Reconstruction along a geodesic from sphere data in Finsler geometry and anisotropic elasticity
RRECONSTRUCTION ALONG A GEODESIC FROM SPHERE DATA INFINSLER GEOMETRY AND ANISOTROPIC ELASTICITY
MAARTEN V. DE HOOP, JOONAS ILMAVIRTA, AND MATTI LASSASA
BSTRACT . Dix formulated the inverse problem of recovering an elastic bodyfrom the measurements of wave fronts of point scatterers. We geometrize thisproblem in the framework of linear elasticity, leading to the geometrical inverseproblem of recovering a Finsler manifold from certain sphere data in a given opensubset of the manifold. We solve this problem locally along any geodesic throughthe measurement set.
1. I
NTRODUCTION
Consider the following problem in local or global seismology: There are pointsources with known onset times within the earth and we can detect the wave frontsin a small measurement device. Without any assumptions isotropy, can we use thissphere data to find the material parameters (the stiffness tensor field) within theplanet if densely many point sources are available?The propagation of the fastest wave fronts or singularities to the elastic waveequation follows the geodesic flow of a Finsler geometry. This Finsler geometry isnot Euclidean or even Riemannian unless we make strong symmetry assumptionson the stiffness tensor. A geometric version of the physical problem was solvedin [4] in Riemannian geometry, but it is only applicable in an isotropic or ellipticallyanisotropic situation. Without assuming any kind of isotropy, we are forced to studya similar geometric problem on Finsler manifolds.The earth or a subset thereof is modelled as a smooth Finsler manifold ( 𝑀, 𝐹 ) and the measurement device is an open subset 𝑈 ⊂ 𝑀 . For a point 𝑥 ∈ 𝑀 anda radius 𝑟 > a sphere on the manifold is the image of the sphere on 𝑇 𝑥 𝑀 un-der the exponential map. The sphere may intersect itself and fail to be smooth incomplicated ways, as we make no assumptions on conjugate points. These spheresrepresent the wave fronts of the fastest elastic waves under a mild assumption; seesection 1.4. The data consists of smooth subsets of these spheres without any knowl-edge of whether two sphere segments originate from the same source.The only thing known about the sphere segments is their radius which corre-sponds to the travel time. This travel time and the origin time can be assumed tobe known if the point sources are artificially produced by sending waves from theset 𝑈 and measuring the arrivals of scattered wave fronts from point scatterers.We prove in theorem 5 that this data determines the curvature operator and theJacobi fields backwards along any geodesic hitting 𝑈 . The data consists of spheres, Date : February 23, 2021.Simons Chair in Computational and Applied Mathematics and Earth Science, Rice University,Houston TX, USA. [email protected] .Unit of Computing Sciences, Tampere University, Finland. [email protected] .Department of Mathematics and Statistics, University of Helsinki, Finland. [email protected] . a r X i v : . [ m a t h . DG ] F e b ECONSTRUCTION ALONG A GEODESIC FROM SPHERE DATA 2 so we may use the associated normal coordinates to describe 𝑀 outside 𝑈 . In thesecoordinates the local Riemannian metric is determined by the sphere data as we shoin theorem 6.1.1. Earlier results.
It was shown in [4] that in Riemannian geometry the spheredata (similar to what was considered originally by Dix [3], a prominent seismolo-gist of the 1950s) determines the universal cover of the manifold. Our main goaland contribution is to make this result more applicable by pushing it into the realof Finsler geometry. A crucial component in the proof of both the older Riemann-ian and the present Finslerian result is setting up a closed ODE system along thegeodesic starting normal to the spheres in the data. We do this in lemma 11.The ODE problems are equivalent in the two geometries; the local Riemannianmetric along the reference geodesic makes the Finslerian quantities behave muchlike their Riemannian counterparts. The ODE system in [4] was set up for the firstderivatives few derivatives of the inverse shape operator 𝐾 ( 𝑟, 𝑡 ) . The version of ourlemma 11 is set up with Jacobi fields and their covariant derivatives instead. TheJacobi fields are an item of more direct interest and all curvature quantities can becomputed directly from them. The Jacobi fields written in a parallel orthonormalframe encode the components of the local Riemannian metric or the fundamentaltensor as we shall see in the proof of theorem 6.Our version also suggests a computational algorithm that is expected to behavemore stably than that extracted from [4]. Unlike the earlier version, our ODE systemdoes not suffer from conjugate points at all, and the system can be solved in one goalong all of the relevant geodesic. In this sense we present an improvement andsimplification — in addition to generalization — of the results of [4].Inverse problems in elasticity have recently been posed and solved in the frame-work of Finsler geometry. The determination of a Finsler manifold with suitableproperties has been shown from boundary distance data [5] and broken scatteringdata [6].The geometric point of view and Finsler geometry corresponding to the propa-gation of elastic waves is only beginning to be studied. Nevertheless, inverse prob-lems for anisotropic elasticity have received substantial attention using different ap-proaches in the past. Of these we mention problems of identifying inclusions [12] orcracks [11], tomography for residual stresses [16], and various problems where thestiffness tensor field itself is to be reconstructed [8, 9, 13, 14]. A different geomet-ric approach, using tools of metric rather than smooth geometry, was recently usedto approximately reconstruct a manifold from a finite number of seismic sources atunknown times [7].1.2. The geometric setting.
This subsection sets up the geometric preliminariesneeded to present our results.1.2.1.
Finsler geometry.
A Finsler manifold is a differentiable manifold 𝑀 witha function 𝐹 𝑥 ∶ 𝑇 𝑥 𝑀 → [0 , ∞) for each 𝑥 . This function need not be symmetric,but otherwise it satisfies the assumptions of a norm. Combining the functions onseparate fibers gives rise to the Finsler function 𝐹 ∶ 𝑇 𝑀 → [0 , ∞) , which is con-tinuous on 𝑇 𝑀 and smooth enough on
𝑇 𝑀 ⧵ . Any differentiability assumptionsare only assumed to hold on the punctured tangent bundle 𝑇 𝑀 ⧵ .The Finsler function 𝐹 could be additionally assumed to be symmetric on eachtangent space (reversible), but we do not need to make this assumption. Finsler ECONSTRUCTION ALONG A GEODESIC FROM SPHERE DATA 3 functions arising from elasticity are reversible. It follows from reversibility that thedistance function is symmetric and the reverse of a geodesic is a geodesic. We donot assume reversibility, so these properties do not generally hold.For any 𝑥 ∈ 𝑀 and 𝑣 ∈ 𝑇 𝑥 𝑀 ⧵ we define in local coordinates the matrix 𝑔 𝑖𝑗 ( 𝑥, 𝑣 ) = 12 𝜕 𝑣 𝑖 𝜕 𝑣 𝑗 𝐹 𝑥 ||| 𝑣 . (1)Observe that if 𝐹 𝑥 is given by the square root of a positive definite quadratic form(a Riemannian metric on 𝑇 𝑥 𝑀 ), then 𝑔 ( 𝑥, 𝑣 ) is independent of 𝑣 . In fact, 𝑔 ( 𝑥, 𝑣 ) isindependent of 𝑣 if and only if the metric is Riemannian.We call this 𝑔 ( 𝑥, 𝑣 ) or 𝑔 𝑣 ( 𝑥 ) the local Riemannian metric , and it is also knownas the fundamental tensor. Once we have a preferred direction 𝑣 ∈ 𝑇 𝑥 𝑀 ⧵ , wehave a metric tensor 𝑔 ( 𝑥, 𝑣 ) which gives a natural way to linearly identify 𝑇 𝑥 𝑀 and 𝑇 ∗ 𝑥 𝑀 and give an inner product on both spaces. In this paper we work alonggeodesics, and the preferred direction is the tangent of the geodesic. This directiondependence is what sets Finsler geometry apart from Riemannian geometry.The length of any curve is defined using the Riemannian metric associated withits tangent direction. This gives rise to a distance function 𝑑 𝐹 ∶ 𝑀 × 𝑀 → [0 , ∞) ,and length is minimized locally by geodesics like in Riemannian geometry. If 𝐹 isnot reversible, then the length 𝑑 𝐹 ( 𝑥, 𝑦 ) of the shortest curve from 𝑥 ∈ 𝑀 to 𝑦 ∈ 𝑀 is in general different from 𝑑 𝐹 ( 𝑦, 𝑥 ) .We say that the Finsler function 𝐹 ∶ 𝑇 𝑀 → [0 , ∞) is fiberwise analytic if foreach 𝑥 ∈ 𝑀 the restriction 𝐹 𝑥 ∶ 𝑇 𝑥 𝑀 → [0 , ∞) is real-analytic on 𝑇 𝑥 𝑀 ⧵ . Wedo not need analytic structure on the manifold for this definition; each fiber is afinite dimensional vector space and linear isomorphisms preserve analyticity onsuch spaces.Many concepts familiar from Riemannian geometry have their Finsler counter-parts. We use, in particular, geodesics, Jacobi fields, shape operators, surface nor-mal coordinates, and exponential maps. For more precise definitions and furtherinsights into Finsler geometry, we refer to [17, 1].An important difference between Finsler and Riemannian geometry, besides thedirectional dependence of the local Riemannian metric mentioned above, is thatsome symmetry and regularity is lost. The Finsler distance function is not symmet-ric, and the exponential map typically fails to be smooth at the origin of a tangentspace.1.2.2. Operators along a geodesic.
Let ( 𝑀, 𝐹 ) be a Finsler manifold and 𝛾 ∶ 𝐼 → 𝑀 a unit speed geodesic defined on an open interval 𝐼 ⊂ ℝ . We assume for con-venience that 𝐼 .The directional curvature operator 𝑅 ̇𝛾 ( 𝑡 ) along 𝛾 is a linear operator on 𝑇 𝛾 ( 𝑡 ) 𝑀 .It is also known as the Riemann curvature in the direction ̇𝛾 [17] and we we willdenote later by 𝑅 ( 𝑡 ) for simplicity. In Riemannian geometry it can be defined by 𝑅 ̇𝛾 ( 𝑉 ) = 𝑅 ( 𝑉 , ̇𝛾 ) ̇𝛾, (2)where 𝑅 is the Riemann curvature tensor. In Finsler geometry, the matrix elementsof 𝑅 𝑣 for 𝑣 ∈ 𝑇 𝑥 𝑀 ⧵ are given by [17, (6.4)] ( 𝑅 𝑣 ( 𝑥, 𝑣 )) 𝑖𝑘 = 2 𝜕𝐺 𝑖 𝜕𝑥 𝑘 − 𝑣 𝑗 𝜕 𝐺 𝑖 𝜕𝑥 𝑗 𝜕𝑦 𝑘 + 2 𝐺 𝑗 𝜕𝐺 𝑖 𝜕𝑦 𝑗 𝜕𝑦 𝑘 − 𝜕𝐺 𝑖 𝜕𝑦 𝑗 𝜕𝐺 𝑖 𝜕𝑦 𝑘 , (3) ECONSTRUCTION ALONG A GEODESIC FROM SPHERE DATA 4 where (cf. [17, (5.2) and (5.7)]) 𝐺 𝑖 ( 𝑥, 𝑣 ) = 14 𝑔 𝑖𝑙 ( 𝑥, 𝑣 ) ( 𝜕 𝐹 𝜕𝑥 𝑘 𝜕𝑣 𝑙 ( 𝑥, 𝑣 ) 𝑣 𝑘 − 𝜕𝐹 𝜕𝑥 𝑙 ( 𝑥, 𝑣 ) ) = 14 𝑔 𝑖𝑙 ( 𝑥, 𝑣 ) ( 𝜕𝑔 𝑗𝑙 𝜕𝑥 𝑘 ( 𝑥, 𝑣 ) − 𝜕𝑔 𝑗𝑘 𝜕𝑥 𝑙 ( 𝑥, 𝑣 ) ) 𝑣 𝑗 𝑣 𝑘 (4)are the geodesic coefficients. A Finsler metric is called Berwald if the geodesic co-efficients are given by 𝐺 𝑖 ( 𝑣 ) = Γ 𝑖𝑗𝑘 ( 𝑥 ) 𝑣 𝑗 𝑣 𝑘 for some local functions Γ 𝑖𝑗𝑘 ( 𝑥 ) . Rie-mannian metrics are Berwald and the local functions in question are the Christoffelsymbols.For any 𝑡 ∈ 𝐼 , we denote by 𝑁 𝑡 = { 𝑣 ∈ 𝑇 𝛾 ( 𝑡 ) 𝑀 ; 𝑔 ̇𝛾 ( ̇𝛾, 𝑣 ) = 0} (5)the set of vectors normal to ̇𝛾 ( 𝑡 ) in the sense of the inner product 𝑔 ̇𝛾 ( 𝑡 ) on 𝑇 𝛾 ( 𝑡 ) 𝑀 .Since 𝑅 ̇𝛾 ( 𝑡 ) is self-adjoint with respect to 𝑔 ̇𝛾 and 𝑅 ̇𝛾 ( 𝑡 ) ̇𝛾 ( 𝑡 ) = 0 (see e.g. [2, Section6.1]), we have 𝑅 ̇𝛾 ( 𝑡 ) ( 𝑁 𝑡 ) ⊂ 𝑁 𝑡 . Therefore we may consider the curvature operatoras a map 𝑅 ( 𝑡 ) ∶ 𝑁 𝑡 → 𝑁 𝑡 (6)for all 𝑡 ∈ 𝐼 .A Jacobi field along 𝛾 is a vector field 𝐽 along 𝛾 satisfying the Jacobi equation 𝐷 𝑡 𝐽 ( 𝑡 ) + 𝑅 ( 𝑡 ) 𝐽 ( 𝑡 ) = 0 , (7)where 𝐷 𝑡 is the covariant derivative along 𝛾 . As in Riemannian geometry, Jacobifields correspond to geodesic variations. The basic properties of Jacobi fields arethe same as in Riemannian geometry, including the way they split in parallel andnormal components. We will only study normal Jacobi fields, and they can becharacterized as those Jacobi fields 𝐽 for which 𝐽 ( 𝑡 ) ∈ 𝑁 𝑡 for all 𝑡 ∈ 𝐼 . Then also 𝐷 𝑡 𝐽 ( 𝑡 ) ∈ 𝑁 𝑡 for all 𝑡 ∈ 𝐼 .We can thus define the solution operator to the Jacobi equation as the map 𝑈 ( 𝑡, 𝑠 ) ∶ 𝑁 𝑡 → 𝑁 𝑠 (8)for which 𝑈 ( 𝑡, 𝑠 )( 𝑉 , 𝑊 ) = ( 𝐽 ( 𝑡 ) , 𝐷 𝑡 𝐽 ( 𝑡 )) for the Jacobi field 𝐽 with initial condi-tions 𝐽 ( 𝑠 ) = 𝑉 and 𝐷 𝑡 𝐽 ( 𝑡 ) | 𝑡 = 𝑠 = 𝑊 . Clearly 𝑈 ( 𝑡, 𝑡 ) is the identity and 𝑈 ( 𝑡, 𝑠 ) −1 = 𝑈 ( 𝑠, 𝑡 ) .1.2.3. Parallel frames.
Consider again the geodesic 𝛾 ∶ 𝐼 → 𝑀 and recall that 𝐼 . Take any orthonormal basis 𝑤 , … , 𝑤 𝑛 −1 , 𝑤 𝑛 = ̇𝛾 (0) of 𝑇 𝛾 (0) 𝑀 with respectto 𝑔 ̇𝛾 (0) . Let 𝑓 𝑘 ( 𝑡 ) be the parallel translation of 𝑤 𝑘 along 𝛾 . The dual frame has thebasis covectors 𝑓 𝑘 ( 𝑡 ) ∈ 𝑇 ∗ 𝛾 ( 𝑡 ) 𝑀 satisfying 𝑓 𝑖 ( 𝑡 )[ 𝑓 𝑗 ( 𝑡 )] = 𝛿 𝑖𝑗 for all 𝑡 ∈ 𝐼 . See e.g.[2, Chapter 4] for more details on parallel transport in Finsler geometry.These frames induce a natural bijection 𝑁 𝑡 → 𝑁 given by ∑ 𝑛 −1 𝑘 =1 𝑓 𝑘 (0) 𝑓 𝑘 ( 𝑡 ) .This is an isometry for the inner products 𝑔 ̇𝛾 .Using this identification, we get the curvature operator ̂𝑅 ( 𝑡 ) ∶ 𝑁 → 𝑁 (9)and the Jacobi field evolution operator ̂𝑈 ( 𝑡, 𝑠 ) ∶ 𝑁 → 𝑁 . (10) ECONSTRUCTION ALONG A GEODESIC FROM SPHERE DATA 5
These operators describe the curvature and the evolution of Jacobi field along all ofthe geodesic in terms of the space 𝑁 at 𝑡 = 0 .1.3. The inverse problem.
With the geometric prerequisites in place, we can pro-ceed to describe the inverse problem and its solution in detail.1.3.1.
Sphere data.
Consider a Finsler manifold 𝑀 without boundary, and an openset 𝑈 ⊂ 𝑀 . We measure data on the known set 𝑈 and aim to determine the un-known remainder of the manifold, 𝑀 ⧵ 𝑈 . The data consists of all smooth subsetsof spheres intersecting 𝑈 together with their radii. We next describe the data indetail. We assume that we know the geometry fully in 𝑈 .Take any point 𝑥 ∈ 𝑀 . A sphere on 𝑇 𝑥 𝑀 is a level set of of the Finsler func-tion 𝐹 𝑥 on this fiber, and the radius is the value of 𝐹 𝑥 . A generalized sphere on 𝑀 is the image of a sphere on 𝑇 𝑥 𝑀 under the exponential map exp 𝑥 ∶ 𝑇 𝑥 𝑀 → 𝑀 based at 𝑥 .A forward sphere of radius 𝑟 > centered at 𝑥 is the set { 𝑦 ∈ 𝑀 ; 𝑑 𝐹 ( 𝑥, 𝑦 ) = 𝑟 } . (11)Since the distance function 𝑑 𝐹 may not be symmetric, it is important that the dis-tance is measured from 𝑥 to 𝑦 , not vice versa. For sufficiently small radii the gen-eralized sphere is in fact a forward sphere, but we include all radii.A generalized sphere might not be a smooth hypersurface due to conjugate points.The differential of the exponential map exp 𝑥 fails to be bijective precisely at theconjugate locus. The differential may not exist at zero — unlike in Riemanniangeometry — but if it does, it is bijective.Let 𝐶 ( 𝑥 ) ⊂ 𝑇 𝑥 𝑀 denote the conjugate locus of 𝑥 . It consists of all points con-jugate to 𝑥 , not only the first ones along each geodesic. Definition 1. A visible smooth sphere on 𝑇 𝑥 𝑀 of radius 𝑟 > is an open (in therelative topology of 𝑇 𝑥 𝑀 ) connected subset of 𝑆 𝑛 −1 𝑇 𝑥 𝑀 (0 , 𝑟 ) ⧵ [ exp −1 𝑥 ( 𝑀 ⧵ 𝑈 ) ∪ 𝐶 ( 𝑥 ) ] . (12)Here 𝑆 𝑛 −1 𝑇 𝑥 𝑀 (0 , 𝑟 ) = { 𝑣 ∈ 𝑇 𝑥 𝑀 ; 𝐹 𝑥 ( 𝑣 ) = 𝑟 } is the Finsler sphere of radius 𝑟 .Removal of the complement of 𝑈 corresponds to “visibility” and that of the con-jugate locus to “smoothness”. This smoothness does not prohibit self-intersections. Definition 2. A visible smooth sphere on 𝑀 of radius 𝑟 > centered at 𝑥 ∈ 𝑀 isthe oriented surface exp 𝑥 ( 𝑆 ) , where 𝑆 is a visible smooth sphere on 𝑇 𝑥 𝑀 and sosmall that exp 𝑥 | 𝑆 is injective. This hypersurface inherits an orientation from theFinsler sphere on 𝑇 𝑥 𝑀 , telling which way is outward.We will work on visible smooth spheres locally, so the smallness assumptionon 𝑆 will be left implicit from now on. This assumption guarantees that exp 𝑥 ( 𝑆 ) isindeed a smooth surface and we need not worry about cut points.By 𝜈 we always denote the outward unit normal. If 𝑥 ∈ 𝑀 is a point on a visiblesmooth sphere of radius 𝑟 > and 𝜈 ( 𝑥 ) is the normal vector to the sphere, then thecenter of the sphere is 𝛾 𝑥,𝜈 ( 𝑥 ) (− 𝑟 ) . Because the metric is not assumed reversible,it may be that 𝛾 𝑥,𝜈 ( 𝑥 ) (− 𝑟 ) ≠ 𝛾 𝑥, − 𝜈 ( 𝑥 ) ( 𝑟 ) since reverse geodesics are not necessarilygeodesics. ECONSTRUCTION ALONG A GEODESIC FROM SPHERE DATA 6
Definition 3.
The sphere data in the set 𝑈 is the set 𝑆𝐷 ( 𝑈 , 𝑀, 𝐹 ) = {(Σ , 𝑡 ); 𝑡 > and Σ ⊂ 𝑈 is a visible smooth sphere on 𝑀 of radius 𝑡 } (13)of pairs of oriented hypersurfaces and their radii. Knowledge of the center pointsof the spheres is not included in the data.The centers of the spheres can be considered virtual point sources. Our data isgiven as sphere data, and the aim is to reconstruct a Finsler manifold. The structureis described by a family of coordinate local systems and the Finsler function onthem. It is important that the coordinate systems are based on the sphere data. Definition 4.
Let ( 𝑀 𝑖 , 𝐹 𝑖 ) , 𝑖 = 1 , , be two Finsler manifolds, and 𝑈 𝑖 ⊂ 𝑀 𝑖 opensubsets. If 𝜓 ∶ 𝑈 → 𝑈 is a diffeomorphism, the pullback of the sphere data is 𝜓 ∗ 𝑆𝐷 ( 𝑈 , 𝑀 , 𝐹 ) = {( 𝜓 −1 (Σ) , 𝑡 ); (Σ , 𝑡 ) ∈ 𝑆𝐷 ( 𝑈 , 𝑀 , 𝐹 )} . (14)The two manifolds ( 𝑀 𝑖 , 𝐹 𝑖 ) are said to have the same sphere data if there is a dif-feomorphism 𝜓 ∶ 𝑈 → 𝑈 so that 𝑆𝐷 ( 𝑈 , 𝑀 , 𝐹 ) = 𝜓 ∗ 𝑆𝐷 ( 𝑈 , 𝑀 , 𝐹 ) .We will assume that the diffeomorphism 𝜓 ∶ 𝑈 → 𝑈 is actually an isometry.Physically, it makes sense to assume that the measurement area is fully known andonly other regions of the manifold (planet) are unknown.1.3.2. Theorems.
Our main result is the next theorem. It states that the sphere datadetermines the curvature operator and Jacobi fields along any geodesic through themeasurement set. The setting is illustrated in figure 1.
Theorem 5.
Let ( 𝑀 𝑖 , 𝐹 𝑖 ) for 𝑖 = 1 , be two Finsler manifolds without bound-ary and 𝑈 𝑖 ⊂ 𝑀 𝑖 open subsets. Suppose there is an isometry 𝜓 ∶ 𝑈 → 𝑈 so that up to identification by 𝜓 the two manifolds have the same sphere data: 𝑆𝐷 ( 𝑈 , 𝑀 , 𝐹 ) = 𝜓 ∗ 𝑆𝐷 ( 𝑈 , 𝑀 , 𝐹 ) .Let 𝐼 ⊂ ℝ be an open interval containing . Take any ( 𝑥, 𝑣 ) ∈ 𝑇 𝑀 with 𝐹 ( 𝑥, 𝑣 ) = 1 and a geodesic 𝛾 ∶ 𝐼 → 𝑀 with the initial data ( 𝑥, 𝑣 ) . Let 𝛾 be ageodesic 𝐼 → 𝑀 with the initial data d 𝜓 ( 𝑥, 𝑣 ) . Identify the spaces 𝑁 𝑖 = { 𝑣 ∈ 𝑇 𝛾 𝑖 (0) ; 𝑔 ̇𝛾 𝑖 (0) ( ̇𝛾 𝑖 (0) , 𝑣 ) = 0} for 𝑖 = 1 , by d 𝜓 ∶ 𝑁 → 𝑁 .Then for all 𝑡, 𝑠 ∈ 𝐼 ∩ (−∞ , we have ̂𝑅 ( 𝑡 ) = ̂𝑅 ( 𝑡 ) and ̂𝑈 ( 𝑡, 𝑠 ) = ̂𝑈 ( 𝑡, 𝑠 ) . As the map 𝜓 was assumed to be isometric, the conclusion is trivial as long asthe geodesic 𝛾 remains in 𝑈 . The theorem states that the curvatures and Jacobifields agree along all of the geodesic. The only constraint is that the two geodesicsmight only be defined a finite amount of time into the past if the geodesics are notmaximal or the manifolds are not geodesically complete. Notice also that as 𝜓 isan isometry 𝑈 → 𝑈 , its differential d 𝜓 ∶ 𝑇 𝑈 → 𝑇 𝑈 maps unit vectors to unitvectors.Suppose now that the Finsler manifold is geodesically complete. Given a vis-ible smooth sphere Σ ⊂ 𝑈 , we can define the surface normal exponential map 𝜑 ∶ Σ × ℝ → 𝑀 by following geodesics in the normal direction; see section 3 andespecially (54) for more details. Given any coordinates 𝛼 ∶ ℝ 𝑛 −1 ⊃ Ω → Σ , thereis an induced map 𝜑 𝛼 ∶ Ω × ℝ → 𝑀,𝜑 𝛼 ( 𝑧, 𝑡 ) = 𝜑 ( 𝛼 ( 𝑧 ) , 𝑡 ) . (15) ECONSTRUCTION ALONG A GEODESIC FROM SPHERE DATA 7 F IGURE
1. The setting of theorem 5: We start from a point 𝑥 ∈ 𝑈 and a unit vector 𝑣 ∈ 𝑇 𝑥 𝑀 at time 𝑡 = 0 and follow the geodesicbackwards into the unknown 𝑀 ⧵ 𝑈 . At all times 𝑡 we have thenormal plane 𝑁 𝑡 to ̇𝛾 ( 𝑡 ) as a subset of 𝑇 𝛾 ( 𝑡 ) 𝑀 . We can identify allthese planes canonically with 𝑁 through parallel transport. Thisidentification makes the operators act on just 𝑁 instead of a bundleof planes along 𝛾 . The data we use is the visible smooth surfacesnormal to the geodesic near the initial point 𝑥 , pictured in lightgray.This map helps give local coordinates on 𝑀 in terms of the sphere data.When 𝜑 𝛼 is a diffeomorphism from some subset Ω ′ ⊂ Ω× ℝ to the image 𝜑 𝛼 (Ω ′ ) ,it gives rise to a natural local vector field 𝐺 on 𝑀 by pushing forward the constantvector field (0 ,
1) ∈ ℝ 𝑛 −1 × ℝ on Ω ′ . The integral curves of 𝐺 are the geodesicsalong which 𝜑 is defined and which come orthogonally to Σ .The next theorem states that the sphere data determines when 𝜑 𝛼 gives validlocal coordinates (we call these the surface normal coordinates) and determinessome properties of the metric in these coordinates. Theorem 6.
Let ( 𝑀 𝑖 , 𝐹 𝑖 ) for 𝑖 = 1 , be two Finsler geodesically complete mani-folds without boundary and 𝑈 𝑖 ⊂ 𝑀 𝑖 open subsets. Suppose there is an isometry 𝜓 ∶ 𝑈 → 𝑈 so that up to identification by 𝜓 the two manifolds have the samesphere data: 𝑆𝐷 ( 𝑈 , 𝑀 , 𝐹 ) = 𝜓 ∗ 𝑆𝐷 ( 𝑈 , 𝑀 , 𝐹 ) .Let Σ ⊂ 𝑈 be a visible smooth surface and 𝛼 ∶ ℝ 𝑛 −1 ⊃ Ω → Σ any local co-ordinates. Let 𝛼 = 𝜓 ◦ 𝛼 the the corresponding coordinates on the correspondingvisible smooth surface Σ = 𝜓 (Σ ) ⊂ 𝑈 . Define the maps 𝜑 𝛼 𝑖 ∶ Ω × ℝ → 𝑀 𝑖 asin (15) .For any 𝑧 ∈ Ω and 𝑡 < , the differential d 𝜑 𝛼 ( 𝑧, 𝑡 ) is bijective if and only if d 𝜑 𝛼 ( 𝑧, 𝑡 ) is too. In this case, both 𝜑 𝛼 𝑖 | Ω ′ define local coordinates on 𝑀 𝑖 for someneighborhood Ω ′ ⊂ Ω × (−∞ , of ( 𝑧, 𝑡 ) . Let 𝐺 𝑖 be the vector field correspondingto the surface normal coordinate system as defined above and let 𝑔 𝑖 = 𝑔 𝑖𝐺 𝑖 be theRiemannian metric tensor corresponding to it.Then the local Riemannian metrics agree in these local coordinates: 𝑔 = 𝑔 on Ω ′ . If the manifolds 𝑀 and 𝑀 are Riemannian, then their metric tensors agree inboundary normal coordinates. In Finsler geometry we can only ever hope to recover ECONSTRUCTION ALONG A GEODESIC FROM SPHERE DATA 8 F IGURE
2. The setting of theorem 6: The surface normal expo-nential map of a visible smooth sphere Σ follows geodesics from Σ backwards into the manifold. If the geodesics are lifted to the bun-dle (indicated with gray arrows), the “lifted normal exponentialmap” 𝜑 𝛼 ∶ Ω × ℝ → 𝑇 𝑀 always an immersion. The so defined(local) submanifold of
𝑇 𝑀 is always smooth but the usual nor-mal exponential map 𝜑 𝛼 fails to be a local diffeomorphism at thecenter point of the sphere (on the right) and points conjugate to it(between the center and Σ ). These points are focal to Σ . At theseconjugate points the submanifold of 𝑇 𝑀 has a small projection tothe base but has many points in the same fiber, illustrated by sev-eral arrows at the intersection points. The gray arrows define thevector field 𝐺 , and it is a local vector field on 𝑀 when the surfacenormal coordinates actually give local coordinates.the Finsler functions 𝐹 𝑖 in directions close to the vector field 𝐺 𝑖 , as other directionsmight not correspond to geodesics that come to the measurement sets 𝑈 𝑖 ⊂ 𝑀 𝑖 . Itis this directional nature of Finsler geometry that makes it natural to state the resultsalong geodesics rather than on the manifold 𝑀 . Remark . The singular set of 𝜑 𝛼 corresponds to focal points of the visible smoothsphere Σ . Therefore the sphere data determines all the focal distances at all pointson all the visible smooth spheres. See section 3 for details and figure 2 for anillustration.We assumed geodesic completeness for technical convenience. It can be left out,but then more care is needed in the statement of the theorem to ensure that therelevant geodesics exist on both manifolds.We point out that the surface normal coordinates of a visible smooth sphere cor-respond to the normal coordinates of the center point of that sphere. If we choosethe center to be within the known set 𝑈 ⊂ 𝑀 , then theorem 6 gives the local Rie-mannian metric in these normal coordinates when they are valid.
ECONSTRUCTION ALONG A GEODESIC FROM SPHERE DATA 9
Remark . Our main result, theorem 5, is a result along geodesics, and is corollary,theorem 6, gives a very local result in small open sets. We have chosen to leave outquestions of globalization from the present paper as they present a substantial hurdlein Finsler geometry. If the manifold were Riemannian, then theorem 6 would givethe metric tensor in some local coordinate system, thus giving a full description ofthe local geometry. In Finsler geometry the information given by the theorem isfar from knowing the Minkowski norm on every tangent space in every direction— it does not even give the norm in a neighborhood of the direction of 𝐺 in everytangent space. The vector field 𝐺 is illustrated in figure 2; we have only access tomultiple directions at a point if the point is focal to the visible smooth sphere, butthen we do not have a neighborhood in the base. Therefore the promotion froma single geodesic to a global conclusion is a far bigger step than in Riemanniangeometry and it is best taken separately. Additional assumptions are needed for theconclusion of [4], unique determination of the metric universal cover, to hold.1.4. Finsler geometry from seismology.
A certain class of elastic waves, namelyquasi-compressional waves ( qP waves) follow the geodesics of a Finsler metric.The Finsler metric arises from the elastic tensor. In each fiber the Finsler functionis given by finitely many elastic parameters and is in fact reversible and fiberwiseanalytic. This is very similar to the way every Riemannian metric is fiberwise an-alytic. We do not make these assumptions for a few reasons: they turn out to beunimportant, and there may be other physical models where they do not hold.If one measures in a small open set the wave fronts of qP -waves arriving froma point-like event inside the Earth and knows the travel times, then one obtains thesphere data as given in definition 3. Elastic Finsler metrics are in fact reversible,but out of pure geometrical interest we do not make this assumption.For more details on elastic Finsler geometry, we refer to the discussions in [5, 6].For a geophysical discussion of the state of the art of Dix’s problem in anisotropicmedia, see [10, 4]. 2. P ROOF OF THE MAIN THEOREM
Curvature, shape, and Jacobi fields.
To set up the tools for proving ourresults, we expand on the presentation of sections 1.2.2 and 1.2.3. To keep notationas simple as possible, we denote covariant derivatives along the geodesic 𝛾 by 𝐷 𝑡 instead of ∇ ̇𝛾 ( 𝑡 ) .In terms of the frame, the directional curvature operator 𝑅 ( 𝑡 ) along the Finslergeodesic is given by 𝑅 ( 𝑡 ) = 𝑛 −1 ∑ 𝑖,𝑘 =1 ( 𝑅 ̇𝛾 ( 𝑡 ) ( 𝛾 ( 𝑡 ) , ̇𝛾 ( 𝑡 ))) 𝑖𝑘 𝑓 𝑖 ( 𝑡 ) 𝑓 𝑘 ( 𝑡 ) . (16)Using the fixed frame, we can also regard 𝑅 ( 𝑡 ) simply as a matrix depending on 𝑡 .This corresponds to the operator ̂𝑅 ( 𝑡 ) of (9) upon identifying 𝑁 with ℝ 𝑛 −1 throughan orthonormal basis. But we shall drop these identifications and the hat and simplyconsider 𝑅 ( 𝑡 ) as a time-dependent matrix.Our reconstruction algorithm works in the reverse direction along a geodesicbecause we want the original measured signal to come in the forward direction.The signs will be different from the related Riemannian result [4]. ECONSTRUCTION ALONG A GEODESIC FROM SPHERE DATA 10
Consider the forward sphere of radius 𝑟 − 𝑡 > centered at 𝛾 ( 𝑡 ) . If 𝛾 ( 𝑡 ) and 𝛾 ( 𝑟 ) arenot conjugate along 𝛾 , then this defines locally a smooth surface Σ( 𝑟, 𝑡 ) near 𝛾 ( 𝑟 ) andvarying 𝑟 with 𝑡 fixed foliates a neighborhood of 𝛾 ( 𝑟 ) with the hypersurfaces Σ( 𝑟, 𝑡 ) .Let 𝜈 be the normal vector field of these surfaces oriented in the same directionas ̇𝛾 ( 𝑟 ) . It is a well defined smooth vector field near 𝛾 ( 𝑟 ) and has unit length at eachpoint.We define a Riemannian metric ̂𝑔 in this small neighborhood of 𝛾 ( 𝑟 ) by letting ̂𝑔 ( 𝑥 ) = 𝑔 𝜈 ( 𝑥 ) ( 𝑥 ) . With respect to this Riemannian metric we can define the shapeoperator 𝑆 ( 𝑟, 𝑡 ) of Σ( 𝑟, 𝑡 ) at 𝛾 ( 𝑟 ) as we would in the Riemannian case. More explic-itly, the action of 𝑆 ( 𝑟, 𝑡 ) on 𝑤 ∈ 𝑇 𝛾 ( 𝑟 ) Σ( 𝑡, 𝑟 ) = ̇𝛾 ( 𝑟 ) ⟂ ⊂ 𝑇 𝛾 ( 𝑟 ) 𝑀 is 𝑆 ( 𝑟, 𝑡 ) 𝑤 = ̂ ∇ 𝑤 𝜈 ,the covariant derivative of the normal field with respect to the Riemannian metriccorresponding to the normal vector field. Expressed in terms of the frame, 𝑆 ( 𝑟, 𝑡 ) is a matrix depending on two parameters.Notice that the very construction of 𝑆 ( 𝑟, 𝑡 ) depends on the two points not beingconjugate. Difficulties brought by conjugate points are an important aspect of ourinverse problem. However, we have diminished the role of conjugate points in ourapproach in comparison to the Riemannian one in [4]. In lemma 11 we will solveJacobi fields from a system of ODEs and the Jacobi fields are well-defined every-where, whereas in [4] the inverse shape operator 𝐾 was solved, and it is not definedeverywhere if there are conjugate or focal points.The operators 𝑅 ( 𝑡 ) and 𝑆 ( 𝑟, 𝑡 ) are related via the Riccati equation [17, Theo-rem 14.4.2] ∇ ̇𝛾 ( 𝑟 ) 𝑆 ( 𝑟, 𝑡 ) + 𝑆 ( 𝑟, 𝑡 ) + 𝑅 ( 𝑟 ) = 0 , (17)which holds for any 𝑟, 𝑡 ∈ ℝ for which the corresponding points on 𝛾 are not con-jugate. Here the covariant derivative ∇ of the local Riemannian metric associatedwith the family of geodesics that give rise to the surface 𝑆 ( 𝑟, 𝑡 ) . For simplicity, wewill write ∇ ̇𝛾 ( 𝑟 ) 𝑆 ( 𝑟, 𝑡 ) = 𝐷 𝑟 𝑆 ( 𝑟, 𝑡 ) .Let us then study the asymptotics of 𝑆 ( 𝑟, 𝑡 ) near 𝑟 = 𝑡 . We denote 𝐾 ( 𝑟, 𝑡 ) = 𝑆 ( 𝑟, 𝑡 ) −1 whenever 𝑆 ( 𝑟, 𝑡 ) is invertible. Invertibility can indeed fail: If one equipsthe sphere 𝑆 𝑛 with the usual round Riemannian metric, then 𝑆 ( 𝑟, 𝑡 ) = 0 when 𝑟 = 𝑡 + 𝜋 , and this occurs before the first conjugate point ( 𝑟 = 𝑡 + 𝜋 ). Lemma 9.
For 𝑟 sufficiently close to 𝑡 but 𝑟 ≠ 𝑡 , the shape operator 𝑆 ( 𝑟, 𝑡 ) isinvertible and we have 𝐾 ( 𝑟, 𝑡 ) = ( 𝑟 − 𝑡 ) id + ( 𝑟 − 𝑡 ) 𝑅 ( 𝑟 ) + 𝑜 ( | 𝑟 − 𝑡 | ) . (18) Proof.
By [17, Theorem 14.4.3] we have 𝑆 ( 𝑟, 𝑡 ) = ( 𝑟 − 𝑡 ) −1 id − ( 𝑟 − 𝑡 )3 𝑅 ( 𝑟 ) + 𝑜 ( | 𝑟 − 𝑡 | ) . (19)We may replace 𝑅 ( 𝑟 ) with 𝑅 ( 𝑡 ) , as this only induces an error ( | 𝑡 − 𝑟 | ) . The invert-ibility result follows immediately. Using Neumann series for the inverse gives (18). (cid:3) Determination of curvature operators and Jacobi fields.
For any 𝑡 ∈ ℝ ,let 𝐽 ( ⋅ , 𝑡 ) be a linearly independent family of 𝑛 − 1 normal Jacobi fields along 𝛾 which vanish at 𝛾 ( 𝑡 ) . We can think of 𝐽 ( 𝑟, 𝑡 ) as an ( 𝑛 − 1) × ( 𝑛 − 1) matrix whoseevery column is a Jacobi field with respect to the variable 𝑟 . Geometrically, theseJacobi fields correspond to geodesics emanating from 𝛾 ( 𝑡 ) in directions close to ̇𝛾 ( 𝑡 ) . ECONSTRUCTION ALONG A GEODESIC FROM SPHERE DATA 11
Every Jacobi field satisfies the Jacobi equation, and that can be written collec-tively as 𝐷 𝑟 𝐽 ( 𝑟, 𝑡 ) + 𝑅 ( 𝑟 ) 𝐽 ( 𝑟, 𝑡 ) = 0 . (20)The Jacobi fields also satisfy a first-order equation as the next lemma shows.These requirements do not determine the matrix-valued function 𝐽 ( 𝑟, 𝑡 ) uniquely.It is only unique up to multiplication from the right by an invertible matrix. Thematrix nature of our operators will be explained in more details soon. Lemma 10.
The Jacobi fields satisfy 𝐷 𝑟 𝐽 ( 𝑟, 𝑡 ) = 𝑆 ( 𝑟, 𝑡 ) 𝐽 ( 𝑟, 𝑡 ) (21) whenever the shape operator 𝑆 ( 𝑟, 𝑡 ) is defined.Proof. The shape operator was only defined when 𝛾 ( 𝑡 ) and 𝛾 ( 𝑟 ) are not conjugatealong 𝛾 . The curvature operator 𝑅 ( 𝑡 ) and the shape operator 𝑆 ( 𝑟, 𝑡 ) are related viathe Riccati equation (17), which holds for any 𝑟, 𝑡 ∈ ℝ for which the correspondingpoints on 𝛾 are not conjugate.It follows from standard ODE theory that for any fixed 𝑡 there is a solution to (21)and the solution is unique up to change of basis if 𝐽 is assumed invertible at somepoint. The shape operator annihilates vectors parallel to the reference geodesic,whence the rows of a solution 𝐽 stay normal to ̇𝛾 ( 𝑟 ) if they are normal at somepoint.If a matrix ̃𝐽 ( 𝑟, 𝑡 ) whose columns are linearly independent and normal to ̇𝛾 ( 𝑟 ) sat-isfies (21), then it is easy to check using (17) that it also satisfies (20). Since Jacobifields stay bounded as 𝑟 → 𝑡 and the asymptotic formula (19) indicates that 𝑆 ( 𝑟, 𝑡 ) blows up in this limit, we must have ̃𝐽 ( 𝑟, 𝑡 ) = ( | 𝑟 − 𝑡 | ) and thus ̃𝐽 ( 𝑟, 𝑟 ) = 0 . Forany 𝑡 ∈ ℝ there is an ( 𝑛 − 1) -dimensional space of normal Jacobi fields vanish-ing at 𝛾 ( 𝑡 ) and 𝐽 ( ⋅ , 𝑡 ) was defined so that it spans this space. Therefore, up to thefreedom of changing basis on this space, the functions ̃𝐽 ( ⋅ , 𝑡 ) and 𝐽 ( ⋅ , 𝑡 ) agree. Inparticular, the Jacobi fields 𝐽 ( 𝑟, 𝑡 ) solve (21). (cid:3) Let us now make the use of the frames explicit for the sake of concreteness.There are unique matrix-valued functions 𝐣 ( 𝑟, 𝑡 ) , 𝐬 ( 𝑟, 𝑡 ) and 𝐫 ( 𝑟 ) so that 𝐽 ( 𝑟, 𝑡 ) = 𝑛 −1 ∑ 𝑗,𝑘 =1 𝐣 𝑗𝑘 ( 𝑟, 𝑡 ) 𝑓 𝑗 ( 𝑟 ) 𝑒 𝑘 , (22)where 𝑒 𝑘 ∈ ℝ 𝑛 −1 are the Euclidean basis vectors, and 𝑆 ( 𝑟, 𝑡 ) = 𝑛 −1 ∑ 𝑗,𝑘 =1 𝐬 𝑗𝑘 ( 𝑟, 𝑡 ) 𝑓 𝑗 ( 𝑟 ) 𝑓 𝑘 ( 𝑟 ) (23)and 𝑅 ( 𝑟 ) = 𝑛 −1 ∑ 𝑗,𝑘 =1 𝐫 𝑗𝑘 ( 𝑟 ) 𝑓 𝑗 ( 𝑟 ) 𝑓 𝑘 ( 𝑟 ) . (24)Similarly, when the shape operator is invertible, we let 𝐤 ( 𝑟, 𝑡 ) = 𝐬 ( 𝑟, 𝑡 ) −1 , corre-sponding to the operator 𝐾 ( 𝑟, 𝑡 ) defined above. The matrix-valued shape opera-tor 𝐬 ( 𝑟, 𝑡 ) is not defined when the corresponding two points on the geodesic areconjugate, just like with 𝑆 ( 𝑟, 𝑡 ) . ECONSTRUCTION ALONG A GEODESIC FROM SPHERE DATA 12
One could freely identify 𝑅 ( 𝑡 ) = 𝐫 ( 𝑡 ) = ̂𝑅 ( 𝑡 ) and similarly for the other quanti-ties, but we find that the distinction adds some clarity to the proof.Let us reiterate the geometrical interpretation of 𝐽 ( 𝑟, 𝑡 ) : For any 𝑡 ∈ ℝ and 𝑣 ∈ ℝ 𝑛 −1 , the vector field 𝑟 ↦ ∑ 𝑛 −1 𝑗,𝑘 =1 𝐣 𝑗𝑘 𝑓 𝑗 ( 𝑟 ) 𝑣 𝑘 is a normal Jacobi field along 𝛾 whichvanishes at 𝑟 = 𝑡 . The Jacobi fields corresponding to linearly independent choicesof 𝑣 are linearly independent. Since the Jacobi fields are linearly independent andvanish at 𝑟 = 𝑡 , it follows that the matrix 𝜕 𝑟 𝐣 ( 𝑟, 𝑡 ) is invertible at and near 𝑟 = 𝑡 .Let us rewrite our key identities in matrix form. The second-order equation forthe Jacobi fields (cf. (20)) is 𝜕 𝑟 𝐣 ( 𝑟, 𝑡 ) + 𝐫 ( 𝑟 ) 𝐣 ( 𝑟, 𝑡 ) = 0 . (25)The first-order equation for the Jacobi fields (cf. (21)) is 𝜕 𝑟 𝐣 ( 𝑟, 𝑡 ) = 𝐬 ( 𝑟, 𝑡 ) 𝐣 ( 𝑟, 𝑡 ) . (26)The asymptotic expansions for 𝐤 near 𝑟 = 𝑡 (cf. (18)) is 𝐤 ( 𝑟, 𝑡 ) = ( 𝑟 − 𝑡 ) 𝐼 + ( 𝑟 − 𝑡 ) 𝐫 ( 𝑟 ) + 𝑜 ( | 𝑟 − 𝑡 | ) . (27)Now we are ready to prove a uniqueness result for Jacobi fields and curvature op-erators. Lemma 11.
Let 𝐼 ⊂ 𝐼 ⊂ ℝ be two nested open intervals. Knowing the shapeoperator 𝑆 ( 𝑟, 𝑡 ) for all 𝑟 ∈ 𝐼 and 𝑡 ∈ 𝐼 for which it is defined determines uniquely (1) the Jacobi fields 𝐽 ( 𝑟, 𝑡 ) up to multiplication by a 𝑡 -dependent invertible ma-trix from the right, (2) the shape operator 𝑆 ( 𝑟, 𝑡 ) when it is defined, (3) the inverse shape operator 𝐾 ( 𝑟, 𝑡 ) when it is defined, and (4) the curvature operator 𝑅 ( 𝑡 ) for all 𝑟, 𝑡 ∈ 𝐼 .Proof. We will prove the result in matrix form as it is more convenient. We assumethat 𝐼 = (− 𝜀, 𝜀 ) and 𝐼 = (− 𝑇 , 𝜀 ) for < 𝜀 < 𝑇 , so that we prove “uniqueness tothe reverse direction”. This direction is relevant for our application, and the prooffor the other direction is identical. Combining the two directions gives the fullstatement. By translation, we may assume 𝐼 and 𝐼 to be of the given form, and atthe end one may let 𝑇 → ∞ to cover the case when 𝐼 is unbounded.By making 𝜀 > to be sufficiently small we can ensure that the shape opera-tor 𝑆 ( 𝑟, 𝑡 ) is well defined when ( 𝑟, 𝑡 ) ∈ (− 𝜀, 𝜀 ) and 𝑟 ≠ 𝑡 . This amounts to sayingthat the initial data is set on such a short interval that no focal points to the relevantspheres appear. Step 1: Setting up a system of ODEs.
For 𝑘 ∈ {0 , , , , let us denote 𝐣 𝑘 ( 𝑟, 𝑡 ) = 𝜕 𝑘𝑡 𝐣 ( 𝑟, 𝑡 ) (28)and 𝐲 𝑘 ( 𝑟, 𝑡 ) = 𝜕 𝑟 𝜕 𝑘𝑡 𝐣 ( 𝑟, 𝑡 ) . (29)Using (25) we get the equations { 𝜕 𝑟 𝐣 𝑘 ( 𝑟, 𝑡 ) = 𝐲 𝑘 ( 𝑟, 𝑡 ) 𝜕 𝑟 𝐲 𝑘 ( 𝑟, 𝑡 ) = − 𝐫 ( 𝑟 ) 𝐣 𝑘 ( 𝑟, 𝑡 ) (30)for all four values of 𝑘 . ECONSTRUCTION ALONG A GEODESIC FROM SPHERE DATA 13
The crucial step is to compute the curvature matrix 𝐫 in terms of 𝐣 𝑘 and 𝐲 𝑘 .Recall that the matrix 𝐲 ( 𝑟, 𝑡 ) is invertible in a neighborhood of the diagonal 𝑟 = 𝑡 .Combining (26) and the definition of 𝐤 ( 𝑟, 𝑡 ) as the inverse of 𝐬 ( 𝑟, 𝑡 ) when invertible,we find 𝐤 ( 𝑟, 𝑡 ) = 𝐣 ( 𝑟, 𝑡 ) 𝐲 ( 𝑟, 𝑡 ) −1 . (31)On the other hand, the asymptotic formula (27) indicates that 𝜕 𝑡 𝐤 ( 𝑟, 𝑡 ) | 𝑡 = 𝑟 = −2 𝐫 ( 𝑟 ) . (32)Combining (31) and (32), we find 𝐫 ( 𝑟 ) = − 12 𝑄 ( 𝐣 ( 𝑟, 𝑟 ) , 𝐣 ( 𝑟, 𝑟 ) , 𝐣 ( 𝑟, 𝑟 ) , 𝐲 ( 𝑟, 𝑟 ) , 𝐲 ( 𝑟, 𝑟 ) , 𝐲 ( 𝑟, 𝑟 )) , (33)where the function 𝑄 is defined via 𝑄 ( 𝐴 , 𝐴 , 𝐴 , 𝐵 , 𝐵 , 𝐵 )= 𝐴 𝐵 −10 − 3 𝐴 𝐵 −10 𝐵 𝐵 −10 + 6 𝐴 𝐵 −10 𝐵 𝐵 −10 𝐵 𝐵 −10 − 3 𝐴 𝐵 −10 𝐵 𝐵 −10 . (34)We used the fact that 𝐣 ( 𝑟, 𝑟 ) = 0 to simplify this expression. This function is onlywell defined when the matrix 𝐵 is invertible, and 𝐲 ( 𝑟, 𝑡 ) is indeed invertible in aneighborhood of the diagonal. Observe that this 𝑄 is invariant under multiplicationof all the input matrices from the right by the same invertible matrix.The ODE system (30) can now be recast as { 𝜕 𝑟 𝐣 𝑘 ( 𝑟, 𝑡 ) = 𝐲 𝑘 ( 𝑟, 𝑡 ) 𝜕 𝑟 𝐲 𝑘 ( 𝑟, 𝑡 ) = 𝑄 ( 𝐣 ( 𝑟, 𝑟 ) , 𝐲 ( 𝑟, 𝑟 )) 𝐣 𝑘 ( 𝑟, 𝑡 ) , (35)where we have abbreviated the notation for 𝑄 for legibility. This is a closed systemof ODEs for the matrix-valued functions 𝐣 𝑘 and 𝐲 𝑘 . We will prove unique solvabilityof this system. Step 2: Setting up initial conditions.
Let us now see why the data deter-mines 𝐫 | 𝐼 . For 𝑟, 𝑡 ∈ 𝐼 , the shape operator matrix 𝐬 ( 𝑟, 𝑡 ) is known when it ex-ists, and it is always existent invertible when 𝑟 and 𝑡 are close enough but different;see lemma 9. The inverse is denoted by 𝐤 ( 𝑟, 𝑡 ) . When extended continuously to thediagonal 𝑟 = 𝑡 where 𝐬 is not defined, it is not invertible, but it is continuous and sev-eral times differentiable across the diagonal. The data therefore determines 𝐤 ( 𝑟, 𝑡 ) for 𝑟, 𝑡 ∈ 𝐼 when | 𝑟 − 𝑡 | is small enough. By (32) this determines 𝐫 | 𝐼 .Let us then study setting initial conditions at 𝑟 = 0 . We denote 𝐶 𝑟 = { 𝑡 ∈ 𝐼 ; 𝛾 ( 𝑡 ) and 𝛾 ( 𝑟 ) are conjugate or 𝑡 = 𝑟 } . (36)For any 𝑟 ∈ 𝐼 the set 𝐶 𝑟 ⊂ 𝐼 is discrete since conjugate points cannot accumulate.We are free to choose initial conditions for the Jacobi fields 𝐣 ( 𝑟, 𝑡 ) under someconstraints: ∙ Because 𝐣 ( 𝑟, 𝑟 ) = 0 , we need to have 𝐣 ( 𝑟, 𝑡 ) = ( | 𝑟 − 𝑡 | ) . ∙ The matrix 𝐣 ( 𝑟, 𝑡 ) is invertible if and only if 𝑡 ∉ 𝐶 𝑟 . ∙ We want the initial conditions be regular enough so that 𝐣 ∈ 𝐶 is possible.To achieve this, we will first set our initial values in a set that avoids conjugatepoints. ECONSTRUCTION ALONG A GEODESIC FROM SPHERE DATA 14
Let us denote 𝑈 = 𝐼 × 𝐼 and 𝐶 = ⋃ 𝑟 ∈ 𝐼 { 𝑟 }× 𝐶 𝑟 ⊂ 𝑈 . The set 𝐶 ⊂ 𝑈 is closed,which follows from continuity of the Jacobi field matrix 𝐣 ( 𝑟, 𝑡 ) . In the statement ofthe theorem, we assumed that 𝐬 | 𝑈 ⧵ 𝐶 is known.We will now construct a set of line segments in 𝑈 ⧵ 𝐶 on which we can set initialvalues easily. There are numbers 𝑡 > 𝑡 > ⋯ ∈ ̄𝐼 so that 𝐶 = { 𝑡 , 𝑡 , … , 𝑡 𝑁 } .Here the number 𝑁 of points may be one, infinity, or any number in between. Since 𝐶 , we have 𝑁 ≥ . By making 𝜀 > smaller if necessary, we may assumethat − 𝜀 ∉ 𝐶 . Let us denote 𝑡 = − 𝜀 . For 𝜀 > small enough we have in fact 𝑡 = 0 , but we need not assume this.If 𝑁 < ∞ , we let 𝑡 𝑁 +1 = − 𝑇 . By replacing 𝑇 with 𝑇 − 𝛿 for some small 𝛿 > if necessary, we may assume that − 𝑇 ∉ 𝐶 . The full result then follows by letting 𝛿 → . For convenience and clarity, we simply assume that − 𝑇 ∉ 𝐶 .For any 𝑖 ≥ , there is 𝑟 𝑖 ∈ 𝐼 so that ( 𝑟 𝑖 , 𝑡 𝑖 ) ∉ 𝐶 and the points 𝛾 (0) and 𝛾 ( 𝑟 𝑖 ) are not conjugate. The second non-conjugacy condition can also be achieved bytaking 𝜀 > so small that there are no conjugate points on 𝛾 | 𝐼 . There are numbers 𝑎 ′ 𝑖 , 𝑏 ′ 𝑖 ∈ ℝ so that
12 ( 𝑡 𝑖 −1 + 𝑡 𝑖 ) > 𝑎 ′ 𝑖 > 𝑡 𝑖 > 𝑏 ′1 >
12 ( 𝑡 𝑖 + 𝑡 𝑖 +1 ) (37)and { 𝑟 𝑖 } × [ 𝑏 ′ 𝑖 , 𝑎 ′ 𝑖 ] ∩ 𝐶 = ∅ . Then we pick any 𝑏 𝑖 −1 ∈ ( 𝑡 𝑖 , 𝑎 ′ 𝑖 ) and 𝑎 𝑖 ∈ ( 𝑏 ′ 𝑖 , 𝑡 𝑖 ) .We set 𝑎 = 𝑡 = − 𝜀 . If 𝑁 < ∞ , we let 𝑏 𝑁 = − 𝑇 .With this construction we have found disjoint line segments 𝐿 𝑖 = {0} × [ 𝑏 𝑖 , 𝑎 𝑖 ] , ≤ 𝑖 < 𝑁 + 1 , and 𝐿 ′ 𝑖 = { 𝑟 𝑖 } × [ 𝑏 ′ 𝑖 , 𝑎 ′ 𝑖 ] , < 𝑖 < 𝑁 + 1 . Each of these segmentshas positive distance to the conjugate set 𝐶 ⊂ 𝑈 , and the union of projections tothe 𝑡 -axis is ̄𝐼 . Any 𝑡 ∈ 𝐼 is in the projection of one or two such segments. Theshape operator is well defined in a neighborhood of each of these line segments.Now we can start setting the initial values for our Jacobi fields. First, we simplylet 𝐣 | 𝐿 = 𝐼 . Since 𝐬 is defined in a neighborhood of 𝐿 , the condition (26) deter-mines 𝜕 𝑟 𝐣 | 𝐿 . With the initial value and derivative given, the Jacobi equation (25)determines 𝐣 on 𝐼 × [ 𝑏 , 𝑎 ] , as 𝐫 | 𝑖 is determined from the data as explained above.In particular, the Jacobi field matrix 𝐣 is determined on part of 𝐿 ′1 , namely { 𝑟 } ×[ 𝑏 , 𝑎 ′1 ] . This part is uniquely determined, invertible and smooth because the twopoints 𝛾 (0) and 𝛾 ( 𝑟 ) are not conjugate. Now we extend 𝐣 smoothly to the wholeline segment 𝐿 ′1 , keeping it invertible at every point. The derivative 𝜕 𝑟 𝐣 | 𝐿 ′1 is thendetermined by (26). Using the Jacobi equation (25) again, this information on 𝐿 ′1 determines 𝐣 on an initial part of 𝐿 , and we continue it smoothly and invertibly tothe whole 𝐿 . Continuing iteratively and using the knowledge of the shape operatorin 𝑈 ⧵ 𝐶 , we find the values of 𝐣 (0 , 𝑡 ) and 𝜕 𝑟 𝐣 (0 , 𝑡 ) for all 𝑡 ∈ 𝐼 .Now that we have fixed our consistent initial data corresponding to a well be-haved Jacobi field matrix 𝐣 ( 𝑟, 𝑡 ) , we no longer need to avoid conjugate points. Step 3: Uniqueness of the solution.
We then set out to prove uniqueness ofsolutions to the ODE system (35) with our initial data for 𝐣 𝑘 (0 , 𝑡 ) and 𝐲 𝑘 (0 , 𝑡 ) forall 𝑘 ∈ {0 , , , and 𝑡 ∈ 𝐼 . We already know existence, so it suffices to showuniqueness. One could show existence with similar arguments.To abbreviate our notation, let us denote 𝑍 ( 𝑟, 𝑡 ) = ( 𝐣 ( 𝑟, 𝑡 ) , 𝐣 ( 𝑟, 𝑡 ) , 𝐣 ( 𝑟, 𝑡 ) , 𝐣 ( 𝑟, 𝑡 ) , 𝐲 ( 𝑟, 𝑡 ) , 𝐲 ( 𝑟, 𝑡 ) , 𝐲 ( 𝑟, 𝑡 ) , 𝐲 ( 𝑟, 𝑡 )) ∈ ( ℝ ( 𝑛 −1)×( 𝑛 −1) ) . (38) ECONSTRUCTION ALONG A GEODESIC FROM SPHERE DATA 15
We use operator norms for the individual matrices, and the norm of 𝑍 ( 𝑟, 𝑡 ) is thesum of norms of the eight matrices.Suppose 𝑍 ′ ( 𝑟, 𝑡 ) is another solution with the same initial conditions, with all thematrices decorated with a prime. We know that 𝑍 (0 , 𝑡 ) = 𝑍 ′ (0 , 𝑡 ) for all 𝑡 ∈ 𝐼 .Let 𝜌 = inf { 𝑟 ≤ 𝑍 ( 𝑠, 𝑡 ) = 𝑍 ′ ( 𝑠, 𝑡 ) for all 𝑡 ∈ 𝐼 and 𝑠 ∈ ( 𝑟, . (39)To show that 𝑍 = 𝑍 ′ , assume for a contradiction that 𝜌 > − 𝑇 . By continuity, 𝑍 ( 𝜌 , 𝑡 ) = 𝑍 ′ ( 𝜌 , 𝑡 ) for all 𝑡 ∈ 𝐼 .Using the fundamental theorem of calculus, we find that for any 𝜌 ∈ (− 𝑇 , 𝜌 ) and 𝑡 ∈ 𝐼 , we have 𝑍 ( 𝜌, 𝑡 ) − 𝑍 ′ ( 𝜌, 𝑡 ) = ∫ 𝜌𝜌 ( 𝜕 𝑟 𝑍 ( 𝑟, 𝑡 ) − 𝜕 𝑟 𝑍 ′ ( 𝑟, 𝑡 )) d 𝑟. (40)Using the equation (35), we find formulas for 𝜕 𝑟 𝑍 ( 𝑟, 𝑡 ) − 𝜕 𝑟 𝑍 ′ ( 𝑟, 𝑡 ) : ⎧⎪⎨⎪⎩ 𝜕 𝑟 𝐣 𝑘 ( 𝑟, 𝑡 ) − 𝜕 𝑟 𝐣 ′ 𝑘 ( 𝑟, 𝑡 ) = 𝐲 𝑘 ( 𝑟, 𝑡 ) − 𝐲 ′ 𝑘 ( 𝑟, 𝑡 ) 𝜕 𝑟 𝐲 𝑘 ( 𝑟, 𝑡 ) − 𝜕 𝑟 𝐲 ′ 𝑘 ( 𝑟, 𝑡 ) = 𝑄 ( 𝑍 ( 𝑟, 𝑟 )) 𝐲 𝑘 ( 𝑟, 𝑡 )− 𝑄 ( 𝑍 ′ ( 𝑟, 𝑟 )) 𝐲 ′ 𝑘 ( 𝑟, 𝑡 ) , (41)where we have abbreviated 𝑄 ( 𝑍 (′) ( 𝑟, 𝑟 )) = 𝑄 ( 𝐣 (′) ( 𝑟, 𝑟 ) , 𝐲 (′) ( 𝑟, 𝑟 )) . (42)We will estimate these derivatives in terms of a suitable norm of 𝑍 − 𝑍 ′ .The function 𝑄 defined in (34) is smooth and well defined in a neighborhood ofany point where 𝐵 (which plays the role of 𝐲 ) is invertible. Therefore there are 𝜂, 𝐿 > so that 𝑄 is 𝐿 -Lipschitz in the closed ball ̄𝐵 ( 𝑍 ( 𝜌 , 𝜌 ) , 𝜂 ) .For any 𝛿 > , let us denote 𝐹 𝛿 = [ 𝜌 − 𝛿, 𝜌 ] . Due to continuity, for sufficientlysmall 𝛿 > we have max ( 𝑟,𝑡 )∈ 𝐹 𝛿 || 𝑍 ( 𝑟, 𝑡 ) − 𝑍 ( 𝜌 , 𝜌 ) || ≤ 𝜂 (43)and max ( 𝑟,𝑡 )∈ 𝐹 𝛿 || 𝑍 ′ ( 𝑟, 𝑡 ) − 𝑍 ( 𝜌 , 𝜌 ) || ≤ 𝜂. (44)This ensures that the function 𝑄 is 𝐿 -Lipschitz in its parameters when ( 𝑟, 𝑡 ) ∈ 𝐹 𝛿 .It follows that | 𝑄 ( 𝑍 ( 𝑟, 𝑡 )) | ≤ || 𝑄 ( 𝑍 ( 𝜌 , 𝜌 )) || + 𝐿𝜂 =∶ 𝛼 (45)for all ( 𝑟, 𝑡 ) ∈ 𝐹 𝛿 and similarly for || 𝑄 ( 𝑍 ′ ( 𝑟, 𝑡 )) || .By (40) and (41), we have 𝐣 𝑘 ( 𝜌, 𝜏 ) − 𝐣 ′ 𝑘 ( 𝜌, 𝜏 ) = ∫ 𝜌𝜌 ( 𝐲 𝑘 ( 𝑟, 𝜏 ) − 𝐲 ′ 𝑘 ( 𝑟, 𝜏 ) ) d 𝑟, (46)and so max 𝐹 𝛿 || 𝐣 𝑘 − 𝐣 ′ 𝑘 || ≤ 𝛿 max 𝐹 𝛿 || 𝑍 − 𝑍 ′ || . (47) ECONSTRUCTION ALONG A GEODESIC FROM SPHERE DATA 16
Similarly, we find || 𝐲 𝑘 ( 𝜌, 𝜏 ) − 𝐲 ′ 𝑘 ( 𝜌, 𝜏 ) || = 12 ||||| ∫ 𝜌𝜌 ( 𝑄 ( 𝑍 ( 𝑟, 𝑟 )) 𝐣 𝑘 ( 𝑟, 𝜏 ) − 𝑄 ( 𝑍 ′ ( 𝑟, 𝑟 )) 𝐣 ′ 𝑘 ( 𝑟, 𝜏 ) ) d 𝑟 ||||| ≤ ∫ 𝜌𝜌 ( || 𝑄 ( 𝑍 ( 𝑟, 𝑟 ))( 𝐣 𝑘 ( 𝑟, 𝜏 ) − 𝐣 ′ 𝑘 ( 𝑟, 𝜏 )) || + || ( 𝑄 ( 𝑍 ( 𝑟, 𝑟 )) − 𝑄 ( 𝑍 ′ ( 𝑟, 𝑟 ))) 𝐣 ′ 𝑘 ( 𝑟, 𝜏 ) || ) d 𝑟 ≤ ∫ 𝜌𝜌 ( 𝛼 || 𝐣 𝑘 ( 𝑟, 𝜏 ) − 𝐣 ′ 𝑘 ( 𝑟, 𝜏 ) || + 𝐿 || 𝑍 ( 𝑟, 𝑟 ) − 𝑍 ′ ( 𝑟, 𝑟 ) || || 𝐣 ′ 𝑘 ( 𝑟, 𝜏 ) || ) d 𝑟. (48)The ODE (35) with naive estimates and Grönwall’s inequality can be used to estab-lish an a priori estimate max 𝐹 𝛿 ||| 𝐣 ′ 𝑘 ||| ≤ 𝛽 < ∞ . We may then estimate max 𝐹 𝛿 || 𝐲 𝑘 − 𝐲 ′ 𝑘 || ≤ 𝛿 𝛼 + 𝐿𝛽 ) max 𝐹 𝛿 || 𝑍 − 𝑍 ′ || . (49)Combining (47) and (49) for all four values of 𝑘 gives max 𝐹 𝛿 || 𝑍 − 𝑍 ′ || ≤ 𝛿 (4 + 2( 𝛼 + 𝐿𝛽 )) max 𝐹 𝛿 || 𝑍 − 𝑍 ′ || . (50)If we choose 𝛿 = min { 𝛿 ,
18 + 4( 𝛼 + 𝐿𝛽 ) } , (51)the estimate (50) gives max 𝐹 𝛿 || 𝑍 − 𝑍 ′ || = 0 . Thus 𝑍 = 𝑍 ′ in 𝐹 𝛿 .This implies that 𝑄 ( 𝑍 ( 𝑟, 𝑟 )) = 𝑄 ( 𝑍 ′ ( 𝑟, 𝑟 )) for all 𝑟 ∈ [ 𝜌 − 𝛿, 𝜌 ] . Since 𝑍 ( 𝜌 , 𝑡 ) = 𝑍 ′ ( 𝜌 , 𝑡 ) for all 𝑡 ∈ 𝐼 , unique solvability of the ODE system (30)with fixed 𝑅 | [ 𝜌 − 𝛿,𝜌 ] shows that the solutions 𝑍 ( 𝑟, 𝑡 ) and 𝑍 ′ ( 𝑟, 𝑡 ) must coincide forall ( 𝑟, 𝑡 ) ∈ [ 𝜌 − 𝛿, 𝜌 ] × 𝐼 . This contradicts the definition of 𝜌 and shows thatindeed 𝜌 = − 𝑇 , whether it is a real number or −∞ . Step 4: Finding 𝑆 , 𝐾 , and 𝑅 . We have now proven that the Jacobi fields 𝐣 areuniquely determined in 𝐼 × 𝐼 . The matrix 𝐬 ( 𝑟, 𝑡 ) exists when the two points 𝛾 ( 𝑟 ) and 𝛾 ( 𝑡 ) are not conjugate, and this happens precisely when 𝐣 ( 𝑟, 𝑡 ) is invertible. Thematrix 𝐬 ( 𝑟, 𝑡 ) may be computed from (26): 𝐬 ( 𝑟, 𝑡 ) = ( 𝜕 𝑟 𝐣 ( 𝑟, 𝑡 )) 𝐣 ( 𝑟, 𝑡 ) −1 . (52)Therefore the matrix 𝐬 is uniquely determined on 𝐼 × 𝐼 when it exists from thedata. The same holds immediately for the inverse 𝐤 as well.The matrix 𝐤 is defined near the diagonal 𝑟 = 𝑡 and from it the curvature ma-trix 𝐫 ( 𝑡 ) can be computed by (32). The proof is complete. (cid:3) Remark . The function 𝐣 corresponding to a geodesic passing through 𝑈 is deter-mined by the data uniquely up to a 𝑡 -dependent change of basis. One can define thematrix 𝐬 ( 𝑟, 𝑡 ) by (52) and it exists precisely when 𝐣 ( 𝑟, 𝑡 ) is invertible. It is sometimespossible to define the matrix 𝐤 ( 𝑟, 𝑡 ) in a limit sense (extension by continuity) evenwhen 𝐬 ( 𝑟, 𝑡 ) is not invertible. This is possible if and only if 𝜕 𝑟 𝐣 ( 𝑟, 𝑡 ) is invertible andis given by 𝐤 = 𝐣 ( 𝜕 𝑟 𝐣 ) −1 . For example, this happens at the diagonal 𝑟 = 𝑡 . ECONSTRUCTION ALONG A GEODESIC FROM SPHERE DATA 17
Remark . The proof suggests an algorithm for solving the Jacobi fields. One firstneeds to set up initial values of the Jacobi fields. This was done in the proof bysetting initial values away from conjugate points and propagating the initial data to 𝑟 = 0 by the Jacobi equation. This is possible because 𝐹 | 𝑈 determines the curvatureoperator 𝑅 on 𝐼 . Having initial data on the open interval 𝐼 was only needed toavoid conjugate points in setting up the initial conditions for Jacobi fields.Then one solves the ODE (35) iteratively. If it has been solved for 𝑟 ∈ [ 𝜌 , and 𝑡 ∈ 𝐼 , it is then solved in a small square [ 𝜌 − 𝛿, 𝜌 ] with a corner at ( 𝜌 , 𝜌 ) by means of fixed point iteration of the integral formulation { 𝐣 𝑘 ( 𝜌, 𝑡 ) = 𝐣 𝑘 ( 𝜌 , 𝑡 ) + ∫ 𝜌𝜌 𝐲 𝑘 ( 𝑟, 𝑡 ) d 𝑟 𝐲 𝑘 ( 𝜌, 𝑡 ) = 𝐲 𝑘 ( 𝜌 , 𝑡 ) + ∫ 𝜌𝜌 𝑄 ( 𝐣 ( 𝑟, 𝑟 ) , 𝐲 ( 𝑟, 𝑟 )) 𝐣 𝑘 ( 𝑟, 𝑡 ) d 𝑟. (53)For small enough step size this will converge; the proof essentially shows that thecorresponding integral operator is a contraction for small enough step size 𝛿 > .As a consistency check, one needs to enforce 𝐣 ( 𝑟, 𝑟 ) = 0 . Once a solution hasbeen found, the curvature operator 𝑄 ( 𝐣 ( 𝑟, 𝑟 ) , 𝐲 ( 𝑟, 𝑟 )) is uniquely determined for 𝑟 ∈ [ 𝜌 − 𝛿, 𝜌 ] . This can be used to propagate the Jacobi fields for other valuesof 𝑡 ∈ 𝐼 through the area 𝑟 ∈ [ 𝜌 − 𝛿, 𝜌 ] . This extends the solution from ( 𝑟, 𝑡 ) ∈[ 𝜌 ,
0] × 𝐼 to ( 𝑟, 𝑡 ) ∈ [ 𝜌 − 𝛿,
0] × 𝐼 .2.3. Proof of the main theorem.
The bulk of the proof of our main result wasdone in lemma 11, and it remains to put the pieces together.
Proof of theorem 5.
Let 𝜀 > be such that [− 𝜀, 𝜀 ] ⊂ 𝐼 and 𝛾 ([− 𝜀, 𝜀 ]) ⊂ 𝑈 . Letus write 𝐼 = (− 𝜀, 𝜀 ) and 𝐼 = 𝐼 ∩ (−∞ , 𝜀 ) .Up to identifying with the diffeomorphism 𝜓 , the sphere data on the two mani-folds agree. With this identification, Σ ( 𝑟, 𝑡 ) = Σ ( 𝑟, 𝑡 ) whenever 𝑟 ∈ 𝐼 and 𝑡 ∈ 𝐼 with 𝑡 < 𝑟 . As the two surfaces Σ 𝑖 ( 𝑟, 𝑡 ) coincide, so do their shape operators 𝑆 𝑖 ( 𝑟, 𝑡 ) .This holds whenever 𝑡 < 𝑟 .When 𝑡 > 𝑟 and both times are in 𝐼 , the spheres have “negative radius”. Thissimply means that in the definition of spheres (see definition 2) the geodesics arefollowed backwards, corresponding to the exponential map of the reversed Finslerfunction ⃖𝐹 . The relevant parts of these spheres are contained in 𝑈 𝑖 by the choice of 𝜀 > , and the spheres agree simply because 𝜓 is isometric.Therefore we have 𝑆 ( 𝑟, 𝑡 ) = 𝑆 ( 𝑟, 𝑡 ) whenever 𝑟 ∈ 𝐼 , 𝑡 ∈ 𝐼 , and the shapeoperators are defined. This allows us to use lemma 11. It immediately shows that 𝑅 ( 𝑡 ) = 𝑅 ( 𝑡 ) .The operator ̂𝑈 ( 𝑡, 𝑠 ) is fully determined by ̂𝑅 ( 𝑡 ) through the Jacobi equation, sothe equality of the solution operators follows from the equality of the curvatureoperators. (cid:3) Remark . The last step of the uniqueness proof was backwards in light of theproof of lemma 11. The Jacobi fields 𝐽 ( 𝑟, 𝑡 ) were the main object whose uniquenesswas established and everything else was derived from them. Indeed, one couldextract 𝑈 ( 𝑡, 𝑠 ) from 𝐽 ( 𝑡, 𝑠 ) , but this is a detour in comparison to arguing by theJacobi equation as we did. The lemma gives 𝐽 ( 𝑟, 𝑡 ) = 𝐽 ( 𝑟, 𝑡 ) 𝐴 ( 𝑡 ) for some smoothpointwise invertible matrix-valued function 𝐴 ∶ 𝐼 → ℝ ( 𝑛 −1)×( 𝑛 −1) . ECONSTRUCTION ALONG A GEODESIC FROM SPHERE DATA 18
3. S
URFACE NORMAL COORDINATES
Let Σ ⊂ 𝑀 be a smooth oriented submanifold of codimension , and let 𝜈 ( 𝑥 ) be the positively oriented unit normal vector to Σ at 𝑥 ∈ Σ . We define the surfacenormal exponential map for Σ as 𝜑 ∶ Σ × ℝ → 𝑀,𝜑 ( 𝑥, 𝑡 ) = 𝛾 𝑥,𝜈 ( 𝑥 ) ( 𝑡 ) , (54)where 𝛾 𝑥,𝑣 is the unique geodesic with 𝛾 𝑥,𝑣 (0) = 𝑥 and ̇𝛾 𝑥,𝑣 (0) = 𝑣 . This map isillustrated in figure 2. The surface normal exponential map of (54) might not bedefined for all 𝑡 ∈ ℝ if the manifold is not geodesically complete.Note that typically 𝛾 𝑥,𝜈 ( 𝑥 ) ( 𝑡 ) ≠ exp 𝑥 ( 𝑡𝜈 ( 𝑡 )) when 𝑡 < if the Finsler geometryis not reversible. This map 𝜑 will be relevant for 𝑡 < , as we follow the arrivinggeodesics backwards from the spheres in the data towards the sources.The point 𝜑 ( 𝑥, 𝑡 ) ∈ 𝑀 is said to be a focal point to Σ if the differential d 𝜑 ( 𝑥, 𝑡 ) is not bijective. Therefore every non-focal point has a neighborhood where 𝜑 is adiffeomorphism and thus gives coordinates on 𝑀 in terms of coordinates on Σ . Wecall these the surface normal coordinates associated with Σ . These coordinates areparticularly natural for our problem, as the manifold 𝑀 is mostly unknown and thesurfaces are given as the data.Transforming coordinates on the sphere Σ in the data into coordinates on 𝑀 wasdescribed in more detail in (15). Lemma 15.
Let Σ ⊂ 𝑀 be a smooth oriented submanifold of codimension on ageodesically complete Finsler manifold 𝑀 . Define the surface normal exponentialmap 𝜑 ∶ Σ × ℝ → 𝑀 as in (54) . This map is smooth.Take any 𝑥 ∈ Σ and 𝑡 ∈ ℝ , and any 𝑣 ∈ 𝑇 𝑥 Σ and 𝜏 ∈ ℝ = 𝕋 𝑡 ℝ . Then [d 𝜑 ( 𝑥, 𝑡 )]( 𝑣, 𝜏 ) = 𝐽 ( 𝑡 ) , (55) where 𝐽 is the Jacobi field along the geodesic 𝛾 𝑥,𝜈 ( 𝑥 ) with the initial conditions 𝐽 (0) = 𝑣 + 𝜏𝜈 ( 𝑥 ) and 𝐷 𝑡 𝐽 (0) = 𝑆𝑣, (56) where 𝑆 is the shape operator of Σ .Proof. Smoothness follows from the smoothness of the submanifold and the geo-desic flow.As the direction of the relevant geodesic starting at 𝑥 normal to Σ is 𝜈 ( 𝑥 ) , theshape operator is a map 𝑆 ∶ 𝑇 𝑥 Σ → 𝑇 𝑥 Σ . Therefore the initial condition 𝐷 𝑡 𝐽 (0) given above has no component parallel to the geodesic 𝛾 ∶= 𝛾 𝑥,𝜈 ( 𝑥 ) .The derivative in the variable 𝑡 is trivial, as it corresponds to moving the endpointalong the geodesic 𝛾 . The claim holds for this directional derivative due to [d 𝜑 ( 𝑥, 𝑡 )](0 , 𝜏 ) = 𝜏 ̇𝛾 ( 𝑡 ) = 𝐽 ( 𝑡 ) , (57)where 𝐽 is the (constant!) Jacobi field with initial conditions 𝐽 (0) = 𝜏 ̇𝛾 (0) and 𝐷 𝑡 𝐽 (0) = 0 . We thus set 𝜏 = 0 and focus on differentiation in the variable 𝑥 .Let 𝜎 ∶ (− 𝜀, 𝜀 ) → Σ be a smooth curve so that 𝜎 (0) = 𝑥 and 𝜎 ′ (0) = 𝑣 . Now 𝛾 𝑠 = 𝛾 𝜎 ( 𝑠 ) ,𝜈 ( 𝜎 ( 𝑠 )) is a family of geodesics parametrized smoothly by 𝑠 ∈ (− 𝜀, 𝜀 ) . Thevariation of this family at 𝛾 = 𝛾 is the Jacobi field 𝐽 ( 𝑡 ) = 𝜕 𝑠 𝛾 𝑠 ( 𝑡 ) | 𝑠 =0 .Let us find the initial conditions of this Jacobi field. We have 𝐽 (0) = 𝜎 ′ (0) = 𝑣 .As all the geodesics in the family start normal to Σ , the covariant derivative is ECONSTRUCTION ALONG A GEODESIC FROM SPHERE DATA 19 given by 𝐷 𝑡 𝐽 (0) = 𝑆𝐽 (0) — a more complicated case was covered in lemma 10.Therefore 𝐽 has the initial conditions as claimed (with 𝜏 = 0 ), and so [d 𝜑 ( 𝑥, 𝑡 )]( 𝑣,
0) = 𝜕 𝑠 𝜑 ( 𝜎 ( 𝑠 ) , 𝑡 ) | 𝑠 =0 = 𝐽 ( 𝑡 ) (58)as claimed. (cid:3) We are now ready to prove our second theorem.
Proof of theorem 6.
Consider a visible smooth sphere Σ ⊂ 𝑈 , a point 𝑥 ∈ Σ andlet 𝑆 be the shape operator of Σ at 𝑥 . By lemma 15 the surface normal exponentialmap 𝜑 has a bijective differential at ( 𝑥, 𝑡 ) if and only if the map 𝛽 ∶ 𝑣 → 𝐽 ( 𝑡 ) isbijective, where 𝑣 ∈ 𝑇 𝑥 Σ and 𝐽 is the Jacobi field along 𝛾 𝑥,𝜈 ( 𝑥 ) with the initialconditions 𝐽 (0) = 𝑣 and 𝐷 𝑡 𝐽 (0) = 𝑆𝑣 . (Both conditions are equivalent with 𝑡 notbeing a focal distance to Σ at 𝑥 by definition.) The local coordinate maps 𝛼 𝑖 ∶ Ω → Σ 𝑖 do not change these properties.As the shape operator and the solution operator to the Jacobi equation (with re-spect to a parallel frame) are determined by the sphere data due to theorem 5, thetwo maps 𝜑 𝛼 𝑖 for 𝑖 = 1 , have bijective differentials at the same points. This wasthe first claim.We will now drop the index 𝑖 and show that the sphere data determines the Rie-mannian metric tensor 𝑔 = 𝑔 𝐺 along the vector field 𝐺 in the surface normal coor-dinates given by 𝜑 𝛼 . To that end, pick any indices 𝑗, 𝑘 ∈ {1 , … , 𝑛 } .The last coordinate vector field is 𝜕 𝑥 𝑗 = 𝐺 . Thus if 𝑗 = 𝑘 = 𝑛 , then 𝑔 𝑗𝑘 = | 𝐺 | =1 . If 𝑘 < 𝑛 , the coordinate vector field 𝜕 𝑥 𝑘 coincides with the value of a Jacobifield 𝐽 𝑘 , for which 𝐽 𝑘 (0) is the 𝑘 th basis vector in the local coordinates 𝛼 on Σ and 𝐷 𝑡 𝐽 𝑘 (0) = 𝑆𝐽 𝑘 (0) . Here 𝑆 is again the shape operator on Σ , determined by thedata. This Jacobi field 𝐽 𝑘 ( 𝑡 ) is normal to the geodesic at all times and so 𝑔 𝑗𝑘 = 𝑔 ( 𝜕 𝑥 𝑗 , 𝜕 𝑥 𝑘 ) = 𝑔 ( 𝐺, 𝐽 𝑘 ) = 0 . (59)Here we let the points of evaluation implicit to reduce clutter.If 𝑗 and 𝑘 are both in {1 , … , 𝑛 − 1} , then the coordinate vector fields both cor-respond to Jacobi fields and 𝑔 𝑗𝑘 = 𝑔 ( 𝐽 𝑗 , 𝐽 𝑘 ) . (60)As the data determines the two Jacobi fields 𝐽 𝑗 and 𝐽 𝑘 in an orthonormal frame(theorem 5), we know the map 𝑡 ↦ 𝑔 ̇𝛾 ( 𝑡 ) ( 𝐽 𝑗 ( 𝑡 ) , 𝐽 𝑘 ( 𝑡 )) . Evaluation of this map givesthe components of the metric tensor in directions normal to the vector field 𝐺 .The metric tensor in the direction parallel to 𝐺 is determined more straightfor-wardly as we saw above. This completes the proof. (cid:3) Acknowledgements.
MVdH gratefully acknowledges the support from the SimonsFoundation under the MATH + X program, and the National Science Foundationunder grant DMS-1559587. JI was supported by the Academy of Finland (decisions295853, 332890, and 336254). Much of the work was completed during JI’s visitsto Rice University, and he is grateful for hospitality and support from the SimonsFoundation. ML was supported by Academy of Finland (decisions 320113, 303754,318990, and 312119). The authors thank Teemu Saksala for numerous remarks anddiscussions and Negar Erfanian for help with drawing figures 1 and 2.
ECONSTRUCTION ALONG A GEODESIC FROM SPHERE DATA 20 R EFERENCES [1] D. B AO , S.-S. C HERN AND
Z. S
HEN , An Introduction to Riemann–Finsler Geometry , GraduateTexts in Mathematics, 200, Springer-Verlag, New York, 2000.[2] S.-S. C
HERN AND
Z. S
HEN , Riemann–Finsler Geometry , Nankai Tracts in Mathematics, 6,World Scientific, Singapore, 2005.[3] C.H. D IX , Seismic velocities from surface measurements , Geophysics, 20 (1955), pp. 68–86.[4] M. V. DE H OOP , S. H
OLMAN , E. I
VERSEN , M. L
ASSAS , B. U
RSIN , Recovering the isom-etry type of a Riemannian manifold from local boundary diffraction travel times , Journal deMathématiques Pures et Appliquées, 103: 830–848, 2015.[5] M. V. DE H OOP , J. I
LMAVIRTA , M. L
ASSAS , T. S
AKSALA , Determination of a compact Finslermanifold from its boundary distance map and an inverse problem in elasticity . Preprint.[6] M. V. DE H OOP , J. I
LMAVIRTA , M. L
ASSAS , T. S
AKSALA , A foliated and reversible Finslermanifold is determined by its broken scattering relation . Preprint.[7] M. V. DE H OOP , J. I
LMAVIRTA , M. L
ASSAS , T. S
AKSALA , Stable reconstruction of simpleRiemannian manifolds from unknown interior sources . In preparation.[8] M. V. DE H OOP , G. N
AKAMURA , J. Z
HAI , Unique Recovery of Piecewise Analytic Densityand Stiffness Tensor from the Elastic-Wave Dirichlet-To-Neumann Map , SIAM J. Appl. Math.,79(6): 2359–2384, 2019.[9] M. V. DE H OOP , G. U
HLMANN , A. V
ASY , Recovery of material parameters in transverselyisotropic media.
Preprint.[10] V. G
RECHKA , I. T
SVANKIN , J. K. C
OHEN , Generalized Dix equation and analytic treatment ofnormal-moveout velocity for anisotropic media , Geophysical Prospecting, 47: 117–148, 1999.[11] M. I
KEHATA AND
H. I
TOU , An inverse problem for a linear crack in an anisotropic elastic bodyand the enclosure method , Inverse Problems, 24(2): 025005, 2008.[12] M. I
KEHATA , G. N
AKAMURA , K. T
ANUMA , Identification of the shape of the inclusion in theanisotropic elastic body , Applicable Analysis, 72(1–2): 17–26, 1999.[13] G. N
AKAMURA , K. T
ANUMA , G. U
HLMANN , Layer Stripping for a Transversely IsotropicElastic Medium , SIAM J. Appl. Math., 59(5): 1879–1891, 2015.[14] G. N
AKAMURA , G. U
HLMANN , Global uniqueness for an inverse boundary problem arising inelasticity , Inventiones mathematicae, 118: 457–474, 1994.[15] G. P
ATERNAIN , M. S
ALO , G. U
HLMANN , H. Z
HOU , The geodesic X-ray transform with matrixweights.
Preprint.[16] V. S
HARAFUTDINOV , J. W
ANG , Tomography of small residual stresses , Inverse Problems, 28:065017, 2012.[17] Z. S