ M isa sufficiently small open neighborhood of ( t , γ ( t , ξ ′ )), and ˜ U := ι − ( T ∗ V ) = ι − − ( T ∗ V ),then clearly Λ ± ∩ T ∗ V = ccl( ι ± ( ˜ U )) . The codimension-one hypersurface S ⊂ V of the statement is given by S := { ( t, γ ( t, ξ ′ )) | ( t, ξ ′ ) ∈ ˜ U } . If J , J are two linearly independent Jacobi fields along the geodesic γ ( · , ξ ′ ) that areorthogonal to it and satisfy J (0) = J (0) = 0, then the tangent spaces of S are givenby T ( t,γ ( t,ξ ′ )) S = span { ∂ t + ˙ γ ( t, ξ ′ ) , J ( t ) , J ( t ) } . We claim that ι ± ( ˜ U ) ⊂ N ∗ S . Indeed( ∓ dt ± ˙ γ ( t, ξ ′ ) ♭ )( ∂ t + ˙ γ ( t, ξ ′ )) = ∓ ± k ˙ γ ( t, ξ ′ ) k g = 0 , ( ∓ dt ± ˙ γ ( t, ξ ′ ) ♭ )( J i ( t )) = g ( ± ˙ γ ( t, ξ ′ ) , J i ( t )) = 0 . Since the conormal bundle N ∗ S has rank one, we conclude ccl( ι + ( ˜ U ) ∪ ι − ( ˜ U )) = N ∗ S .Finally, since ι ± has image inside the characteristic set Char g , S is a lightlike hypersur-face. (cid:3) Lemma 2.5.
Let S j ⊂ M , j = 1 , , , be lightlike hypersurfaces with a triple transverseintersection at (0 , ) , i.e. Dim (cid:16) N ∗ (0 , ) S + N ∗ (0 , ) S + N ∗ (0 , ) S (cid:17) = 3 . (2.6) Then there is an open set O ⊂ M containing (0 , ) so that K := S ∩ S ∩ S ∩ O is aspacelike segment. Furthermore, N ∗ ( t,x ) K = N ∗ ( t,x ) S + N ∗ ( t,x ) S + N ∗ ( t,x ) S , ∀ ( t, x ) ∈ K. (2.7) Proof.
The transversality condition (2.6) ensures that near (0 , ) the triple intersection S ∩ S ∩ S is an open curve segment K and that (2.7) holds.It remains to check that K is spacelike. Observe that due to (2.7) if V ∈ T (0 , ) K ⊂ T (0 , ) M then there are three independent covectors ξ j ∈ T ∗ (0 , ) M ∩ Char g , j = 1 , ,
3, sothat ξ j ( V ) = 0. Without loss of generality ξ j = − dt + ξ ′ j with ξ ′ j ∈ S ∗ M . By linearindependence we have that ξ ′ = ξ ′ (2.8)Without loss of generality we can write V = a∂ t + V ′ for some V ′ ∈ S M . Using ξ ( V ) = ξ ( V ) = 0, we see that ξ ′ ( V ′ ) = ξ ′ ( V ′ ) = a . Due to (2.8) we have that | a | < V is spacelike and this implies that the segment K is spacelike near (0 , ). (cid:3) Lemma 2.6.
Let S ⊂ M be a lightlike hypersurface and let ξ ∈ N ∗ ( t,x ) S . If V ∈ T ( t,x ) S ∩ Char g then V ∈ span { ξ ♯ } where ♯ is taken with respect to the Lorentzian metric − dt + g . ETERMINING RIEMANNIAN MANIFOLDS FROM MEASUREMENTS AT A SINGLE POINT 7
Proof.
Write ξ = − dt + ξ ′ with ξ ′ ∈ S ∗ x M then ξ ♯ = ∂ t + ξ ′ ♯ where ξ ′ ♯ is the raising ofindex with respect to the Riemannian metric g on M . Since V ∈ T ( t,x ) M is lightlike itis of the form (modulo multiplication by R ), V = ∂ t + V ′ for some V ′ ∈ S x M . By thefact that ξ ∈ N ∗ ( t,x ) S , ξ ( V ) = 0 meaning that1 = ξ ′ ( V ′ ) = h ξ ′ ♯ , V ′ i g . Since both ξ ′ ♯ and V ′ are unit vectors, we must have that V ′ = ξ ′ ♯ . That is V = ξ ♯ . (cid:3) Lemma 2.7.
For all linearly independent covectors η ′ , η ′ ∈ S ∗ y M , there exists η ′ ∈ span { η ′ , η ′ } ∩ S ∗ y M arbitrarily close to η ′ such that Dim (cid:0) span {− dt − η ′ , − dt − η ′ , − dt − η ′ } (cid:1) = 3 . (2.9) Proof.
We choose orthonormal coordinates on T ∗ y M so that − η ′ = (1 , , , − η ′ = ( s, p − s , , for some s ∈ [ − , η ′ and η ′ are linearly independent, we have s = ±
1. Wechoose η ′ of the form − η ′ = ( r, p − r , r ∈ ( − , \ { s } near s , so that η ′ ∈ span { η ′ , η ′ } . The three vectors − dt − η ′ i , i = 1 , ,
3, are linearly independent if and only if the matrix − − s √ − s − r √ − r is non-singular, that is, if and only if − s s = − r r . Since t − t t is monotonic for t ∈ ( − , r ∈ ( − , \ { s } . (cid:3) Analysis of PDE and Wave Front.
We first construct sources f which arecompactly supported near a point in M with wavefront set microlocalized near a singledirection.Let ξ ′ ∈ S ∗ M and write ξ = − dt + ξ ′ ∈ T ∗ (0 , ) M as an element of Char g . Using localcoordinates we can construct a homogeneous degree zero symbol ω ,h ( t, x, ξ ) ∈ S ( M )such that ω ,h ( t, x, ξ ) = 1 for ξ near ± ξ and ω ,h ( t, x, D ) δ (0 , ) ∈ I ( M ; ccl B h ( ξ ) ∪ ccl B h ( − ξ )) . For a Lagrangian submanifold Λ ⊂ T ∗ M we define I ( M , Λ) := [ m ∈ N I m ( M , Λ)to be the space of Lagrangian distributions associated to Λ. We refer the reader toDefinition 4.2.1 of [Dui96] for definition of I m ( M , Λ).Set χ ,h ∈ C ∞ ( M ) such that χ ,h (0 , ) = 1 but supp( χ ,h ) ⊂ B G ((0 , ); h ). Define thecompactly supported conormal distribution f h ∈ I ( M ; ccl B h ( ξ ) ∪ ccl B h ( − ξ )) by f h ( t, x ) := χ ,h ( t, x ) ω ,h ( t, x, D ) h D i − N δ (0 , ) (2.10) LEO TZOU for large fixed
N > h D i of order 1.We will often drop the h subscript to simplify notation. We choose N > f h ( t, x ) ∈ C c ( M ).Let u ∈ I ( M , Λ) be a Lagrangian distribution for some conic Lagrangian Λ and σ ( u ) ∈ S µ (Λ; Ω / ⊗ L ) /S µ − (Λ; Ω / ⊗ L )be its principal symbol where Ω / is the half-density bundle and L → Λ is the Maslovcomplex line bundle with local trivializations defined on open conic neighborhoods U j ⊂ Λ. The transition functions of two local trivialization of L on two intersecting openconic sets U j and U k are given by i m j,k where m j,k ∈ Z . This observation allows us todefine the notion of homogeneity of sections of L . Let ρ > m ρ : ( t, x, ξ ) ( t, x, ρξ ) mapping T ∗ M to itself. If ℓ is a section of L whose trivialization in the openconic neighborhood U j is given by the C valued function ℓ j , we can define m ∗ ρ ℓ by thesection whose local trivialization on U j is m ∗ ρ ℓ j . Note that since the transition functionsare constants (and thus remain unchanged under pullback by m ρ ), this definition iscoordinate independent.The fact that transition functions are constants also allows us to define the notion ofa Lie derivative and pullback by a flow. Indeed if X is a section of T Λ and Φ X denotesits flow, we define (Φ Xs ) ∗ ℓ for s small by (Φ Xs ) ∗ ℓ j ( t, x, ξ ) := ℓ j (Φ Xs ( t, x, ξ )) provided thatboth Φ Xs ( t, x, ξ ) and ( t, x, ξ ) are in U j . Using the group property of s Φ xs we canextend this definition for any s ∈ R by observing that transition functions between localtrivializations are constants. The Lie derivative can then be defined by differentiatingthe pullback with respect to the parameter s .It will also be useful to have local representations for sections of the half-densitybundle Ω / on Λ. Locally, a conic neighborhood U ⊂ Λ can be written as U = { ( t, x, d t,x φ ( t, x, θ ) | ( t, x, θ ) ∈ C φ } for some V ⊂ M , Θ ⊂ R N a conic subset, φ : V × Θ → R a homogeneous degree 1nondegenerate phase function, and C φ := { ( t, x, θ ) ∈ V × Θ | d θ φ ( t, x, θ ) = 0 } . Denote by T φ : C φ → U the diffeomorphism ( t, x, θ ) ∈ C φ ( t, x, d t,x φ ( t, x, θ )). De-fine d C φ := ( d θ φ ) ∗ δ , R N to be a smooth section of Ω on C φ . Here δ , R N is the deltadistribution on R N at the origin.To see that d C φ is degree N , it is useful to write down an explicit expression for d C φ . Let λ , . . . , λ n be homogeneous degree 1 local coordinates on C φ which we extendsmoothly to a neighourbhood of C φ in V × R N . In these coordinates d C φ = (cid:12)(cid:12)(cid:12)(cid:12) D ( λ , . . . , λ n , φ θ , . . . , φ θ N ) D ( t, x, θ ) (cid:12)(cid:12)(cid:12)(cid:12) − dλ . . . dλ n . (2.11)The determinant is homogeneous of order N − n and dλ . . . dλ n is order n . So d C φ ishomogeneous of degree N .If [ a ] ∈ S µ (Λ , Ω / ⊗ L ) /S µ − (Λ , Ω / ⊗ L ) has homogeneous principal symbol and( t, x, ξ ) ∈ Λ, we say that [ a ]( t, x, ξ ) = 0 ETERMINING RIEMANNIAN MANIFOLDS FROM MEASUREMENTS AT A SINGLE POINT 9 if there is a an element a ∈ [ a ] homogeneous of degree µ such that a ( t, x, ξ ) = 0 . If ℓ φ is a non-vanishing homogeneous degree 0 section of the pullback bundle T ∗ φ L and[ a ]( t , x , ξ ) = 0 then under pullback by T φ , T ∗ φ a ( t , x , θ ) = a φ ( t , x , θ ) q d C φ ⊗ ℓ φ for some a φ ( t, x, θ ) ∈ S µ − N/ ( V × Θ) which satisfies | a φ ( t , x , ρθ ) | ≥ C t ,x ,θ | ρ | µ − N/ for ρ > Lemma 2.8.
Suppose K ⊂ M is a smooth submanifold whose conormal bundle N ∗ K istransverse to H . Let Λ = FLO + g ( N ∗ K ∩ Char g ) and let a ∈ S µ (Λ , Ω / ⊗ L ) solve thetransport equation L H a = 0 on Λ , a | ∂ Λ = a ∂ Λ for some a ∂ Λ ∈ C ∞ ( ∂ Λ , Ω / ⊗ L ) .Suppose ( m ∗ ρ a )( t , x , ξ ) = ρ µ a ( t , x , ξ ) and a ( t , x , ξ ) = 0 for some ( t , x , ξ ) ∈ ∂ Λ then ( m ∗ ρ a )( t, x, ξ ) = ρ µ a ( t, x, ξ ) and a ( t, x, ξ ) = 0 for all ( t, x, ξ ) ∈ FLO + g ( t , x , ξ ) .Proof. Assume that ( t , x , ξ ) , ( t, x, ξ ) ∈ Λ ∪ ∂ Λ can be represented by ( t , x , θ ) , ( t, x, θ ) ∈ C φ via the mapping T φ and ( t, x, ξ ) = e + ( s , t , x , ξ ). In this proof we adopt the nota-tion e s ( t, x, ξ ) := e + ( s, t, x, ξ ) . To prove the lemma it suffices to show that if a ( t , x , ξ ) = 0 and ( m ∗ ρ a )( t , x , ρξ ) = ρ µ a ( t , x , ξ ) then m ∗ ρ a ( t, x, ξ ) = ρ µ a ( t, x, ξ ) , a ( t, x, ξ ) = 0 . (2.12)To this end let ( t ( s ) , x ( s ) , ξ ( s )) := e s ( t , x , ξ ) for s ∈ (0 , s ). Note that( t ( s ) , x ( s ) , ρξ ( s )) = e s/ρ ( t , x , ρξ ) (2.13)for s ∈ (0 , s /ρ ). Since L H a = 0 we have that a ( t ( s ) , x ( s ) , ξ ( s )) = ( e ∗ s a )( t ( s ) , x ( s ) , ξ ( s )) (2.14)for all s . So using the rescaling formula (2.13) together with the invariance formula(2.14), we see that (2.12) is equivalent to m ∗ ρ e ∗− s /ρ a ( t, x, ξ ) = ρ µ e ∗− s a ( t, x, ξ ) , e ∗− s a ( t, x, ξ ) = 0 (2.15)Define ˜ e s := T − φ ◦ e s ◦ T φ as the pullback flow defined on C φ and write ( t ( s ) , x ( s ) , θ ( s )) =˜ e s ( t , x , θ ). Using these coordinates we see that (2.15) is equivalent to m ∗ ρ ˜ e ∗− s /ρ (cid:16) ( a φ q d C φ ⊗ ℓ φ )( t, x, θ ) (cid:17) = ρ µ ˜ e ∗− s (cid:16) a φ q d C φ ⊗ ℓ φ (cid:17) ( t, x, θ ) . (2.16)We compute the LHS of (2.16). By definition m ∗ ρ ˜ e ∗− s /ρ ( a φ )( t, x, ξ ) = ˜ e ∗− s /ρ a φ ( t, x, ρξ ) = a φ ( t , x , ρξ ) = ρ µ − N/ a φ ( t , x , ξ ) . (2.17) The last equality comes from assumption that the symbol is homogeneous at ( t , x , ξ ).The second last equality comes from the rescaling formula (2.13) and its analogousrelation for the ˜ e s flow.The half-density p d C φ is given by (2.11) where the coordinate functions λ j are ho-mogeneous of degree 1. So using (2.13) we have that m ∗ ρ ˜ e ∗− s /ρ q d C φ = ρ N/ ˜ e ∗− s q d C φ (2.18)The section of the Maslov bundle ℓ φ is homogeneous of degree 0 so m ∗ ρ ˜ e ∗− s /ρ ℓ φ = ˜ e ∗− s ℓ φ . (2.19)Substituting (2.17), (2.18), and (2.19) into the LHS of (2.16) we have that (2.16) isequivalent to ρ µ a φ ( t , x , ξ )˜ e ∗− s (cid:16)q d C φ ⊗ ℓ φ (cid:17) ( t, x, θ ) = ρ µ ˜ e ∗− s (cid:16) a φ q d C φ ⊗ ℓ φ (cid:17) ( t, x, θ ) . (2.20)Observing that ˜ e ∗− s a φ ( t, x, θ ) = a φ ( t , x , θ ) verifies (2.20). (cid:3) Two Lagrangians Λ , Λ ⊂ T ∗ M are said to be an intersecting pair if Λ ∩ Λ = ∂ Λ and T λ Λ ∩ T λ Λ = T λ ∂ Λ , ∀ λ ∈ ∂ Λ In this case for all m ∈ Z , Definition 3.1 of [MU79] defined the class of paired Lagrangiandistributions I m ( M , Λ , Λ ). We denote by I ( M , Λ , Λ ) := S m ∈ Z I m ( M , Λ , Λ ). Proposition 2.9.
Let K ⊂ M be a smooth submanifold and f ∈ I ( M , N ∗ K ) where N ∗ K is transverse to H and assume that f has homogeneous principal symbol. Let u bea solution of ✷ g u = f, u | t< − = 0 . (2.21) Then N ∗ K and Λ := FLO + g ( N ∗ K ∩ Char g ) form an intersecting pair of Lagrangians. Thesolution u belongs to u ∈ I ( M , N ∗ K, Λ) with homogeneous principal symbol on Λ . Fur-thermore if ( t , x , ξ ) ∈ WF( f ) ∩ Char g with σ ( f )( t , x , ξ ) = 0 , then FLO + g ( t , x , ξ ) ⊂ WF( u ) with σ ( u )( t, x, ξ ) = 0 for all ( t, x, ξ ) ∈ FLO + g (( t , x , ξ )) .Proof. It suffices to construct a parametrix for (2.21) satisfying ✷ g u − f ∈ C ∞ . Thiswas done in Proposition 6.6 of [MU79] where the homogeneous principal symbol σ ( u )satisfies L H σ ( u ) = 0 o n Λ , σ ( u ) | N ∗ K ∩ Λ = R (cid:0) ( τ − k ξ ′ k g ) − σ ( f ) (cid:1) | N ∗ K ∩ Λ . Here R is the operator defined via Definition (4.7) in [MU79]. By Lemma 2.8 it sufficesto show that if ( t , x , ξ ) ∈ N ∗ K ∩ Char g satisfies σ ( f )( t , x , ξ ) = 0 then R (cid:0) ( τ − k ξ ′ k g ) − σ ( f ) (cid:1) ( t , x , ξ ) = 0 . To this end observe that H is the Hamiltonian vector field of the Hamiltonian h ( t, x, τ dt + ξ ′ ) := τ − k ξ ′ k g whose zero energy set is Char g . So since the Hamiltonian vector field H is transverse to the Lagrangian N ∗ K , N ∗ K ∩ Char g is a smooth three-dimensionalsubmanifold of T ∗ M . So near ( t , x , ξ ) we can choose h , . . . h to be a coordinatesystem for N ∗ K ∩ Char g . We then have that { h , . . . , h } is a coordinate system for ETERMINING RIEMANNIAN MANIFOLDS FROM MEASUREMENTS AT A SINGLE POINT 11 N ∗ K in a neighbourhood of ( t , x , ξ ). Note that since H is the Hamiltonian flow of h , h = 0 on FLO + g ( N ∗ K ∩ Char g ) and N ∗ K ∩ Char g = N ∗ K ∩ Λ = ∂ Λ . The Lagrangian Λ is diffeomorphic to R + × ∂ Λ via the map ( s, t, x, ξ ) e + ( s, t, x, ξ ).Define h ∈ C ∞ (Λ) by h ( e + ( s, t, x, ξ )) = s with ( t, x, ξ ) ∈ ∂ Λ. Clearly { h , h } = Hh = 1 > { h , h , h , h } forms a local coordinate system for Λ near FLO + g ( t , x , ξ ). By (4.7)of [MU79], in these coordinates if σ ( f ) = ˆ f | dh ∧ · · · ∧ dh | / ⊗ ℓ then R (cid:0) h − σ ( f ) (cid:1) = ˆ f | dh ∧ dh ∧ dh ∧ dh | / ⊗ ℓ. So by our assumption that σ ( f )( t , x , ξ ) = 0, we can conclude that R (cid:0) h − σ ( f ) (cid:1) ( t , x , ξ ) = 0thus completing the proof. (cid:3) Lemma 2.10.
Let K j ⊂ M , j = 1 , , be submanifolds such that K is transverse to K and K is transverse to K ∩ K . Let u j ∈ I ( M ; N ∗ K j ) for j = 1 , , .i) Let ω ( t, x, D ) be a pseudodifferential operator whose wavefront is disjoint from N ∗ K ∪ N ∗ K . Then ω ( t, x, D )( u u ) ∈ I ( M , N ∗ ( K ∩ K )) with principal symbol given by σ ( u u )( t, x, ξ ) = ω ( t, x, ξ ) σ ( u )( t, x, ξ (1) ) σ ( u )( t, x, ξ (2) ) where ξ = ξ (1) + ξ (2) .ii) Let ω ( t, x, D ) be a pseudodifferential operator whose wavefront is disjoint from [ j =1 N ∗ K j ∪ [ j,k =1 ,j = k N ∗ ( K j ∩ K k ) then ω ( x, D )( u u u ) ∈ I ( M , N ∗ ( K ∩ K ∩ K )) with principal symbol given by σ ( u u u )( ξ ) = ω ( x, ξ ) σ ( u )( ξ (1) ) σ ( u )( ξ (2) ) σ ( u )( ξ (3) ) where ξ = ξ (1) + ξ (2) + ξ (3) .Proof. Part i) is Lemma 2 of [CMOP19]. Part ii) is proof of equation (56) in [CMOP19].Note that both are based on the work of [GU93]. (cid:3)
Lemma 2.11.
Suppose that S and S are lightlike hypersurfaces of M of codimensionone and intersects transversely, then N ∗ ( S ∩ S ) ∩ Char g ⊂ ( N ∗ S ∪ N ∗ S ) . Proof.
This is a trivial linear algebra calculation. See e.g. (30) in [CMOP19]. (cid:3)
Lemma 2.12. If u ∈ I ( M , N ∗ S ) is a conormal distribution for some lightlike hypersur-face S containing the point (0 , ) . Suppose σ ( u )(0 , , ξ ) = 0 , then for all timelike curve α ( · ) which intersects S transversely at (0 , ) the distribution α ∗ u ∈ D ′ ( R ) is singular atthe point ∈ R .Proof. Since everything is local we may assume that M = R endowed with Lorentzianmetric − dt + g where g is a time-independent Riemannian metric on R . We chooselocal coordinates z = ( z , z , z , z ) ∈ R so that S = { z = 0 } , ξ = dz , and α ( t ) = ( α ( t ) , α ( t ) , α ( t ) , α ( t )) with α (0) = (0 , , , α (0) = 0 (2.22)In these coordinates, since σ ( u )(0 , , ξ ) = 0, u ( z ) = Z R e iθz a ( z, θ ) dθ where for some N ∈ R , the symbol a ( z, θ ) satisfies | a (0 , θ ) | ≥ C | θ | N (2.23)for all θ >
1. The pullback α ∗ u ( t ) is then α ∗ u ( t ) = Z R e iθα ( t ) ˜ a ( t, θ ) dθ (2.24)where ˜ a ( t, θ ) = a ( α ( t ) , θ ). By (2.22) the map ( t, θ ) α ( t ) θ is a non-degenerate phasefunction. So (2.24) is a oscillatory distribution with nondegenerate phase. For any χ ∈ C ∞ c ( R ) supported in a neighborhood of 0, we can take the Fourier Transform of χα ∗ u . Apply Thm 2.3.1 of [Dui96] we have that due to (2.23) | F ( χα ∗ u )( τ ) | ≥ C | τ | N ′ for all τ >
1. Therefore 0 ∈ singsupp( α ∗ u ). (cid:3) Remark 2.13.
Lemma 2.24 states that an observer along a timelike world line is able tosee a the singularity of distribution whose wavefront is the conormal bundle of a lightlikehypersurface. This is not true in general for lightlike Lagrangians.Consider the the Minknowski metric − dt + e on R and the corresponding Green’sfunction ✷ R G ( t, x ) = 0 , G (0 , x ) = 0 , ∂ t G (0 , x ) = δ ( x )where δ ( x ) is the delta distribution at the origin ∈ R . The distribution G ( · , · ) actson ϕ ∈ C ∞ c ( R ) by h ϕ, G ( · , · ) i R = Z ∞−∞ t − Z ∂B e ( ,t ) ϕ ( x )dVol t ( x ) dt where dVol t is the volume form of the round sphere of radius t . This is Langrangiandistribution on R which is not conormal at (0 , ) ∈ R . Take the curve α ( s ) = ( s, ) , s ∈ R . Then ( α ∗ G )( s ) = 0 for all s ∈ R . ETERMINING RIEMANNIAN MANIFOLDS FROM MEASUREMENTS AT A SINGLE POINT 13
We conclude the section with a lemma about the linearization of solutions of (1.1)along timelike curves
Lemma 2.14.
Let u solve (1.1) with source give by P l =0 ǫ l f l where f l ∈ C Nc (( − , T +1) × U ) for large but finite N . Then u = u ans + u r where u ans := X β ∈ N , | β |≤ ǫ β u β and k u r k C ([ − , T +1] × M ) ≤ C P ≤| β |≤ ǫ β .i) Let β j ∈ N be the multi-index which is in the j th slot and elsewhere. Then ✷ g u β j = f j u β j | t< − = 0 . ii) If | β | = 2 then u β = 0 We denote, for distinct j, k ∈ { , . . . , } , β j,k to be the elementwith in the j th and k th slot and zero elsewhere .iii) Let j , k , and l be distinct and β j,k,l ∈ N be the element which is in the j th, k th,and l th spot and elsewhere. Then ✷ g u β j,k,l = − u j u k u l u β j,k,l | t< − = 0 . iv) If | β | = 4 then u β = 0 . We define β j,k,l,m as above.v) Let ˆ β = (1 , , , , . Then u ˆ β solves ✷ g u ˆ β = − X j,k,l,m,n =0 j = k = l = m = n u β j,k,l u β m u β n , u ˆ β | t< − = 0 . Remark 2.15. If u is as in Lemma 2.14, we denote by u j := u β j as in i) of the lemma, u jk := u β j,k as in ii) of lemma, u jkl := u β j,k,l as in iii) of the lemma, u jklm := u β j,k,l,m asin iv) of lemma, and u := u ˆ β as in v) of the Lemma. Proof.
For each multiindex β ∈ N satisfying | β | ≤ u β so that the ansatz u ans := X β ∈ N , | β |≤ ǫ β u β solve the approximate (1.1): ✷ g u ans + u = X l =0 ǫ l f l + X ≤| β |≤ ǫ β R β . where each R β ∈ C N ′ ([ − , T + 1] × M ) for some large N ′ . Direct (but cumbersome)calculation shows that i) - v) are satisfied.Using the result for solving (172) in Appendix C of [KLU13] we can find u r ∈ C ([ − , T + 1] × M ) solving ✷ g u r + 3 u r u + u + 3 u u ans = X ≤| β |≤ ǫ β R β . Furthermore we have the well-posedness estimate k u r k C ([ − , T +1] × M ) ≤ X ≤| β |≤ ǫ β k R β k C N ′ ([ − , T +1] × M )) . (2.25)Simple calculation shows that u = u ans + u r solves (1.1). (cid:3) It is useful to have an explicit formula for u in terms of the solution u of (1.1)with source P l =0 ǫ l f l . To this end we write ǫ := ( ǫ , . . . , ǫ ) and adopt the notation K ( ǫ ) := (0 , ǫ , . . . , ǫ ). That is, we replace the first entry of ǫ by 0. Define K j ( ǫ )accordingly for j = 0 , · · · ,
4. Similarly for distinct j, k ∈ { , . . . , } , K j,k ( ǫ ) replaces the j th and k th entry of ǫ with 0 and leaves the rest unchanged. Define the operators K j,k,l and K j,k,l,m accordingly for distinct j, k, l, m ∈ { , . . . , } .To highlight the dependence of u on the parameter ǫ we write u = u ( ǫ ) in the followingdiscussion. Using the estimate for u r in Lemma 2.14 and the formal expansion of u ans ,we see that Lemma 2.16. If α : [ − , T + 1] → M is a smooth timelike curve then α ∗ u = lim ǫ → ǫ = ··· = ǫ =0 α ∗ u ( ǫ ) ǫ .α ∗ u ... = lim ǫ → ,ǫ = ··· = ǫ ,ǫ = ǫ =0 α ∗ u ( ǫ ) − P j =0 α ∗ u ( K j ǫ ) + P j,k =0 ,j = k α ∗ u ( K j,k ǫ ) ǫ . . . ǫ . α ∗ u = lim ǫ → ,ǫ = ··· = ǫ α ∗ u ( ǫ ) − P j =0 α ∗ u ( K j ǫ ) + P j,k =0 ,j = k α ∗ u ( K j,k ǫ ) − P j,k,l =0 ,j = k = l α ∗ u ( K j,k,l ǫ ) + P j,k,l,m =0 ,j = k = l = m α ∗ u ( K j,k,l,m ǫ ) ǫ . . . ǫ . We formally write the limit on the right hand side as ∂ ǫ ( α ∗ u ) | ǫ = ··· = ǫ =0 , ∂ ǫ ...ǫ ( α ∗ u ) | ǫ = ··· = ǫ =0 , ∂ ǫ ...ǫ ( α ∗ u ) | ǫ = ··· = ǫ =0 etc. Formally, we can view the result of Lemma 2.16 as the commutation of the restrictionoperator with differentiation with respect to ǫ .3. Analysis of Multilinear Wave Interaction
Lightlike Flowout from Spacelike Curves.
In this subsection we deduce someconormal properties for the Langrangian generated by flowing out by lightlike covectorson the normal bundle of a spacelike curve. The projection of these Lagrangians are ingeneral very irregular objects. However, we deduce sufficient conormal properties for usto carry on our analysis later.Let K ⊂ M be a smooth and short spacelike segment containing the point (0 , y ).Suppose − dt + η ′ ∈ N ∗ (0 ,y ) K is a lightlike covector. Let γ g ([0 , R ] , η ′ ) be the uniquelength minimizing geodesic between the initial point y and the end point x ∈ U whichsatisfies d g ( x , ) < δ / δ is defined in (1.5). Assume the segment does not ETERMINING RIEMANNIAN MANIFOLDS FROM MEASUREMENTS AT A SINGLE POINT 15 contain conjugate points. Note though that there may be other (longer) geodesics joiningthe two end points. Set Λ := FLO + g ( N ∗ K ∩ Char g )to be the flowout and define the point λ ∈ Λ to be λ := ( R, x , − dt + ξ ′ )where ξ ′ = ˙ γ g ( R, η ′ ) ♭ . The set N ∗ K ∩ Char g is a smooth 3-manifold by Lemma 2.2and so Λ is a smooth Lagrangian submanifold of T ∗ M by Lemma 2.3. Observe that thefuture oriented flow (2.2) gives a diffeomorphism R + s × ( N ∗ K ∩ Char g ) → Λ , ( s, t, x, ξ ) e + ( s, t, x, ξ ) . (3.1)Due to their conic nature, N ∗ K ∩ Char g and Λ are sometimes inconvenient to workwith, as the differential of the base projection acting on these manifolds has at leasta one dimensional kernel. For this reason, we will sometimes work with the fiberwisequotients by R + of the complements of the zero-sections of N ∗ K ∩ Char g and Λ; inorder to simplify the notation, we will not indicate the removal of the zero-section in thenotation, and simply write ( N ∗ K ∩ Char g ) / R + and Λ / R + for such quotients. We willsee them by means of the identifications( N ∗ K ∩ Char g ) / R + ∼ = { ( t, x, ξ ) ∈ N ∗ K | ξ = ± dt + ξ ′ , ξ ′ ∈ S ∗ x M } , (3.2)Λ / R + ∼ = { ( t, x, ξ ) ∈ Λ | ξ = ± dt + ξ ′ , ξ ′ ∈ S ∗ x M } . (3.3)The future oriented Hamiltonian flow (3.1) induces a diffeomorphism of the quotients R + × ( N ∗ K ∩ Char g ) / R + → Λ / R + e + ( s, r, x, ∓ dt + η ′ ) = ( r + s, γ g ( s, ± η ′ ) , ∓ dt ± ˙ γ g ( s, ± η ′ ) ♭ ) . (3.4) Proposition 3.1.
There exists sequences Λ ∋ λ j = ( R j , x j , − dt + ξ ′ j ) → λ = ( R, x , − dt + ξ ′ ) , ( N ∗ K ∩ Char g ) / R + ∋ η j = ( r j , y j , − dt + η ′ j ) → (0 , y , − dt + η ′ ) ,s j → R, with x j = x , R j > R , e + ( s j , η j ) = λ j , open neighborhoods O j ⊂ M of ( R j , x j ) , openneighborhoods U j ⊂ ( N ∗ K ∩ Char g ) / R + of η j , quantities δ j > , and open neighborhoods Γ j ⊂ K of ( r j , y j ) such that S j := π ◦ e + (( s j − δ j , s j + δ j ) × U j ) is a lightlike codimension-one hypersurface in M satisfying T ∗ O j ∩ FLO + g ( N ∗ Γ j ∩ Char g ) = N ∗ ( S j ∩ O j ) . Finally we can arrange the sequences λ j , S j , and O j so that π ◦ FLO + g ( N ∗ ( S j ∩ O j )) ∩ ([ R, R + δ / × x ) = ∅ . (3.5) Proof.
Since K is spacelike and short, there is a smooth function t K : M → R such that K = { ( t K ( x ) , x ) | x ∈ K ′ } for some smooth curve K ′ ⊂ M . Furthermore we can assumewithout loss of generality that K is short enough so that K ′ ⊂ π ( U ′ ) (3.6) where U ′ is as in Lemma 2.1.By assumption Λ ⊂ T ∗ M is a conic Lagrangian which contains the element λ =( R, x , − dt + ξ ′ ). The set of points λ ∈ Λ for which π Λ := π | Λ has constant rank in aneighborhood of λ is open and dense. So there is a sequence λ j = ( R j , x j , − dt + ξ ′ j ) , ξ ′ j ∈ S ∗ x j M (3.7)converging to λ with R j > R and x j = x such that π Λ has constant rank in a neigh-borhood of λ j . If j ∈ N is sufficiently large, π ◦ FLO + g ( λ j ) ∩ [ R, R + δ / × { x } = ∅ (3.8)where δ is the radius of injectivity defined in (1.5).As Λ is the flowout of a submanifold transverse to the vector field H , for each λ j thereis a unique ( s j , r j , y j , − dt + η ′ j ), which we denote by ( s j , η j ), so that e + ( s j , η j ) = λ j . Using the constant rank theorem there are conic open sets V j ⊂ T ∗ M containing λ j suchthat π Λ (Λ ∩ V j ) = S j is a smooth codimension k submanifold of M and that Λ ∩ V j is aconic open subset of N ∗ S j .By observing the the fibers of Λ are lightlike covectors, we can deduce the codimension k . Indeed, suppose k ≥ t, x ) ∈ S α . By the fact that Λ ∩ V j is a conic open subsetof N ∗ S α , π − (( t, x )) ∩ Λ is an open subset of N ∗ ( t,x ) S α . Since π − (( t, x )) ∩ Λ is a conic opensubset of a vector space which is of dimension two or greater, there exists two linearlyindependent covectors ξ, ˜ ξ ∈ π − (( t, x )) ∩ Λ such that (1 − a ) ξ + a ˜ ξ ∈ π − (( t, x )) ∩ Λ forall a ∈ (0 , k ≥ k ≥ k = 1. In other words,Dim( S j ) = rank( Dπ Λ ( λ )) = 3 , ∀ λ ∈ V j ∩ Λ . (3.9)Since S j is lightlike we may assume that it is contained in a sufficiently small neighbor-hood of ( R j , x j ) so that S j = { ( t S j ( x ) , x ) | x ∈ S ′ j } (3.10)for some open subset S ′ j ⊂ M and smooth function t S j ( · ) : S ′ j → R .Recall that the quotient of Λ and N ∗ K ∩ Char g by R + is identified with (3.2) and(3.3). The projection π Λ ( λ ) = π Λ / R + ( λ ) so we may assume that V j is chosen sufficientlysmall so that π Λ / R + : (Λ ∩ V j ) / R + → S j is a diffeomorphism . (3.11)So for each λ j of the form (3.7) there is δ j > U j ⊂ ( N ∗ K ∩ Char g ) / R + open,( s j , r j , y j , − dt + η ′ j ) ∈ ( s j − δ j , s j + δ j ) × U j such that the map ( s, η ) π ◦ e + ( s, η ) is a diffeomorphism from( s j − δ j , s j + δ j ) × U j → S j . By (3.6), we may assume that U j is chosen sufficiently small so that U j ⊂ { ( t K ( y ) , y, − dt + η ′ ) | ( y, η ′ ) ∈ U ′ } . (3.12) ETERMINING RIEMANNIAN MANIFOLDS FROM MEASUREMENTS AT A SINGLE POINT 17
By shrinking S j if necessary we may assume that it is the image of ( s j − δ j , s j + δ j ) × U j under π ◦ e + .We now need to show that there exists an open set O j ⊂ M containing ( R j , x j ) andopen segment Γ j ⊂ K containing ( r j , y j ) such that π − ( O j ) ∩ FLO + g ( N ∗ Γ j ∩ Char g ) ⊂ N ∗ S j . We will do this by first showing that
Lemma 3.2.
For each fixed j ∈ N sufficiently large, there exists an open set O j ⊂ M containing ( R j , x j ) and an open segment Γ j ⊂ K containing ( r j , y j ) such that O j ∩ π (Λ j ) = O j ∩ S j . (3.13) Here, Λ j := FLO + g ( N ∗ Γ j ∩ Char g ) .Proof of Lemma 3.2. The map ( s, η ) e + ( s, η ) is a diffeomorphism from R + × N ∗ K ∩ Char g → Λ. So for each λ j of the form (3.7) there exists a unique s j ∈ R and η j =( r j , y j , − dt + η ′ j ) ∈ ( N ∗ K ∩ Char g ) / R + so that λ j = e + ( s j , η j ). We also have that s j → R and η j → (0 , y , − dt + η ′ ). In particular, x j = γ g ( s j , η ′ j ). Furthermore, for j ∈ N sufficiently large, ( y j , η ′ j ) ∈ U ′ ⊂ S ∗ M where U ′ is as in Lemma 2.1. Therefore the unitspeed Riemannian geodesic segment γ g ([0 , s j ] , η ′ j ) satisfies the following condition γ g ([0 , s j ] , η ′ j ) is the unique minimizer between its endpoints x j and y j . (3.14)We will show that (3.14) is contradicted if the lemma fails to hold. To this end, supposethe lemma fails for some j ∈ N large enough so that ( y j , η ′ j ) ∈ U ′ . Then with j ∈ N fixedthere exists a sequence of points { ( r j,l , y j,l ) } l ∈ N ⊂ K converging to ( r j , y j ), a sequenceof future-pointing lightlike covectors η j,l ∈ ( N ∗ ( r j,l ,y j,l ) K ∩ Char g ) / R + , and s j,l so that π ◦ e + ( s j,l , η j,l ) → ( R j , x j ) but π ◦ e + ( s j,l , η j,l ) / ∈ S j . Furthermore since π ◦ e + ( · , · ) : ( s j − δ j , s j + δ j ) × U j → S j is a diffeomorphism, we have that( s j,l , η j,l ) / ∈ ( s j − δ j , s j + δ j ) × U j . (3.15)Using the explicit expression for the flow (3.4) we see that s j,l → s j . This and (3.15)forces η j,l / ∈ U j for all l sufficiently large. Using (3.4) again we have that γ g ( s j,l , η ′ j,l ) → γ g ( s j , ˆ η ′ j ) = x j for some ˆ η ′ j = η ′ j . This contradicts (3.14). Thus the lemma is established. (cid:3) Now that Lemma 3.2 established, we only need to show that there exists Γ j ⊂ K containing ( r j , y j ) so that for all ( t, x ) ∈ O j ∩ π (Λ j ), the fibers of Λ j over ( t, x ) satisfies (cid:0) π − ( t, x ) ∩ Λ j (cid:1) = N ∗ ( t,x ) S j . If this is false, then by Lemma 2.6 there would be a future pointing lightlike bicharacter-istic curve emanating from N ∗ Γ j ∩ Char g whose projection intersects S j transversally.This would contradict Lemma 3.2Finally, (3.5) is a consequence (3.8) if we choose S j and O j sufficiently small. (cid:3) Threefold Interaction Producing Conormal Waves.
In this section we willuse Proposition 3.1 to deduce certain conormal properties produced by interacting waves.To this end consider distinct points x , x , x ∈ U and y ∈ M \ U and assume thateach x l is joined to y by the unit speed geodesic segment γ g ([0 , R l ] , ξ ′ l ) which is theunique minimizer between the end points with no conjugate points along the segment.Observe that as a consequence, d g ( x , γ ( R + t, ξ ′ )) > R − t (3.16)Furthermore we assume that˙ γ g ( R l , ξ ′ l ) = − ˙ γ g ( R k , ξ ′ k ) (3.17)when l = k . We label the opposite of the arrival direction at y of the geodesic segments γ g ([0 , R l ] , ξ ′ l ) to be η ′ l := − ˙ γ g ( R l , ξ ′ l ) ♭ . (3.18)Due to (3.17), any two of { η ′ k , η ′ l } is linearly independent. We assume in addition thatDim span { η ′ , η ′ , η ′ } = 2 (3.19)Furthermore we assume thatDim (cid:0) span {− dt − η ′ , − dt − η ′ , − dt − η ′ } (cid:1) = 3 . (3.20)Note that as a consequence of (3.19) and (3.20) we have that dt ∈ span {− dt − η ′ , − dt − η ′ , − dt − η ′ } which then implies − dt + η ′ ∈ span {− dt − η ′ , − dt − η ′ , − dt − η ′ } . (3.21)Let t = 0 and t , t ∈ R satisfy R = R + t = R + t . Denote by ξ l = − dt + ξ ′ l ∈ T ∗ ( t l ,x l ) M and observe that ( R , y ) ∈ \ l =0 π ◦ FLO + g ( t l , x l , ξ l ) (3.22)and also ( t k , x k ) / ∈ I + g ( t l , x l ) . (3.23)Combining the fact that γ g ([0 , R ] , ξ ′ ) is the unique minimizer between the end pointswith (3.17) we get that (2 R , x ) / ∈ π ◦ FLO + g ( t l , x l , ξ l ) (3.24)for l = 0 , , Proposition 3.3.
Assume that (3.22) , (3.17) , (3.23) , (3.19) , and (3.20) are satisfied.For each l = 0 , , there is a sequence of ξ l ; j ∈ T ∗ ( t l ,x l ) M ∩ Char g converging to ξ l satisfyingthe following properties: ETERMINING RIEMANNIAN MANIFOLDS FROM MEASUREMENTS AT A SINGLE POINT 19
For each element ξ l ; j we may choose an h j > so that if h ∈ (0 , h j ) and f l ; j is a( h -dependent) distribution of the form (2.10) with ξ l ; j ∈ WF( f l ; j ) ⊂ ccl B h ( ξ l ; j ) ∪ ccl B h ( ξ l ; j ) ⊂⊂ ccl B h ( ξ l ) ∪ ccl B h ( − ξ l ) ,σ ( f l ; j )( t l , x l , ξ l ; j ) = 0 , and v j are solutions of (1.1) with source f j := P l =0 ǫ l f l ; j then singsupp( v j ) containsa point ( ˜ T j , ˜ x j ) with ˜ T j > R and ˜ x j = x . The sequence { ( ˜ T j , ˜ x j ) } j ∈ N converges to (2 R , ) .Furthermore for each j ∈ N sufficiently large, there are open sets ˜ O j containing ( ˜ T j , ˜ x j ) with ˜ O j ∩ ( R × { x } ) = ∅ such that for all h ∈ (0 , h j ) , v j | ˜ O j ∈ I ( M , N ∗ ˜ S j ) , σ ( v j )( ˜ T j , ˜ x j , ˜ ξ ) = 0 , for ˜ ξ ∈ N ∗ ( ˜ T j , ˜ x j ) ˜ S j (3.25) for some lightlike hypersurfaces ˜ S j which satisfy (3.5) with R in place of R . Due to the absence of conjugate points along γ g ([0 , R l ] , ξ ′ l ), Lemma 2.4 asserts thatthere is a δ y > h > + g (ccl( B h ( ξ l ) ∪ B h ( − ξ )) ∩ Char g ) ∩ T ∗ B G ( R , y ; δ y ) = N ∗ ˆ S l for some lightlike hypersurface ˆ S l . Due to (3.20) and Lemma 2.5, the triple intersection( R , y ) ∈ K h := B G ( R , y ; δ y ) ∩ \ l =0 π ◦ FLO + g (ccl B h ( ξ l ) ∩ Char g ) (3.26)is a spacelike curve. Lemma 2.5 also states that N ∗ ( R ,y ) K = span {− dt − η ′ , − dt − η ′ , − dt − η ′ } . Due to (3.21), − dt + η ′ ∈ N ∗ ( R ,y ) K ∩ Char g . Therefore inserting the definition of η ′ (see (3.18)) into formula (3.4) we see that(2 R , x , ± dt ± ξ ′ ) ∈ Λ := FLO + g ( N ∗ K ∩ Char g ) . Due to (3.24) there is a δ > h > B G (2 R , x ; δ ) ∩ π ◦ FLO + g (ccl B h ( ξ l ) ∩ Char g ) = ∅ (3.27)for l = 0 , , γ g ([0 , R l ] , η ′ l ) are the unique minimizers from y to x l and con-tains no conjugate points. Therefore, y is the first point along γ g ([0 , R ] , ξ ′ ) that inter-sects γ g ([0 , R l ] , ξ ′ l ) for l = 1 ,
2. Furthermore, uniqueness of the minimizers γ g ([0 , R ] , η ′ )and γ g ([0 , R l ] , η ′ l ) ensures that even if we can extend both γ g ([0 , R ] , η ′ ) and γ g ([0 , R l ] , ξ ′ l )slightly, y is still the only point of intersection. Observe also that due to (3.16), I − g (2 R , x ) ∩ π ◦ FLO + g ( t , x , ξ ) ⊂ { ( t, γ g ( t, ξ ′ )) | t ∈ [0 , R ] } Therefore, we can choose δ > I − g ( B G (2 R , x ; δ )) ∩ \ l =0 π ◦ FLO + g (( t l , x l , ξ l )) = ( R , y ) . (3.28) If h > I − g ( B G (2 R , x ; δ )) ∩ \ l =0 π ◦ FLO + g (ccl B h ( ξ l ) ∩ Char g ) = K. (3.29)We now evoke Proposition 3.1 to produce a sequence of elements ( ˜ R j , ˜ y j ) ∈ K con-verging to ( R , y ) and elements ( ˜ T j , ˜ x j , − dt + ˜ ξ ′ j ) ∈ T ∗ B G (2 R , x ; δ ) ∩ Char g convergingto (2 R , x , − dt − ξ ′ ) with ˜ T j > R and ˜ x j = x such that( ˜ T j , ˜ x j , − dt + ˜ ξ ′ j ) ∈ FLO + g ( N ∗ ( ˜ R j , ˜ y j ) K ∩ Char g ) . (3.30)By Lemma 2.1, within N ∗ ( ˜ R j , ˜ y j ) K there is a unique lightlike covector − dt + ˜ η ′ j ∈ N ∗ ( ˜ R j , ˜ y j ) K ∩ Char g and unique ˜ s j > T j , ˜ x j , ∓ dt ± ˜ ξ ′ j ) = e + (˜ s j , ˜ R j , ˜ y j , ∓ dt ± ˜ η ′ j ) (3.31)with (˜ s j , ˜ R j , ˜ y j , − dt + ˜ η ′ j ) → ( R , R , y , − dt + η ′ ) . (3.32)Since ( ˜ R j , ˜ y j ) ∈ K and for l = 0 , , γ g ([0 , R l ] , ξ ′ l ) are unique minimizing segments con-taining no conjugate pints, for each j ∈ N there exists a unique (modulo multiplicationby R ), ξ l ; j ∈ B h ( ξ l ) ∩ Char g such that ( ˜ R j , ˜ y j ) ∈ T l =0 π ◦ FLO + g ( t l , x l , ξ l ; j ). We mayassume that ξ l ; j is of the form ξ l ; j = − dt + ξ ′ l ; j , ξ ′ l ; j ∈ S ∗ x l M. For each fixed j ∈ N there is an h j < h such that all h ∈ (0 , h j ) we have that B h ( ± ξ l ; j ) ⊂ B h ( ± ξ l ). The triple intersectionΓ h ; j := \ l =0 π ◦ FLO + g (ccl B h ( ξ l ; j ) ∩ Char g ) ⊂ K (3.33)is transverse and that T h> Γ h ; j = ( ˜ R j , ˜ y j ). \ l =0 π ◦ FLO + g ( B h ( ξ l ; j ) ∩ Char g )Furthermore, if we set ˜ η l ; j ∈ T ∗ M by( ˜ R j , ˜ y j , ˜ η l ; j ) := e + ( ˜ R j − t l , t l , x l , ξ l ; j ) (3.34)then by Lemma 2.5 N ∗ ( ˜ R j , ˜ y j ) Γ h = span { ˜ η j , ˜ η j , ˜ η j } because the triple intersection (3.33) is transverse. Observe that with η ′ l , l = 0 , , R j , ˜ y j , ˜ η l ; j ) → ( R , y , − dt − η ′ l ) . (3.35) ETERMINING RIEMANNIAN MANIFOLDS FROM MEASUREMENTS AT A SINGLE POINT 21
In particular, − dt + ˜ η ′ j ∈ N ∗ ( ˜ R j , ˜ y j ) Γ h defined by the relation (3.31) obeys − dt + ˜ η ′ j ∈ span { ˜ η j , ˜ η j , ˜ η j } (3.36)and − dt + ˜ η ′ j / ∈ span { ˜ η l,j , ˜ η k,j } k, l = 0 , , . (3.37)To conclude (3.37) we first observe that due to (3.17), − dt + η ′ / ∈ span {− dt − η ′ l } for l = 0 , , j ∈ N sufficientlylarge, − dt + ˜ η ′ j span { ˜ η l,j } for l = 0 , ,
2. Finally we use Lemma 2.11, while observingthat − dt + ˜ η ′ j , ˜ η l ; j , and ˜ η k ; j are all lightlike covectors, to conclude (3.37). Lemma 3.4.
For each j ∈ N sufficiently large there exists an open conic neighourbhoods P j ⊂ T ∗ M containing span {− dt + ˜ η ′ j } ⊂ N ∗ ( ˜ R j , ˜ y j ) Γ h ; j such that P j ∩ FLO + g (ccl B h ( ± ξ l ; j ) ∩ Char g ) = ∅ and P j ∩ (cid:0) FLO + g (ccl( B h ( ξ l ; j ) ∪ B h ( − ξ l ; j )) ∩ Char g ) + FLO + g (ccl( B h ( ξ k ; j ) ∪ B h ( − ξ k ; j )) ∩ Char g ) (cid:1) = ∅ for any k, l = 0 , , and h > sufficiently small.Proof. We identify T ∗ M / R + ∼ = M × S locally and for each η ∈ T M denote by η/ R + the corresponding element in the quotient space. We first use (3.36) to find open sets N j ⊂ S containing both ± ( − dt + ˜ η ′ j ) / R + ∈ S so that ¯ N j ∩ span { ˜ η l ; j , ˜ η k ; j } / R + = ∅ .Now we can choose h > R, y, ˜ η l ) ∈ FLO + g (ccl( B h ( ξ l ; j ) ∪ B h ( − ξ l ; j ))) ∩ T ∗ B G ( R , y ; δ y )and ( R, y, ˜ η k ) ∈ FLO + g (ccl( B h ( ξ k ; j ) ∪ B h ( − ξ k ; j ))) ∩ T ∗ B G ( R , y ; δ y )then we also have N j ∩ span { ˜ η k , ˜ η l } / R + = ∅ . Recall that B G ( R , y ; δ y ) ⊂ M is the openset chosen so that (3.26) holds. We now choose P j = B G ( R , y ; δ y ) × N j × R + . (cid:3) Proposition 3.1 also states that for each j ∈ N there is an open set O j ⊂⊂ B G (2 R , x ; δ )containing ( ˜ T j , ˜ x j ) such that if h > + g ( N ∗ Γ h ; j ∩ Char g ) ∩ T ∗ O j ⊂ N ∗ ( S j ∩ O j ) (3.38)for some lightlike hypersurface S j ∼ = (˜ s j − δ j , ˜ s j + δ j ) × U j where the diffeomorphism is given by the map π ◦ e + ( · , · ) and U j is an open subset of( N ∗ K ∩ Char g ) / R + containing ( ˜ R j , ˜ y j , − dt + ˜ η ′ j ). Proposition 3.1 also states that O j and S j can be chosen to satisfy the flowout condition (3.5). Without loss of generality wemay assume that U j ⊂⊂ P j / R + where P j ⊂ T ∗ M is the open conic subset constructedin Lemma 3.4.Since x = ˜ x j , we can require that O j satisfies( R × x ) ∩ O j = ∅ . (3.39) We also choose here ˜ U j ⊂⊂ U j ⊂⊂ P j / R + containing ( ˜ R j , ˜ y j , − dt + ˜ η ′ j ) and ˜ δ j < δ j so that ˜ S j := π ◦ e + ((˜ s j − ˜ δ j , ˜ s j + ˜ δ j ) × ˜ U j ) (3.40)is compactly contained in S j ∩ O j . Choose ˜ O j ⊂⊂ O j ⊂⊂ B G (2 R , x ; δ ) open andcontaining ( ˜ T j , ˜ x j ) so that ˜ O j ∩ S j ⊂⊂ ˜ S j (3.41)(see(3.27) for criterion on choice of B G (2 R , x ; δ )).In this setup we have thatFLO + g ( N ∗ Γ h ; j ∩ Char g ) ∩ T ∗ ˜ O j ⊂ N ∗ ˜ S j (3.42) Proof of Proposition 3.3.
Fix j ∈ N large and for l = 0 , , f l ; j ∈ I ( M , ccl( B h ( ξ l ; j ) ∪ B h ( − ξ l ; j )))to be sufficiently smooth and supported in small neighborhoods of ( t l , x l ) so that σ ( f l ; j )( t l , x l , ± ξ l ; j ) = 0 . (3.43)For each j we have that v j solves ✷ g v j = − v j v j v j , v j | t< − = 0where ✷ g v jl = f l ; j , v jl | t< − = 0 . So by Proposition 2.9 v jl ∈ I (cid:0) M , ccl ( B h ( ξ l ; j ) ∪ B h ( − ξ l ; j )) , FLO + g (ccl ( B h ( ξ l ; j ) ∩ B h ( − ξ l ; j )) ∩ Char g ) (cid:1) with σ ( v jl )( t, x, ξ ) = 0 (3.44)for all ( t, x, ξ ) ∈ FLO + g ( ± ξ l ; j ). This means in particular σ ( v jl )( ˜ R j , ˜ x j , ± ˜ η l ; j ) = 0 (3.45)where ( ˜ R j , ˜ x j , ˜ η l ; j ) are as in (3.34).Combining Thm 23.2.9 of [H¨07] and Lemma 2.11, we can deduce from the nonhomo-geneous linear equation for v j thatWF( v j ) ∩ Char g ⊂ [ l =0 WF( v jl ) ∪ FLO + g X l =0 WF( v jl ) ! ∩ Char g ! . If j ∈ N is sufficiently large, h ∈ (0 , h j ) is sufficiently small, and ˜ O j ⊂⊂ B G (2 R , x ; δ ) ischosen to be a sufficiently small neighborhood of ( ˜ T j , ˜ x j ), for any χ j ∈ C ∞ c ( ˜ O j ) we canassert that WF( χ j v j ) ⊂ FLO + g X l =0 WF( v jl ) ! ∩ Char g ! due to (3.27). ETERMINING RIEMANNIAN MANIFOLDS FROM MEASUREMENTS AT A SINGLE POINT 23
For h > π − ( I − g ( B G (2 R , x ; δ )) ∩ X l =0 WF( v jl ) ! ⊂ N ∗ Γ h ; j due to (3.33) and (3.29). Furthermore due to (3.45) we have that σ ( v jl )( ˜ R j , ˜ y j , ± ˜ η l ; j ) = 0where ˜ η l ; j is defined by (3.34). Let ω j ( x, t, ξ ) be a homogenous degree 0 symbol whosesupport is contained in the open set P j constructed in Lemma 3.4. Since ˜ U j ⊂⊂ U j ⊂⊂ P j / R + , we can also ask that 1 − ω j = 0 (3.46)in an open conic set containing the closure of ˜ U j × R + where ˜ U j was defined via (3.40).We have then by Lemma 2.10 ω j ( t, x, D ) (cid:16) v j v j v j (cid:17) ∈ I ( M , N ∗ Γ h ; j ) , σ (cid:16) ω j ( t, x, D ) (cid:16) v j v j v j (cid:17)(cid:17) (cid:16) ˜ R j , ˜ y j , ∓ dt ± ˜ η ′ j (cid:17) = 0 (3.47) where − dt + ˜ η ′ j ∈ T ∗ ( ˜ R j , ˜ y j ) M ∩ Char g is as in (3.36). At the same time˜ U j ∩ WF((1 − ω j ( t, x, D ))( v j v j v j )) = ∅ (3.48)We now write v j = v reg + v sing where ✷ g v sing = − ω j ( t, x, D ) (cid:16) v j v j v j (cid:17) , v sing | t< − = 0and ✷ g v reg = − − ω j ( t, x, D )) (cid:16) v j v j v j (cid:17) , v reg | t< − = 0 . Lemma 3.5.
We have that for any χ j ∈ C ∞ c ( ˜ O j ) , χ j v reg ∈ C ∞ c ( M ) . We assuming Lemma 3.5 for the time being and focus on the microlocal analysis of v sing . Combining the flow condition (3.31) and the wavefront property (3.47) allows usto use Proposition 2.9 to deduce that σ ( v sing )( ˜ T j , ˜ x j , ∓ dt ± ˜ ξ ′ j ) = 0 . Furthermore, WF( v sing ) ∩ Char g ⊂ FLO + g (cid:16) WF (cid:16) ω j ( t, x, D ) (cid:16) v j v j v j (cid:17)(cid:17) ∩ Char g (cid:17) so by(3.47) we can use (3.38) to deduce that WF( v sing ) ∩ T ∗ ˜ O j ⊂ N ∗ ˜ S j where ˜ S j is givenin (3.40) and ˜ O j ⊂⊂ B G (2 R , δ ) is chosen to satisfy (3.41). Therefore if ˜ O j ⊂⊂ B G (2 R , δ ) in (3.41) is chosen small enough, v j | ˜ O j ∈ I ( M , N ∗ ˜ S j ) and σ ( v j )( ˜ T j , ˜ x j , ˜ ξ ) =0 for all ˜ ξ ∈ N ∗ ( ˜ T j , ˜ x j ) ˜ S j . So we have verified (3.25) for ˜ S j satisfying (3.5) (cid:3) It remains to give a
Proof of Lemma 3.5.
Let ( t, x, ± dt + η ′ ) ∈ T ∗ ˜ O j ∩ Char g and assume that it is an elementof WF( χ j v reg ). By Thm 23.2.9 [H¨07]FLO − g (( t, x, ± dt + η ′ )) ∩ WF (cid:16) (1 − ω j ( t, x, D )) (cid:16) v j v j v j (cid:17)(cid:17) = ∅ . (3.49) We have that by Lemma 2.11WF( v j v j v j ) ∩ Char g ⊂ [ l =0 WF( v jl ) ! ∪ X l =0 WF( v jl ) ! ∩ Char g ! . Since WF( v jl ) ∩ T ∗ ˜ O j ⊂ FLO + g (ccl( B h ( ξ l ; j ) ∪ B h ( − ξ l ; j )) ∩ Char g ) , an element in FLO − g (( t, x, ± dt + η ′ )) ∩ WF( v jl ) would violate (3.27) and the fact that˜ O j ⊂⊂ B G (2 R , x ; δ ).So in order for (3.49) to hold, we must have thatFLO − g (( t, x, ± dt + η ′ )) ∩ X l =0 WF( v jl ) ! ∩ supp(1 − ω j ) = ∅ . However, ˜ O j ⊂⊂ B G (2 R , x ; δ ) and I − g ( B G (2 R , x ; δ )) ∩ \ l =0 π (WF( v jl )) = Γ h ; j by (3.29), so we must haveFLO − g (( t, x, ± dt + η ′ )) ∩ N ∗ Γ h ; j ∩ Char g ∩ supp(1 − ω j ( t, x, ξ )) = ∅ . (3.50)This means that ( t, x, ± dt + η ′ ) ∈ FLO + g ( N ∗ Γ h ; j ∩ Char g ) so by the fact that ( t, x, ± dt + η ′ ) ∈ T ∗ ˜ O j , we can apply (3.42) to conclude that ( t, x, ± dt + η ′ ) ∈ N ∗ ˜ S j . By (3.40) wecan conclude that FLO − g (( t, x, ± dt + η ′ )) ∩ N ∗ Γ h ; j ∩ Char g ⊂ ˜ U j . Meanwhile by (3.46)1 − ω j = 0 in ˜ U j . This contradicts (3.50). So we conclude that (3.49) must be false andtherefore WF( χ j v reg ) = ∅ . (cid:3) Fivefold Interaction Producing Singularity along R × We will analyzefive-fold interaction of solutions of (1.1) with source of the form P l =0 ǫ l f l ; j where f l ; j for l = 0 , , x = .Before we construct sources f j and f j we need to analyze further the geometry of N ∗ ˜ S j produced by Proposition 3.3. We denote by T j := R × { ˜ x j } (3.51)and by the fact that ˜ x j = , T j ∩ ( R × { } ) = ∅ .Since ( ˜ T j , ˜ x j , − dt + ˜ ξ ′ j ) ∈ N ∗ ( ˜ S j ∩ ˜ O j ) converges to (2 R , , − dt − ξ ′ ) as j → ∞ and(3.5) holds, there is a unique unit covector ξ ′ ; j ∈ S ∗ ˜ x j M , ξ ′ ; j = ˜ ξ ′ j such that γ g ( d g ( , ˜ x j ) , ξ ′ ; j ) = . (3.52)The elements of N ∗ ( ˜ T j , ˜ x j ) T j belong to Ker( ∂ t ) and N ∗ ( ˜ T j , ˜ x j ) ˜ S j is spanned by a singlelightlike covector. Therefore it is easily checked by a dimension count that ± ( − dt + ξ ′ ; j ) ∈ T ∗ ( ˜ T j , ˜ x j ) M = N ∗ ( ˜ T j , ˜ x j ) T j ⊕ N ∗ ( ˜ T j , ˜ x j ) ˜ S j . (3.53) ETERMINING RIEMANNIAN MANIFOLDS FROM MEASUREMENTS AT A SINGLE POINT 25
Denote by δ T j ∈ D ′ ( M ) the distribution obtained by integrating along T j and for0 < a << h small choose χ a ; j ∈ C ∞ c ( ˜ O j )such that χ a ; j = 1 in a neighborhood of ( ˜ T j , ˜ x j )and supp( χ a ; j ) ⊂⊂ B G ( ˜ T j , ˜ x j ; a ).Clearly WF( χ a ; j δ T j ) ⊂ N ∗ T j is spacelike. Choose N a large positive number anddefine the conormal distributions f j and f j by f j := ✷ g (cid:0) χ a ; j h D i − N δ T j (cid:1) ∈ I ( M , N ∗ T j ) , f j = ✷ g χ a ; j ( t, x ) ∈ C ∞ c ( ˜ O j ) (3.54)Note that while both depend on the parameter a >
0, we suppress its dependence in thenotation until we start manipulating this parameter in later sections. The distribution (cid:0) χ a ; j h D i − N δ T j (cid:1) is clearly conormal and σ (cid:0) χ a ; j h D i − N δ T j (cid:1) ( ˜ T j , ˜ x j , ξ ′ ) = 0 (3.55)for all ξ ′ ∈ N ∗ ( ˜ T j , ˜ x j ) T j ⊂ Ker( ∂ t ). Due to the definition of (3.54), uniqueness of the linearwave equation states that if l = 3 , v jl solves ✷ g v jl = f l ; j , v jl | t< − = 0then v j = χ a ; j ∈ C ∞ c ( ˜ O j ) , v j = χ a ; j h D i − N δ T j ∈ I ( M , N ∗ T j ) . (3.56) Note that this holds for both g = g and g = g since g = g in U by assumption (1.3).Due to (3.55) σ ( v j )( ˜ T j , ˜ x j , ξ ′ ) = 0 (3.57)for all ξ ′ ∈ N ∗ ( ˜ T j , ˜ x j ) T j ⊂ Ker( ∂ t ) Lemma 3.6.
For each j ∈ N one can find ω j ( t, x, ξ ) a homogenous degree zero symbolwhich is identically in small conic neighborhoods of − dt + ξ ′ ; j ∈ T ∗ ( ˜ T j , ˜ x j ) M such that χ a ; j ω j ( t, x, D ) (cid:16) v j v j (cid:17) ∈ I ( M , T ∗ ( ˜ T j , ˜ x j ) M ) . Furthermore, σ ( ω j ( t, x, D )( v j v j ))( ˜ T j , ˜ x j , − dt + ξ ′ ; j ) = 0 and ( ˜ T j , ˜ x j , ± ( − dt + ξ ′ ; j )) / ∈ WF (1 − ω j ( t, x, D )) . Proof.
By (3.25), for a ∈ (0 , h ) sufficiently small, χ a ; j v j ∈ I ( M , N ∗ ˜ S j ) for some lightlikehypersurface ˜ S j satisfying (3.5) and that σ ( v j )( ˜ T j , ˜ x j , − dt + ˜ ξ ′ j ) = 0 (3.58)if − dt + ˜ ξ ′ j ∈ N ∗ ( ˜ T j , ˜ x j ) ˜ S j . By (3.52) the lightlike covector − dt + ξ ′ ; j satisfies( ˜ T j + d g (˜ x j , ) , ) ∈ π ◦ FLO + g ( ˜ T j , ˜ x j , − dt + ξ ′ ; j )so by (3.5), − dt + ξ ′ ; j = − dt + ˜ ξ ′ j . So we can find a homogenous degree zero symbol ω j ( t, x, ξ ) which is identically 1 in aconic neighbourhood containing ( ˜ T j , ˜ x j , − dt + ξ ′ ; j ) but( ˜ T j , ˜ x j , − dt + ˜ ξ ′ j ) / ∈ supp( ω j ) . The set T j defined in (3.51) intersects ˜ S j transversely at the point ( ˜ T j , ˜ x j ) ∈ M andthe conormal bundle N ∗ T j is spacelike. So by (3.56) we can choose ω j ( t, x, ξ ) so thatWF( v j ) ∩ supp( ω j ) = ∅ . We are now in a position to apply Lemma 2.10 to conclude that χ a ; j ω j ( t, x, D )( v j v j ) ∈ I ( M , T ∗ ( ˜ T j , ˜ x j ) M ) . Furthermore, due to (3.53), there is a unique b j > ξ ′ j ∈ S ∗ ˜ x j M such that − dt + ξ ′ ; j = − dt + ˜ ξ ′ j + b j ˆ ξ ′ j ∈ T ∗ ( ˜ T j , ˜ x j ) M . So Lemma 2.10 gives σ ( ω j ( t, x, D )( v j v j )( ˜ T j , ˜ x j , − dt + ξ ′ ; j ) = σ ( v j )( ˜ T j , ˜ x j , − dt + ˜ ξ ′ j ) σ ( v j )( ˜ T j , ˜ x j , b ˆ ξ ′ j ) = 0due to (3.58) and (3.57). (cid:3) The main result of this section is
Proposition 3.7.
Let j ∈ N be large, h j > small. For all h ∈ (0 , h j ) and a << h , let v j be the unique solution of (1.1) with source P l =0 ǫ l f l ; j . Then the distribution of onevariable t v j ( · ; ) has a singularity at t = ˜ T j + d g (˜ x j , ) . Observe that for distinct elements k, l, m, n ∈ { , . . . , } , v jk,l = v jklmn = 0 by Lemma2.14. So direct calculation yields that ✷ g v j = X σ ∈ S v jσ (0) σ (1) σ (2) v jσ (3) v jσ (4) , v j | t< − = 0 (3.59)where S is the symmetric group on the five letters { , . . . , } . Denote by S ⊂ S to bethe subgroup which maps { , , } to itself and define v reg as the unique solution to ✷ g v reg = X σ ∈ S \ S v jσ (0) σ (1) σ (2) v jσ (3) v jσ (4) , v reg | t< − = 0 (3.60) Lemma 3.8.
The solution of (3.60) satisfies ( T j + d g (˜ x j , ) , ) / ∈ singsupp( v reg ) . Proof.
Consider v j kl with k, l ∈ { , , } which solves ✷ g v j kl = − v j v jk v jl := f j kl , v j kl | t< − = 0 . (3.61)Since v j is given by (3.56), we have thatsupp( f j kl ) ⊂ supp( v j ) ⊂ B G ( ˜ T j , ˜ x j ; a ) ⊂⊂ B G (2 R , ; δ ) . ETERMINING RIEMANNIAN MANIFOLDS FROM MEASUREMENTS AT A SINGLE POINT 27
By Proposition 2.9, if k ∈ { , , } , v jk ∈ I ( M , T ∗ ( t k ,x k ) M , FLO + g (ccl ( B h ( ξ k ) ∪ B h ( − ξ k )) ∩ Char g )) . (3.62)So by (3.27), if k ∈ { , , } singsupp( v jk ) ∩ supp( v j ) = ∅ for 0 < a < h all sufficientlysmall. Therefore WF( f j kl ) ⊂ WF( v j ) ⊂ N ∗ T j . So by Thm 23.2.9 [H¨07],WF( v j kl ) ⊂ N ∗ T j , singsupp( v j kl ) ⊂ B G ( ˜ T j , ˜ x j ; a ) (3.63)for k, l = 0 , , < a < h small. Similar argument yields v j kl ∈ C ∞ ( M ) (3.64)for k, l = 0 , , < a < h small. Also, observe that sincesupp( v j ) ⊂ B G ( ˜ T j , ˜ x j ; a ) ⊂⊂ B G (2 R , ; δ )we can conclude that for k ∈ { , , } , v jk v j ∈ C ∞ ( M ) due to (3.62) and (3.27). So v j v j v jk ∈ I ( U , N ∗ T j ) with supp( v j v j v jk ) ⊂ B G ( ˜ T j , ˜ x j ; a ). ThereforeWF( v j l ) ⊂ N ∗ T j , singsupp( v j l ) ⊂ B G ( ˜ T j , ˜ x j ; a ) (3.65)for l = 0 , , < a < h small. Therefore, combine (3.63), (3.64), (3.65), and (3.27),for σ ′ ∈ S \ S WF( v jσ ′ (0) σ ′ (1) σ ′ (2) v jσ ′ (3) v jσ ′ (4) ) ⊂ WF( v j ) ∪ [ σ ∈ S (cid:16) WF( v jσ (0) ) + WF( v jσ (1) ) (cid:17) ∪ [ σ ∈ S WF( v jσ (0) ) . So by Lemma 2.11 and the fact that WF( v j ) is spacelike, we get that for σ ∈ S \ S WF( v jσ (0) σ (1) σ (2) v jσ (3) v jσ (4) ) ∩ Char g ⊂ WF( v j ) ∪ WF( v j ) ∪ WF( v j ) . (3.66)In view of the wavefront property given by (3.62), (3.66) becomesWF( v jσ (0) σ (1) σ (2) v jσ (3) v jσ (4) ) ∩ Char g ⊂ [ k =0 FLO + g (ccl ( B h ( ξ k ) ∪ B h ( − ξ k )) ∩ Char g ) . (3.67)So if ξ ∈ T ∗ ( ˜ T j + d g (˜ x j , ) , ) M ∩ Char g , by the wavefront property (3.67) and flowout property(3.27), we must have that for any σ ∈ S \ S ,FLO − g ( ˜ T j + d g (˜ x j , ) , , ξ ) ∩ WF( v jσ (0) σ (1) σ (2) v jσ (3) v jσ (4) ) = ∅ . By Thm 23.2.9 of [H¨07] this means ( ˜ T j + d g (˜ x j , ) , ) / ∈ singsupp( v reg ). (cid:3) In view of Lemma 3.8 v j = v reg + w where v reg is smooth in some neighborhoodof ( T j + d g (˜ x j , ) , ) and w solves ✷ g w = − v j v j v j , w | t< − = 0 (3.68) Proof of Proposition 3.7.
It suffices to show that w solving (3.68) satisfies T j + d g (˜ x j , ) ∈ singsupp( w ( · , )) . (3.69) Let ω j ( t, x, ξ ) be the symbol constructed in Lemma 3.6. We write w = w reg + w sing where w sing solves ✷ g w sing = v j ω j ( t, x, D ) (cid:16) v j v j (cid:17) , w sing | t< − = 0 (3.70)and w reg solves ✷ g w reg = v j (1 − ω j ( t, x, D )) (cid:16) v j v j (cid:17) , w reg | t< − = 0 . (3.71)Inserting v j and v j from (3.56) into the right side of (3.70) we get ✷ g w sing = f source := χ a ; j ω j ( D ) (cid:16) v j (cid:0) χ a ; j h D i − N δ T j (cid:1)(cid:17) w sing | t< − = 0 (3.72)By Lemma 3.6 f source has wavefront set contained in the cotangent space of the point( ˜ T j , ˜ x j ) and that σ ( f source )( ˜ T j , ˜ x j , ∓ dt ± ξ ′ ; j ) = 0 . By (3.52), the element ( ˜ T j + d g (˜ x j , ) , , ∓ dt ± ˙ γ g ( d g (˜ x j , ) , ξ ′ ; j ) ♭ ) belongs to the futureflowout of ( ˜ T j , ˜ x j , ∓ dt ± ξ ′ ; j ). Therefore by Proposition 2.9 w sing ∈ I (cid:16) M , T ∗ ( ˜ T j , ˜ x j ) M , FLO + g (cid:16) T ∗ ( ˜ T j , ˜ x j ) M ∩ Char g (cid:17)(cid:17) and σ ( w sing )( ˜ T j + d g (˜ x j , ) , , ∓ dt ± ˙ γ g ( d g (˜ x j , ) , ξ ′ ; j ) ♭ ) = 0 . (3.73)Since for j ∈ N large ( ˜ T j , ˜ x j ) is close to ( ˜ T j + d g (˜ x j , ) , ), we have that there existsan open set ˆ O j containing ( ˜ T j + d g (˜ x j , ) , ) such thatFLO + g (cid:16) T ∗ ( ˜ T j , ˜ x j ) M ∩ Char g (cid:17) ∩ T ∗ ˆ O j = N ∗ ( ˆ S j ∩ ˆ O j )for some lightlike hypersurface ˆ S j . Therefore w sing | ˆ O j ∈ I ( M , N ∗ ˆ S j ). Due to (3.73) thehypothesis of Lemma 2.12 are satisfied. Therefore we can conclude that ˜ T j + d g (˜ x j , ) ∈ singsupp( w sing ( · , )).To complete the proof we need to verify that ( ˜ T j + d g (˜ x j , ) , ) / ∈ singsupp( w reg ). Thesource term in (3.71) is supported in B G ( ˜ T j , ˜ x j ; a ) so if there is an element ξ ∈ WF( w reg ) ∩ T ∗ ( ˜ T j + d g (˜ x j , ) , ) M (3.74)then by Thm 23.2.9 of [H¨07]FLO − g (( ˜ T j + d g (˜ x j , ) , , ξ )) ∩ WF( v j (1 − ω j ( t, x, D )) (cid:16) v j v j (cid:17) ) = ∅ . (3.75)By (3.56) we have that WF (cid:16) v j (1 − ω j ( t, x, D )) (cid:16) v j v j (cid:17)(cid:17) ⊂ (cid:16) WF( v j ) ∪ WF( v j ) ∪ (cid:16) WF( v j ) + WF( v j ) (cid:17)(cid:17) ∩ WF(1 − ω j ( t, x, D )) and because v j ∈ C ∞ c ( B G ( ˜ T j , ˜ x j ; a ))WF (cid:16) v j (1 − ω j ( t, x, D )) (cid:16) v j v j (cid:17)(cid:17) ⊂ T ∗ B G ( ˜ T j , ˜ x j ; a ) . ETERMINING RIEMANNIAN MANIFOLDS FROM MEASUREMENTS AT A SINGLE POINT 29
Combine this with the fact that WF( v j ) is spacelike, (3.75) becomes T ∗ B G ( ˜ T j , ˜ x j ; a ) ∩ FLO − g (( ˜ T j + d g (˜ x j , ) , , ξ )) ∩ (cid:16) WF( v j ) ∪ (cid:16) WF( v j ) + WF( v j ) (cid:17)(cid:17) ∩ WF(1 − ω j ( t, x, D )) = ∅ (3.76) Within B G ( ˜ T j , ˜ x j ; a ) ⊂⊂ ˜ O j , WF (cid:16) v j | B G ( ˜ T j , ˜ x j ; a ) (cid:17) ⊂ N ∗ ˜ S j and π ◦ FLO + g ( N ∗ ˜ S j ) ∩ ([2 R , R + δ / × )) = ∅ as stated in Proposition 3.3. Recall that ˜ T j > R is a sequence converging to 2 R .Therefore, for j ∈ N sufficiently large that ˜ T j + d g (˜ x j , ) ∈ [2 R , R + δ / − g (( ˜ T j + d g (˜ x j , ) , , ξ )) ∩ WF( v j ) ∩ T ∗ B G ( ˜ T j , ˜ x j ; a ) = ∅ . Therefore, (3.77) becomesFLO − g (( ˜ T j + d g (˜ x j , ) , , ξ )) ∩ (cid:16) WF( v j ) + WF( v j ) (cid:17) ∩ WF(1 − ω j ) = ∅ . (3.77)By Proposition 3.3, inside B G ( ˜ T j , ˜ x j ; a ) the singular support of v j is contained in ˜ S j .Since the singular support of v j is contained in B G ( ˜ T j , ˜ x j ; a ) ∩ T j , we have that (cid:16) WF( v j ) + WF( v j ) (cid:17) ∩ T ∗ B G (( ˜ T j , ˜ x j ) , a ) ⊂ T ∗ ( ˜ T j , ˜ x j ) M . Since ˜ x j is within the injectivity radius of , the only lightlike vectors in T ∗ ( ˜ T j , ˜ x j ) M whosefuture flowout intersects T ∗ ( ˜ T j + d g (˜ x j , ) , ) M are ± ( − dt + ξ ′ ; j ) (see (3.52) for definition of ξ ′ ; j ). However, ω j as constructed by Lemma 3.6 satisfies( ˜ T j , ˜ x j , ± ( − dt + ˜ ξ ′ ; j )) / ∈ WF(1 − ω j ( t, x, D )) . These combined to contradict (3.77). Therefore (3.74) is false andWF( w reg ) ∩ T ∗ ( ˜ T j + d g (˜ x j , ) , ) M = ∅ . (cid:3) Recovering Geometric Data
Throughout this section we assume that L M ,g , T +1 = L M ,g , T +1 .4.1. Geodesics Returning to U . In this subsection we rule out geodesics originatingfrom U hitting after exiting U . For any r ∈ (0 , δ ], let ∂ + S ∗ B ( , r ), ∂ S ∗ B ( , r ),and ∂ − S ∗ B ( , r ) denote the outward, tangential, and inward pointing unit covectorrespectively. We shall prove that Proposition 4.1.
For each r ∈ (0 , δ / , ξ ′ ∈ ∂ + S ∗ B ( , r ) ∪ ∂ S ∗ B ( , r ) , and R ∈ (0 , T ) , we have / ∈ γ g ([0 , R ] , ξ ′ ) if and only if / ∈ γ g ([0 , R ] , ξ ′ ) .Proof. Suppose by contradiction that there exists a ξ ′ ∈ ∂ + S ∗ B ( , r ) ∪ ∂ S ∗ B ( , r ) basedat a point x ∈ ∂B ( , r ) such that γ g ( T ′ , ξ ′ ) = for some T ′ ∈ [0 , R ], but / ∈ γ g ([0 , R ] , ξ ′ ) . (4.1)Choose a generic small t > x := γ g ( t , ξ ′ ) = γ g ( t , ξ ′ ) ∈ U and that the end points of γ g ([ t , T ′ ] , ξ ′ ) and γ g ([ t , T ′ ] , ξ ′ ) are not conjugate to eachother. Define ξ := − dt + ˙ γ g ( t , ξ ′ ) ♭ ∈ T ∗ ( t ,x ) U ∩ Char g , and observe that, for all h > O ⊂ R × U of ( T ′ , ) ∈ R × U such thatFLO + g (ccl( B h ( ξ ) ∪ B h ( − ξ )) ∩ Char g ) ∩ T ∗ O = N ∗ S, ( T ′ , ) ∈ S (4.2)for some lightlike hypersurface S . However, due to (4.1), for h > T ′ , ) / ∈ π ◦ FLO + g (ccl( B h ( ξ ) ∪ B h ( − ξ )) ∩ Char g ) (4.3)Let f ∈ I ( U , ccl( B h ( ξ ) ∪ B h ( − ξ ))) be as constructed in (2.10) with σ ( f )( t , x , ξ ) = 0.Now let u and v be solutions of the non-linear wave equations ✷ g u + u = ✷ g v + v = ǫf,u = v = 0 , on t < − . The distributions u ′ = ∂ ǫ u | ǫ =0 and v ′ = ∂ ǫ u | ǫ =0 solves the linear wave equations ✷ g u = ✷ g v = f. By (4.3) and Thm 23.2.9 of [H¨07],( T ′ , ) / ∈ singsupp( v ′ ) . (4.4)However, due to Proposition 2.9 and (4.2), u ′ | O ∈ I ( O , N ∗ S ) with σ ( u ′ )( T ′ , , ξ ) = 0 for ξ ∈ N ∗ ( T ′ , ) S . So Lemma 2.12 asserts that T ′ ∈ singsupp( u ′ ( · , )) for some T ′ < T .This fact combined with (4.4) contradicts L M ,g , T +1 = L M ,g , T +1 . (cid:3) Geodesics Starting From the Origin.
The goal of this section is to prove thefollowing
Proposition 4.2.
Let R ∈ (0 , T ) and ξ ′ ∈ S ∗ U . The geodesic segment γ g ([0 , R ]; ξ ′ ) ⊂ M is a minimizer between its end points if and only if the geodesic segment γ g ([0 , R ]; ξ ′ ) ⊂ M is a minimizer between its end points. Clearly it suffices to prove one direction of the “if and only if” implication. We willdo so by contradiction. To this end we construct a set of geodesics in case Proposition4.2 fails:
Lemma 4.3.
If Proposition 4.2 fails to hold then one can find ξ ′ long , ξ ′ ∈ S ∗ U and R long , R ∈ R with < R < R long such that (1) The segment γ g ([0 , R long ] , ξ ′ long ) is the unique minimizer between end points anddoes not contain conjugate points. (2) The segments γ g ([0 , R long ] , ξ ′ long ) and γ g ([0 , R ] , ξ ′ ) meet at and only at the endpoints. In addition, ˙ γ g ( R long , ξ ′ long ) = − ˙ γ g ( R , ξ ′ ) . (3) The segment γ g ([0 , R ] , ξ ′ ) is the unique minimizer between the end points anddoes not contain conjugate points. (4) There is a t > R long − R such that γ g ( t , ξ ′ long ) ∈ U and γ g ([ t , R long ] , ξ ′ long ) is the unique minimizer between the end points and does not contain conjugatepoints. ETERMINING RIEMANNIAN MANIFOLDS FROM MEASUREMENTS AT A SINGLE POINT 31
Proof.
Let ξ ′ c ∈ S ∗ U be a covector such that γ g ([0 , R ] , ξ ′ c ) is minimizing but γ g ([0 , R ] , ξ ′ c )is not minimizing. By taking R slightly smaller we may assume without loss of generalitythat γ g ([0 , R ] , ξ ′ c ) is the unique minimizer and does not contain conjugate points. Notethat this implies / ∈ γ g ((0 , R ] , ξ ′ c ) (4.5)Set R c := sup { t > | γ g ([0 , t ] , ξ ′ c ) ⊂ M is a minimizing segment } < R. (4.6)Let t > γ g ([0 , t ] , ξ ′ c ) ⊂ U . Definition (4.6) implies that γ g ([ t , R c ]) is the unique minimizing segment between end points and that it does notcontain conjugate points. This means that for any R ′ c ∈ ( R c , R ) sufficiently close to R c , γ g ([ t , R ′ c ] , ξ ′ c ) is a geodesic segment not containing conjugate points and is the uniqueminimizing segment between end points. We can also arrange that the end points of γ g ([0 , R ′ c ] , ξ ′ c ) are not conjugate to each other (though the geodesic segment is not min-imizing).For any R ′ c ∈ ( R c , R ), the segment γ g ([0 , R ′ c ] , ξ ′ c ) is not minimizing. So for any R ′ c ∈ ( R c , R ) there is a unit covector ξ ′ ∈ S ∗ M and R ′ < R ′ c such that γ g ([0 , R ′ ] , ξ ′ )is a minimizing geodesic between and γ g ( R ′ c , ξ ′ c ). We have the freedom to chooseany R ′ c ∈ ( R c , R ) so we choose it sufficiently close to R c so that the end points of γ g ([0 , R ′ c ] , ξ ′ c ) are not conjugate and R ′ c − R ′ < t / . (4.7)We may also assume that the only intersection of these two segments are at the endpoints. Furthermore, due to (4.5) and Proposition 4.1,˙ γ g ( R ′ c , ξ ′ c ) = − ˙ γ g ( R ′ , ξ ′ ) . (4.8)By the fact that the end points of γ g ([0 , R ′ c ] , ξ ′ c ) are not conjugate to each other, thereis a sequence ξ ′ j ∈ S ∗ U converging to ξ ′ c and R j converging to R ′ c such that such that γ g ( R j , ξ ′ j ) = γ g ( R ′ − /j, ξ ′ ). For any j ∈ N sufficiently large set R long := R j and ξ ′ long := ξ ′ j . Due to (4.8), we can infer that ˙ γ g ( R long , ξ ′ j ) = − ˙ γ g ( R ′ − /j, ξ ′ ) for j ∈ N large enough. So condition (2) is met. The geodesic segment γ g ([0 , R ′ − /j ] , ξ ′ ) is theunique minimizer and does not contain conjugate point since it can be extended slightlyand still be a minimizer. So condition (3) is met by setting R := R ′ − /j . From (4.7),if j ∈ N is chosen large enough R long − R < t . (4.9)Recall that the geodesic segment γ g ([ t , R ′ c ] , ξ ′ c ) does not contain conjugate pointsand is the unique minimizing segment between end points. So by Lemma 2.1, if j ∈ N is chosen sufficiently large γ g ([ t , R long ] , ξ ′ long ) is still the unique minimizer between itsend points and does not contain conjugate points. So condition (4) is met. For j ∈ N chosen large enough Lemma 2.1 asserts that γ g ([0 , R ] , ξ ′ long ) is still the unique minimizerbetween end points and does not contain conjugate points. Since R long < R if j ∈ N is sufficiently large, γ g ([0 , R long ] , ξ ′ long ) is still the unique minimizer between end pointsand does not contain conjugate points. So condition (1) is met. (cid:3) Observe that as a consequence of Condition (3) of Lemma 4.3 we have that / ∈ γ g ((0 , R ] , ξ ′ ). Using Proposition 4.1 we have that / ∈ γ g ((0 , R ] , ξ ′ ) . (4.10)Set y := γ g ( R long , ξ ′ long ) = γ g ( R , ξ ′ ). Furthermore set x = , x := γ g ( t , ξ ′ long ),and ξ ′ := ˙ γ g ( t , ξ ′ long ) ♭ ∈ S ∗ x U . Note that γ g ([0 , R long ] , ξ ′ long ) = γ g ([0 , t ] , ξ ′ long ) ∪ γ g ([0 , R ] , ξ ′ ) , where R := R long − t . (4.11)Condition (4) means that the segment γ g ([0 , R ] , ξ ′ ) is the unique minimizer and doesnot contain conjugate points.By assumption (1.3), g = g in U . Therefore the segment γ g ([0 , R long ] , ξ ′ long ) can bewritten as γ g ([0 , R long ] , ξ ′ long ) = γ g ([0 , t ] , ξ ′ long ) ∪ γ g ([0 , R ] , ξ ′ )= γ g ([0 , t ] , ξ ′ long ) ∪ γ g ([0 , R ] , ξ ′ ) . (4.12)Set η ′ := − ˙ γ g ( R , ξ ′ ) ♭ and η ′ := − ˙ γ g ( R , ξ ′ ) ♭ . By condition (2) of Lemma 4.3, η ′ = − η ′ . For any η ′ ∈ S ∗ y M ∩ span { η ′ , η ′ }\{ η ′ } sufficiently close to η ′ , Lemma 2.1guarantees γ g ([0 , R ] , η ′ ) is the unique minimizing geodesic between the end points anddoes not contain conjugate points. We set x := γ g ( R , η ′ ) and ξ ′ := − ˙ γ g ( R , η ′ ) ♭ .Observe that with this setup, if η ′ is chosen sufficiently close to η ′ ˙ γ g ( R l , ξ ′ l ) = − ˙ γ g ( R k , ξ ′ k ) , k, l ∈ { , , } (4.13)For η ′ chosen sufficiently close to η ′ the segments γ g ([0 , R ] , ξ ′ ) and γ g ([0 , R ] , ξ ′ )are close to each other. Condition (3) of Lemma 4.3 implies / ∈ γ g ((0 , R ] , ξ ′ ). So / ∈ γ g ([0 , R ] , ξ ′ ). Proposition 4.1 then concludes that / ∈ γ g ([0 , R ] , ξ ′ ) . (4.14)The above conditions are satisfied for any η ′ ∈ span { η ′ , η ′ } chosen sufficiently close to η ′ . We now use Lemma 2.7 to give η ′ ∈ span { η ′ , η ′ } the following additional linearalgebra property:Dim (cid:0) span {− dt − η ′ , − dt − η ′ , − dt − η ′ } (cid:1) = 3 , Dim(span { η ′ , η ′ , η ′ } ) = 2 (4.15)Set t and t to be 0 < t = t := R long − R < t (4.16)and observe that ( t , x ) / ∈ I + g ( t , )by the strict inequality t = d g ( x , ) > t − t . And since x is chosen sufficiently closeto , ( t , x ) / ∈ I + g ( t , ) ∪ I + g ( t , x ) . (4.17)Define ξ l ∈ T ∗ ( t l ,x l ) U to be ξ l := − dt + ξ ′ l . In this setting( t + R , y ) ∈ π ◦ FLO + g ( t , , ξ ) ∩ π ◦ FLO + g ( t , x , ξ ) ∩ π ◦ FLO + g ( t , x , ξ ) . (4.18) ETERMINING RIEMANNIAN MANIFOLDS FROM MEASUREMENTS AT A SINGLE POINT 33
However, since γ g ([0 , R long ] , ξ ′ long ) is the unique length minimizing geodesic in M , wehave the following Lemma 4.4. I − g ( t + 2 R , ) ∩ π ◦ FLO + g ( t , , ξ ) ∩ π ◦ FLO + g ( t , x , ξ ) = ∅ . (4.19) Proof.
Note that γ g ([0 , R ] , ξ ′ ) is contained in γ g ([0 , R long ] , ξ ′ long ) by (4.12). So it suf-fices to show that I − g ( t + 2 R , ) ∩ π ◦ FLO + g ( t , , ξ ) ∩ π ◦ FLO + g (0 , , ξ long ) = ∅ (4.20)where ξ long := − dt + ξ ′ long ∈ T ∗ (0 , ) U . We first observe that for t ≤ R long , γ g ( t, ξ ′ long ) ∈ ∂B g ( , t ) since the geodesic is uniquely minimizing up to t = R long by condition (1)of Lemma 4.3. Meanwhile γ g ( t, ξ ′ ) ∈ B g ( , t ). So by writing lightlike geodesics usingexpression (2.3), we see that since t > ( { t ≤ R long } × M ) ∩ π ◦ FLO + g ( t , , ξ ) ∩ π ◦ FLO + g (0 , , ξ long ) = ∅ . So if there is an element in the LHS of (4.20), it must be in I − g ( t + 2 R , ) ∩ (( R long , ∞ ) × M ) = { ( t + 2 R − t, x ) | d g ( , x ) ≤ t, t ∈ (0 , R ) } . (4.21) However, since γ g ([0 , R long ] , ξ ′ long ) is the unique minimizer between end points, γ g ( R long + s, ξ ′ long ) / ∈ B g ( , R long − s ) (4.22)for all s > R long = R + t so for FLO + g (0 , , ξ long ) to intersect { ( t + 2 R − t, x ) | d g ( , x ) ≤ t, t ∈ (0 , R ) } , there must be an s > γ g ( s + R long , ξ ′ long ) ∈ B g ( , R − s ) . This contradicts (4.22) since R < R long . Combine this with the expression (4.21) wecan conclude that I − g ( t + 2 R , ) ∩ (( R long , ∞ ) × M ) ∩ FLO + g (0 , , ξ long ) = ∅ . So (4.20) holds. (cid:3)
We also have
Lemma 4.5. ( t + 2 R , ) / ∈ π ◦ FLO + g (( t , x , ξ )) Proof.
Condition (1) states that γ g ([0 , R long ] , ξ ′ long ) is the unique minimizing segmentbetween and the end point. So we can conclude that / ∈ γ g ((0 , R long ] , ξ ′ long ). By(4.12) we have that / ∈ γ g ([0 , R + R long ] , ξ ′ ) . (4.23)Note that by definition (4.16) and (4.11) R long = t + R = t + R (4.24) Writing explicitly π ◦ FLO + g (( t , x , ξ )) = { ( t + s, γ g ( s, ξ ′ )) | s > } we see that if ( t + 2 R , ) ∈ π ◦ FLO + g (( t , x , ξ )) (4.25)then = γ g ( t + 2 R − t , ξ ′ ) (4.26)By (4.24), t + 2 R − t = R long + R − t < R + R long . So (4.26) would mean that ∈ γ g ([0 , R + R long ] , ξ ′ ), contradicting (4.23). So (4.25)must be false. (cid:3) With these facts established we are now ready to proceed with
Proof of Proposition 4.2.
Suppose the contrary and let ξ l ∈ T ∗ ( t l ,x l ) U be defined as above.Due to (4.19) we can choose h > I − g ( t + 2 R + δ, ) ∩ π ◦ FLO + g ( B h ( ξ ) ∩ Char g ) ∩ π ◦ FLO + g ( B h ( ξ ) ∩ Char g ) = ∅ (4.27)for all δ > h > B G ( t + 2 R , ; δ ) ∩ [ l =0 π ◦ FLO + g ( B h ( ξ l ) ∩ Char g ) ! = ∅ (4.28) for all δ > M . Furthermore due to (4.15), (3.19) and (3.20) hold on M . The condition (3.17)is satisfied by (4.13). In the current setting, (4.18) states that the three g -lightlikegeodesics meet at ( R + t , y ) which is analogous to (3.22) except that the time hasshifted by t .So we are allowed to evoke Proposition 3.3. This means, for each l = 0 , ,
2, thereexists a sequence { ξ l ; j } ∞ j =1 ⊂ T ∗ ( t l ,x l ) U converging to ξ l so that if { f l ; j } j are sequences of( h -dependent) distributions of the form (2.10) with ξ l ; j ∈ WF( f l ; j ) ⊂ B h ( ξ l ; j ) ⊂ B h ( ξ l ) , σ ( f l ; j )( t l , x l , ± ξ l ; j ) = 0and w j are solutions of (1.1) in M with source f j := P l =0 ǫ l f l ; j then each singsupp( w j )contains a point ( ˜ T j , ˜ x j ) with ˜ T j > R + t and ˜ x j = converging to (2 R + t , ) as j → ∞ . Furthermore for each j ∈ N , if h > O j containing ( ˜ T j , ˜ x j ) with ˜ O j ∩ ( R × { } ) = ∅ such that for some lightlike hypersurfaces˜ S j ⊂ M , w j | ˜ O j ∈ I ( M , N ∗ ˜ S j )and σ ( w j )( ˜ T j , ˜ x j , ˜ ξ j ) = 0for all ˜ ξ j ∈ N ∗ ( ˜ T j , ˜ x j ) ˜ S j . ETERMINING RIEMANNIAN MANIFOLDS FROM MEASUREMENTS AT A SINGLE POINT 35
We now choose f j and f j as in (3.54). Recall that these two distributions depend ona parameter a > a << h . Let v j solve (1.1) in M with source f j := P l =0 f l ; j which depends now on the parameters 0 < a << h << h j where h j > j ∈ N . Then according to Proposition 3.7, v j ( · , ) has a singularity at˜ T j + d g (˜ x j , ) for all 0 < a << h << h j . By Lemma 2.16,˜ T j + d g (˜ x j , ) ∈ singsupp (cid:0) ∂ ǫ ...ǫ (cid:0) v j ( · , ) (cid:1) | ǫ = ··· = ǫ =0 (cid:1) . Let u j solve (1.1) in M and source f j = P l =0 f l ; j . By assumption u j ( · ; ) = v j ( · ; )so ˜ T j + d g (˜ x j , ) ∈ singsupp (cid:0) ∂ ǫ ...ǫ (cid:0) u j ( · , ) (cid:1) | ǫ = ··· = ǫ =0 (cid:1) and by Lemma 2.16, ˜ T j + d g (˜ x j , ) ∈ singsupp( u j ( · , )) (4.29)for all 0 < a << h << h j for some h j depending on j ∈ N . The distribution u jl solves ✷ g u jl = f l ; j , u jl | t< − = 0 . Proposition 2.9 states that u jl ∈ I ( M , T ∗ ( t l ,x l ) M , FLO + g (ccl( B h ( ξ l ) ∪ B h ( − ξ l )) ∩ Char g )) (4.30)for l = 0 , ,
2. So due to (4.27) and the fact that ( ˜ T j , ˜ x j ) → (2 R + t , ), for j ∈ N largeenough,singsupp( u j ) ∩ singsupp( u j ) ∩ singsupp( u j ) ∩ I − g (( ˜ T j + d g (˜ x j , ) , )) = ∅ . (4.31)Note the u j solves ✷ g u j = − u j u j u j , u j | t< − = 0 . As only lightlike wavefronts propagate, Lemma 2.11 combined with (4.31) forces singsupp( u j ) ∩ I − g (( ˜ T j + d g (˜ x j , ) , )) ⊂ [ l =0 singsupp( u jl ) ! ∩ I − g (( ˜ T j + d g (˜ x j , ) , )) . (4.32) Combining the above inclusion with (4.28) and (4.27) we have that there exists δ > u j ) ∩ I − g (( ˜ T j + d g (˜ x j , ) , )) ∩ B G (2 R + t , δ ) = ∅ (4.33)for all j ∈ N sufficiently large.By (3.56) we have u j = (cid:0) χ a ; j h D i − N δ T j (cid:1) (4.34)and u j = χ a ; j ( t, x ) . (4.35)We recall for convenience of reader that the definitions of χ a ; j and δ T j are given in (3.54).Note that if 0 < a << h ,supp( χ a ; j ) ⊂ B G (˜ x j , ˜ T j ; a ) ⊂⊂ B G (2 R + t , ; δ ) where ˜ x j → and ˜ T j → t + 2 R . Due to (4.33) we have that for all j ∈ N sufficientlylarge and a > u j u j u j ) ⊂ WF (cid:16) χ a ; j ( t, x ) h D i − k ( χ a ; j δ T j ) (cid:17) ⊂ N ∗ T j . (4.36)Recall that T j = R × { ˜ x j } where ˜ x j = .The distribution u j solves the equation ✷ g u j = − X σ ∈ S u jσ (0) u jσ (1) u jσ (2) σ (3) σ (4) , = − u j u j u j − X σ ∈ S \ S u jσ (0) u jσ (1) u jσ (2) σ (3) σ (4) with initial condition u j | t< − = 0. Recall that we denote by S ⊂ S to be thesubgroup which maps the set { , , } to itself. We have the following Lemma 4.6.
For j ∈ N sufficiently large, let u j reg solve ✷ g u j reg = − X σ ∈ S \ S u jσ (0) u jσ (1) u jσ (2) σ (3) σ (4) with initial condition u j reg | t< − = 0 . Then ( ˜ T j + d g ( , ˜ x j ) , ) / ∈ singsupp( u j reg ) . This lemma implies that ( ˜ T j + d g ( , ˜ x j ) , ) / ∈ singsupp( u j ) since the wavefront ofthe term u j u j u j is given by (4.36) and therefore does not propagate. This contradicts(4.29). The proof is complete. (cid:3) We still need to give a proof 4.6. However, it is essentially the same argument as proofof Lemma 3.8. So we only give a sketch:
Proof of Lemma 4.6.
Use the fact that supp( u j ) and supp( u j ) are contained in B G ( ˜ T j , ˜ x j ; a ) ⊂⊂ B G ( t + 2 R , ; δ )by (4.34) and (4.35). Note that for l = 0 , , u jl ) ⊂ π ◦ FLO + g (ccl B h ( ξ l ) ∩ Char g )by (4.30). So (4.28) implies thatsingsupp( u jl ) ∩ (cid:16) supp( u j ) ∪ supp( u j ) (cid:17) = ∅ . With these facts about how singular supports are disjoint, we proceed as in the proof ofLemma 3.8. (cid:3)
Determining the Distance Function.Proposition 4.7.
Let R < T and suppose for ξ ′ ∈ S ∗ U the geodesic γ g ([0 , R ] , ξ ′ ) isa minimizer between end points and y := γ g ( R , ξ ′ ) ∈ M . Set y := γ g ( R , ξ ′ ) ∈ M . (4.37) Then d g ( x, y ) ≤ d g ( x, y ) (4.38) ETERMINING RIEMANNIAN MANIFOLDS FROM MEASUREMENTS AT A SINGLE POINT 37 for all x ∈ U . Note that thanks to Proposition 4.2, the statement of Proposition 4.7 is actuallysymmetric with the subscript 1 and 2 interchanged. Therefore, it has the following
Corollary 4.8.
For every y ∈ B g ( ; T ) there exists a unique y ∈ B g ( , T ) such that d g ( x, y ) = d g ( x, y ) for all x ∈ U . The statement also holds with and interchanged.Proof. Let ξ ′ ∈ S ∗ U and γ g ([0 , R ] , ξ ′ ) be a minimizing geodesic segment with end points and y . Set y := γ g ( R, ξ ′ ) and by Proposition 4.2, γ g ([0 , R ] , ξ ′ ) is a minimizingsegment between and y . By Proposition 4.7, d g ( x, y ) ≤ d g ( x, y ) for all x ∈ U .Similarly, switching the role of y and y yields d g ( x, y ) ≤ d g ( x, y ) for all x ∈ U .For uniqueness, suppose y , ˜ y ∈ M satisfies d g ( y , x ) = d g (˜ y , x ) for all x ∈ U .Setting R := d g ( y , ) = d g (˜ y , ), we have y = γ g ( R , ξ ′ ), ˜ y = γ g ( R , ˜ ξ ′ ) for some ξ ′ , ˜ ξ ′ ∈ S ∗ M . For all t ∈ (0 , δ ) we then have d g ( γ g ( t, ξ ′ ) , y ) = d g ( γ g ( t, ξ ′ ) , ˜ y ) . We then have, using t = d g ( γ g ( t, ξ ′ ) , ), R = t + d g ( γ g ( t, ξ ′ ) , y ) = t + d g ( γ g ( t, ξ ′ ) , ˜ y )= d g ( , γ g ( t, ξ ′ )) + d g ( γ g ( t, ξ ′ ) , ˜ y ) ≥ d g ( , ˜ y ) = R . So we have, for all t ∈ (0 , δ ), d g ( , γ g ( t, ξ ′ )) + d g ( γ g ( t, ξ ′ ) , ˜ y ) = d g ( , ˜ y ) . This means that a minimizing segment joining and ˜ y contains the entire segment γ g ((0 , δ ) , ξ ′ ), and therefore ˜ y = γ g ( R , ξ ′ ) = y . (cid:3) We now proceed to prove Proposition 4.7. We first make some reductions using thecontinuity of the distance function. By Proposition 4.2, both γ g ([0 , R ] , ξ ′ ) ⊂ M and γ g ([0 , R ] , ξ ′ ) ⊂ M are minimizing segments. Taking R ∈ (0 , T ) slightly smaller wemay assume without loss of generality that both γ g ([0 , R ] , ξ ′ ) , γ g ([0 , R ] , ξ ′ ) are the unique minimizers and have no conjugate points. (4.39) Another observation is that with y ∈ M and y ∈ M fixed, it suffices to prove (4.38)for a dense subset of x ∈ U . As such we only need to prove (4.38) for x ∈ U which isjoined to y ∈ M and y ∈ M by unique minimizing geodesics in M and M which donot contain conjugate points. Furthermore we assume that x / ∈ γ g ([ − δ , δ ] , ξ ′ ) = γ g ([ − δ , δ ] , ξ ′ ) (4.40)To this end let x ∈ U satisfy the above assumptions. Let γ g ([0 , R ] , ξ ′ ) be the uniqueminimizing segment between x and y in M that does not contain conjugate points.The distance d g ( x , y ) is given by R = d g ( x , y ) . (4.41)Again, by density, we may without loss of generality assume that˙ γ g ( R , ξ ′ ) = − ˙ γ g ( R , ξ ′ ) . (4.42)By the fact that γ g ([0 , R ] , ξ ′ ) is the unique minimizer between end points, (4.42) implies / ∈ γ g ((0 , R ] , ξ ′ ) ∪ γ g ([0 , R + R ] , ξ ′ ) . (4.43) Note that since both geodesic segments γ g ([0 , R ] , ξ ′ ) and γ g ([0 , R ] , ξ ′ ) are uniqueminimizers, the two geodesic segments only intersect at the point y .We set η ′ l := − ˙ γ g ( R l , ξ ′ l ) ♭ for l = 0 , η ′ ∈ S ∗ y M ∩ span { η ′ , η ′ } arbitrarily close to η ′ so thatDim (cid:0) span {− dt − η ′ , − dt − η ′ , − dt − η ′ } (cid:1) = 3 , Dim(span { η ′ , η ′ , η ′ } ) = 2 . (4.44)Set R := R . By Lemma 2.1 if η ′ is chosen close to η ′ , we can conclude that γ g ([0 , R ] , η ′ ) is a unique minimizing segment containing no conjugate points. Set ξ ′ := − ˙ γ g ( R , η ′ ) ♭ so that y = γ g ( R , ξ ′ ) = γ g ( R , ξ ′ ) = γ g ( R , ξ ′ ) . If η ′ is chosen sufficiently close to η ′ , condition (4.42) allows us to assert that˙ γ g ( R l , ξ ′ ) = − ˙ γ g ( R k , ξ ′ ) (4.45)for l, k ∈ { , , } .Due to (4.43), if η ′ is chosen sufficiently close to η ′ we have / ∈ γ g ((0 , R ] , ξ ′ ) ∪ γ g ([0 , R + R ] , ξ ′ ) ∪ γ g ([0 , R ] , ξ ′ ) . (4.46)By Proposition 4.1 the same holds for the g geodesics: / ∈ γ g ((0 , R ] , ξ ′ ) ∪ γ g ([0 , R + R ] , ξ ′ ) ∪ γ g ([0 , R ] , ξ ′ ) . (4.47)Set t = t = 0 and t = R − R so that t l + R l = R for all l = 0 , ,
2. Note that by(4.41) this means d g ( x , y ) = R − t . (4.48)Also, by the strict triangle inequality, d g ( x , x ) > | t | and d g ( x , x ) > | t | . So( t l , x l ) / ∈ I + g ( t k , x k ) , k = l. (4.49)Define the lightlike covectors ξ l = − dt + ξ ′ l ∈ T ∗ ( t l ,x l ) M (4.50)for l = 0 , , R , y ) ∈ \ l =0 π ◦ FLO + g ( t l , x l , ξ l ) (4.51)By (4.47) there exists an h > δ > B G (2 R , ; δ ) ∩ [ l =0 π ◦ FLO + g (ccl B h ( ξ l ) ∩ Char g ) ! = ∅ (4.52)Furthermore we can choose the δ > ETERMINING RIEMANNIAN MANIFOLDS FROM MEASUREMENTS AT A SINGLE POINT 39
Lemma 4.9. i)There exists a δ > and h > such that if ξ ∈ B h ( ξ ) ∩ Char g thenthe lightlike geodesic segment I − g ( B G (2 R , ; δ )) ∩ π ◦ FLO + g (0 , , ξ ) is the unique lightlike geodesic segment joining any two points on it.ii) I − g (2 R , ) ∩ π ◦ FLO + g (0 , , ξ ) ⊂ [0 , R ] × M .Proof. i) By assumption γ g ([0 , R ]; ξ ′ ) is the unique minimizer between end points anddoes not have conjugate points. By Lemma 2.1 there exists an open set U ′ ⊂ S ∗ U and δ ′ > ξ ′ ∈ U ′ then γ g ([0 , R + δ ′ ] , ξ ′ ) is the unique minimizer between endpoints without conjugate points. A consequence of this is that d g (cid:0) γ g ( R + δ ′ / t, ξ ′ ) , (cid:1) > R + δ ′ / − t (4.53)for all t ≥ ξ ′ ∈ U ′ . Furthermore for all ξ ′ ∈ U ′ , the lightlike geodesic segment { π ◦ e + ( s, , , − dt + ξ ′ ) | s ∈ [0 , R + δ ′ ] } is the unique lightlike geodesic joining any two points on it. Due to (4.53), for ξ ′ ∈ U ′ I − g (2 R + δ ′ / , ) ∩ FLO + g (0 , , − dt + ξ ′ ) ⊂ { ( t, x ) ∈ M | t ∈ (0 , R + δ ′ / } . So if we choose h > B h ( ξ ) ∩ Char g ) / R + ⊂ {− dt + ξ ′ | ξ ′ ∈ U ′ } , then for any ξ ∈ B h ( ξ ) ∩ Char g , the geodesic segment I − g (2 R + δ ′ / , ) ∩ π ◦ FLO + g (0 , , ξ )is the only lightlike geodesic segment joining any two points on it. Therefore, the lemmais verified if δ > B G (2 R , ; δ ) ⊂⊂ I − g (2 R + δ ′ / , ).ii) This can be seen by taking ξ ′ = ξ ′ in (4.53) and observing that I − g (2 R , ) = { (2 R − t, x ) | d g ( x, ) ≤ t, t ≥ } . (cid:3) The conditions (4.44), (4.51), (4.49),(4.45) allows us to evoke Proposition 3.7 to con-clude
Lemma 4.10.
We have that (1)
For each l = 0 , , there exists a sequence ξ l ; j ∈ T ∗ ( t l ,x l ) M ∩ Char g convergingto ξ l . (2) For each l = 0 , , and j ∈ N large there is an h j > , so that if h ∈ (0 , h j ) thenthere exists sources f l ; j ( · ; h ) of the form (2.10) such that ξ l ; j ∈ WF( f l ; j ( · ; h )) ⊂ ccl( B h ( ξ l ; j ) ∪ B h ( − ξ l ; j )) , σ ( f l ; j ( · ; h ))( t l , x l , ξ l ; j ) = 0 . (3) A sequence ( ˜ T j , ˜ x j ) → (2 R , ) , ˜ T j > R , and ˜ x j = . (4) For any j ∈ N large and h ∈ (0 , h j ) as above, and any a ∈ (0 , h ) sufficiently small,we have sources f j ( · ; a ) and f j ( · ; a ) of the form (3.54) which are supported in B G ( ˜ T j , ˜ x j ; a ) . If ( t, x ) v j ( t, x ; h, a ) are solutions of (1.1) in M with source f j ( · ; h, a ) := X l =0 ǫ l f l ; j ( · ; h ) + X l =3 ǫ l f l ; j ( · ; a ) then for each j ∈ N large ˜ T j + d g (˜ x j , ) ∈ singsupp( v j ( · , ; h, a )) . (4.54) for all < a < h < h j small. Note that we now explicitly write out the dependence of the sources f l ; j and solution v j on the parameters 0 < a < h < h j .Let u j ( · ; h, a ) solve (1.1) with metric g and the same source as v j ( · ; h, a ). The fivefoldinteraction of the nonlinear wave u j ( · ; h, a ) is ✷ g u j ( · ; h, a ) = X σ ∈ S u jσ (0) σ (1) σ (2) ( · ; h, a ) u jσ (3) ( · ; h, a ) u jσ (4) ( · ; h, a ) (4.55)We write the solution u j ( · ; h, a ) of (4.55) as u j reg + u j sing ( · ; h, a ) where ✷ g u j reg = X σ ∈ S \ S u jσ (0) σ (1) σ (2) ( · ; h, a ) u jσ (3) ( · ; h, a ) u jσ (4) ( · ; h, a ) (4.56)and ✷ g u j sing ( · ; h, a ) = 6 u j ( · ; h, a ) u j ( · ; h, a ) u j ( · ; h, a ) . (4.57)Here we denote S ⊂ S as elements of the permutation group which maps the set ofthree letters { , , } to itself. Repeating the argument of Lemma 3.8 we get that Lemma 4.11. If j ∈ N is chosen large enough and a ∈ (0 , h ) is chosen small enough sothat B G ( ˜ T j , ˜ x j ; a ) ⊂⊂ B G (2 R , ; δ ) and B G ( ˜ T j , ˜ x j ; a ) ∩ ( R × { } ) = ∅ , then ( ˜ T j + d g (˜ x j , ) , ˜ x j ) / ∈ singsupp( u reg ) . Proof.
This is similar to proof of Lemma 3.8 so we will only give a brief sketch. Tosimplify notation we define f source := X σ ∈ S \ S u jσ (0) σ (1) σ (2) ( · ; h, a ) u jσ (3) ( · ; h, a ) u jσ (4) ( · ; h, a ) . Note that by (3.56) u j = χ a ; j ∈ C ∞ c ( B G ( ˜ T j , ˜ x j ; a )) , u j = χ a ; j h D i − N δ T j ∈ I ( M , N ∗ T j ) . Meanwhile by Proposition 2.9, for l = 0 , , u jl ∈ I ( M , T ∗ ( t l ,x l ) M , FLO + g (ccl( B h ( ξ l ) ∪ B h ( − ξ l )) ∩ Char g )This combined with flowout condition (4.52) ensures thatsingsupp( u jl ) ∩ (cid:16) supp( u j ) ∪ supp( u j ) (cid:17) = ∅ . Using these facts we can proceed as in Lemma3.8 to conclude that FLO − g (2 R , , ξ ) ∩ WF( f source ) = ∅ for all ξ ∈ T ∗ (2 R , ) U ∩ Char g . The lemma then follows from Thm 23.2.9of [H¨07]. ETERMINING RIEMANNIAN MANIFOLDS FROM MEASUREMENTS AT A SINGLE POINT 41 (cid:3)
With these facts we can now give the
Proof of Proposition 4.7.
From (4.54) we can use Lemma 2.16 to deduce˜ T j + d g (˜ x j , ) ∈ singsupp (cid:0) ∂ ǫ ...ǫ (cid:0) v j ( · , ; h, a ) (cid:1) | ǫ = ··· = ǫ =0 (cid:1) . By the fact that the source-to-solution maps agree, t u j ( t, ; h, a ) is the same as t v j ( t, ; h, a ). So we have that˜ T j + d g (˜ x j , ) ∈ singsupp (cid:0) ∂ ǫ ...ǫ (cid:0) u j ( · , ; h, a ) (cid:1) | ǫ = ··· = ǫ =0 (cid:1) . Lemma 2.16 then implies ˜ T j + d g (˜ x j , ) ∈ singsupp( u j ( · , ; h, a )). By Lemma 4.11we have that ( ˜ T j + d g (˜ x j , ) , ) ∈ singsupp( u j sing ( · ; h, a )) . (4.58)Note that for all a > u j sing ( · ; h, a ) solves (4.57) and the sourceterm of (4.57) is supported in B G ( ˜ T j , ˜ x j ; a ) since supp( u j ( · ; a )) ⊂ B G ( ˜ T j , ˜ x j ; a ) by (3.56).Note that by Condition (3) of Lemma 4.10, x j = . The solution u j ( · ; a ) is smooth andthe wavefront set of u j ( · ; a ) is spacelike. So in order for (4.58) to hold, B G (( ˜ T j , ˜ x j ) , a ) ∩ singsupp( u j ( · ; h )) = ∅ for all a >
0. We remark that u j ( · ; h ) does not depend on the parameter a > f l ; j ( · ; h ) depends only on the parameter h > l = 0 , ,
2. So taking intersection overall a >
0, we conclude that, for each fixed j ∈ N large, if h ∈ (0 , h j ) is sufficiently small,( ˜ T j , ˜ x j ) ∈ singsupp( u j ( · ; h )) . (4.59)Fix j ∈ N large and for each h > u j ( · ; h ) solves ✷ g u j ( · ; h ) = − u j ( · ; h ) u j ( · ; h ) u j ( · ; h ) , u j ( · ; h ) | t< − = 0with each of u jl ( · ; h ) solving linear inhomogeneous wave equation with source f l ; j ( · ; h ) sothat singsupp( u jl ( · ; h )) ⊂ π ◦ FLO + g (WF( f l ; j ( · ; h )) ∩ Char g ) by Proposition 2.9.Take j ∈ N large enough so that ( ˜ T j , ˜ x j ) ∈ B G ((2 R , ) , δ ) where δ > I − g (( ˜ T j , ˜ x j )) ∩ \ l =0 π ◦ FLO + g (WF( f l ; j ( · ; h )) ∩ Char g ) = ∅ (4.60)for all h ∈ (0 , h j ).By Condition (2) of Lemma 4.10, WF( f l ; j ( · ; h )) ⊂ ccl B h ( ± ξ l ; j ) for l = 0 , , I − g (( ˜ T j , ˜ x j )) ∩ \ l =0 π ◦ FLO + g (ccl B h ( ξ l ; j ) ∩ Char g ) = ∅ (4.61) for all h >
0. Furthermore we must have that ∀ h ∈ (0 , h j ) , ∃ η h ∈ T ∗ ( t h ,x h ) M / R + ∩ Char g , ( t h , x h ) ∈ \ l =0 π ◦ FLO + g (ccl B h ( ξ l ; j ) ∩ Char g )(4.62)s . t . ( ˜ T j , ˜ x j ) ∈ π ◦ FLO + g ( t h , x h , η h ) for otherwise ( ˜ T j , ˜ x j ) / ∈ singsupp( u j ( · ; h )) by Thm 23.2.9 of [H¨07].Since this holds for all h >
0, compactness then dictates that there exists ( ˆ R j , ˆ y j )such that ( ˆ R j , ˆ y j ) ∈ I − g (( ˜ T j , ˜ x j )) ∩ \ l =0 π ◦ FLO + g ( t l , x l , ξ l ; j ) . (4.63)By Lemma 4.9, ( ˆ R j , ˆ y j ) is unique element of the triple intersection in the g -pastcausal cone of ( ˜ T j , ˜ x j ). Therefore we have that( ˆ R j , ˆ y j ) = I − g (( ˜ T j , ˜ x j )) ∩ \ l =0 π ◦ FLO + g ( t l , x l , ξ l ; j ) . (4.64)And by (4.62), there is a covector ˆ η j ∈ T ∗ ( ˆ R j , ˆ y j ) M / R + ∩ Char g such that( ˜ T j , ˜ x j ) ∈ π ◦ FLO + g ( ˆ R j , ˆ x j , ˆ η j ) . (4.65)Taking a converging subsequence as j → ∞ in (4.64) and (4.65) we have, by Conditions(1) and (3) of Lemma 4.10,( ˆ R , ˆ y ) = I − g (2 R , ) ∩ \ l =0 FLO + g ( t l , x l , ξ l ) , (4.66)(2 R , ) ∈ π ◦ FLO + g ( ˆ R , ˆ y , ˆ η ) (4.67)for some ( ˆ R , ˆ y ) ∈ M and ˆ η ∈ T ∗ ( ˆ R , ˆ y ) M ∩ Char g . Since ξ ∈ T ∗ (0 , ) M ∩ Char g (see(4.50)), this says that ( ˆ R , ˆ y ) ∈ FLO + g (0 , , ξ ) (4.68)which in turn implies that ˆ y = γ g ( ˆ R , ξ ′ ) . (4.69)Evoke part ii) of Lemma 4.9 combined with (4.68) we concludeˆ R ≤ R . (4.70)By (4.39), the geodesic segment γ g ([0 , R ] , ξ ′ ) is the unique minimizer between and y . So (4.70) and (4.69) combined to give d g (ˆ y , y ) = R − ˆ R . (4.71)By (4.50), ξ ∈ T ∗ ( t ,x ) M ∩ Char g . So since ( ˆ R , ˆ y ) ∈ FLO + g ( t , x , ξ ) by (4.66),we can conclude that ˆ y = γ g ( ˆ R − t , ξ ′ ) , x = γ g (0 , ξ ′ ) . (4.72) ETERMINING RIEMANNIAN MANIFOLDS FROM MEASUREMENTS AT A SINGLE POINT 43
Combining (4.72) and (4.71) we get d g ( x , y ) ≤ d g ( x , ˆ y ) + d g (ˆ y , y ) ≤ ( ˆ R − t ) + ( R − ˆ R ) = R − t . So by (4.48) we get that d g ( x , y ) ≤ d g ( x , y ) . (cid:3) Recover the Metric From Geometric Data.
We now complete the proof ofTheorem 1.1 with the following proposition. Let ( M i , g i ), i = 1 ,
2, be complete Rie-mannian manifolds. For some x i ∈ M i and R >
0, we define V i := (cid:8) ξ ′ ∈ T x i M i (cid:12)(cid:12) d g i ( x i , exp x i ( ξ ′ )) = k ξ ′ k g i < T (cid:9) . Notice that V i is precisely the set of those tangent vectors ξ ′ ∈ T x i M i whose associatedgeodesic γ ( · , ξ ′ ) : [0 , → M i , γ ( t, ξ ′ ) = exp x i ( tξ ′ ) is length-minimizing. Proposition 4.12.
Assume that, for some open neighborhoods U i ⊂ B g i ( x i , T ) of x i ,there exists an isometry ψ : ( U , g ) → ( U , g ) such that (i) dψ ( x ) V = V , (ii) d g ( y, exp x ( ξ ′ )) = d g ( ψ ( y ) , exp x ( dψ ( x ) ξ ′ )) for all y ∈ U and ξ ′ ∈ V .Then, there is an extension of ψ to an isometry ψ : ( B g ( x , T ) , g ) → ( B g ( x , T ) , g ) . Proof.
Since the Riemannian manifolds ( M i , g i ) are complete, every point of B g i ( x i , T )can be connected to x i by means of a length-minimizing geodesic. Namely,exp x i ( V i ) = B g i ( x i , T ) . We claim that there exists a continuous bijection φ : B g ( x , T ) → B g ( x , T ) such that φ (exp x ( ξ ′ )) = exp x ( dψ ( x ) ξ ′ ) , ∀ ξ ′ ∈ V Indeed, since the isometry ψ maps geodesics to geodesics, such a φ is clearly well definedon a Riemannian ball B g ( x , ǫ ) ⊂ U , and indeed φ | B g ( x ,ǫ ) = ψ | B g ( x ,ǫ ) . Assume thatthere exist two distinct vectors ξ ′ , η ′ ∈ V such that y := exp x ( ξ ′ ) = exp x ( η ′ ). All weneed to show in order to have a well defined continuous bijection φ is thatexp x ( dψ ( x ) ξ ′ ) = exp x ( dψ ( x ) η ′ ) . (4.73)We set z := exp x ( δη ′ ) ∈ U for some δ > z := exp x ( dψ ( x ) δη ′ ), and y := exp x ( dψ ( x ) ξ ′ ). We have d g ( x , y ) = k ξ ′ k g = d g ( x , y ) = d g ( x , z ) + d g ( z , y ) = d g ( x , z ) + d g ( z , y ) , where the last equality follows by assumption (ii). This implies that z lies on a length-minimizing geodesic segment joining x and y , and therefore (4.73) holds.We now prove that φ is a diffeomorphism. Fix an arbitrary point y ∈ B g ( x , T ), andconsider the unit-speed length-minimizing geodesic γ : [0 , ℓ ] → M joining x = γ (0)and y = γ ( ℓ ). We fix δ > z := γ ( δ ) ∈ B g ( x , ǫ ). Notice that γ | [ δ,ℓ ] is the unique length-minimizing geodesic segment joining z and y , and does notcontain conjugate points. Analogously, γ := φ ◦ γ | [ δ,ℓ ] is the unique length-minimizinggeodesic segment joining z := φ ( z ) and y := φ ( y ), and does not have conjugate points. Therefore, the Riemannian distances d g i are smooth in an open neighborhood Z i × Y i of ( z i , y i ). We take Z i to be small enough so that Z ⊂ B g ( x , ǫ ) and φ ( Z ) = Z . Foreach w ∈ Z i , the derivative ∂ y i d g i ( w, y i ) ∈ S ∗ y i M is precisely the g i -dual of the tangentvector to the unique length-minimizing geodesic segment joining w and y i . Therefore,for a generic triple of points w , w , w ∈ Z sufficiently close to z , the triple ∂ y d g ( w , y ) , ∂ y d g ( w , y ) , ∂ y d g ( w , y )is a basis of T ∗ y M , and the triple ∂ y d g ( φ ( w ) , y ) , ∂ y d g ( φ ( w ) , y ) , ∂ y d g ( φ ( w ) , y )is a basis of T ∗ y M . This implies that, up to shrinking the open neighborhoods Y i of y i ,the maps κ : Y → R , κ ( y ) = ( d g ( w , y ) , d g ( w , y ) , d g ( w , y )) ,κ : Y → R , κ ( y ) = ( d g ( φ ( w ) , y ) , d g ( φ ( w ) , y ) , d g ( φ ( w ) , y ))are diffeomorphisms onto their images. By assumption (ii), we have κ = κ ◦ φ | Y , andtherefore φ | Y = κ − ◦ κ . This proves that φ is a local diffeomorphism at y . Moreover,for all z ∈ Z , by differentiating the equality d g ( z, y ) = d g ( φ ( z ) , φ ( y )) with respectof y , we obtain ∂ y d g ( z, y ) | {z } ∈ S ∗ y M = ∂ y d g ( φ ( z ) , y ) dφ ( z ) = dφ ( z ) ∗ ∂ y d g ( φ ( z ) , y ) | {z } ∈ S ∗ y M . Since this holds for all z in the open set Z , we obtain non-empty open sets W i ⊂ S ∗ y i M i such that dφ ( y ) ∗ W = W . This implies dφ ( y ) ∗ S ∗ y M = S ∗ y M , and therefore( φ ∗ g ) | y = g | y . Since y is arbitrary, we conclude that φ is a Riemannian isometry asclaimed. (cid:3) References [CMOP19] X. Chen, Lassas M., L. Oksanen, and G. P. Paternain,
Detection of hermitian connectionsin wave equations with cubic non-linearity , arXiv:1902.05711, 02 2019.[dHUW19] Maarten de Hoop, Gunther Uhlmann, and Yiran Wang,
Nonlinear responses from the inter-action of two progressing waves at an interface , Ann. Inst. H. Poincar´e Anal. Non Lin´eaire (2019), no. 2, 347–363. MR 3913189[dHUW20] , Nonlinear interaction of waves in elastodynamics and an inverse problem , Math.Ann. (2020), no. 1-2, 765–795. MR 4055177[Dui96] J. J. Duistermaat,
Fourier integral operators , Progress in Mathematics, vol. 130, Birkh¨auserBoston, Inc., Boston, MA, 1996. MR 1362544[FO19] Ali Feizmohammadi and Lauri Oksanen,
Recovery of zeroth order coefficients in non-linearwave equations .[GU93] Allan Greenleaf and Gunther Uhlmann,
Recovering singularities of a potential from singu-larities of scattering data , Comm. Math. Phys. (1993), no. 3, 549–572. MR 1243710[H¨07] Lars H¨ormander,
The analysis of linear partial differential operators. III , Classics in Math-ematics, Springer, Berlin, 2007, Pseudo-differential operators, Reprint of the 1994 edition.MR 2304165[KLOU14] Yaroslav Kurylev, Matti Lassas, Lauri Oksanen, and Gunther Uhlmann,
Inverse problem foreinstein-scalar field equations .[KLU13] Yaroslav Kurylev, Matti Lassas, and Gunther Uhlmann,
Determination of structures in thespace-time from local measurements: a detailed exposition . ETERMINING RIEMANNIAN MANIFOLDS FROM MEASUREMENTS AT A SINGLE POINT 45 [KLU14] ,
Inverse problems in spacetime i: Inverse problems for einstein equations - extendedpreprint version .[KLU18] Yaroslav Kurylev, Matti Lassas, and Gunther Uhlmann,
Inverse problems for Lorentzianmanifolds and non-linear hyperbolic equations , Invent. Math. (2018), no. 3, 781–857.MR 3802298[LUW17] Matti Lassas, Gunther Uhlmann, and Yiran Wang,
Determination of vacuum space-timesfrom the einstein-maxwell equations .[LUW18] Matti Lassas, Gunther Uhlmann, and Yiran Wang,
Inverse problems for semilinear waveequations on Lorentzian manifolds , Comm. Math. Phys. (2018), no. 2, 555–609.MR 3800791[MU79] R. B. Melrose and G. A. Uhlmann,
Lagrangian intersection and the Cauchy problem , Comm.Pure Appl. Math. (1979), no. 4, 483–519. MR 528633[OSSU20] Lauri Oksanen, Mikko Salo, Plamen Stefanov, and Gunther Uhlmann, Inverse problems forreal principal type operators .[UW18] Gunther Uhlmann and Yiran Wang,
Convolutional neural networks in phase space and in-verse problems .[UW20] Gunther Uhlmann and Yiran Wang,
Determination of space-time structures from gravita-tional perturbations , Comm. Pure Appl. Math. (2020), no. 6, 1315–1367. MR 4156604[UZ19] Gunther Uhlmann and Jian Zhai, On an inverse boundary value problem for a nonlinearelastic wave equation . Leo TzouSchool of Mathematics and Statistics, University of Sydney, NSW 2006, Australia
Email address ::