The Geodesic Ray Transform on Spherically Symmetric Reversible Finsler Manifolds
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This is a self-archived version of an original article. This version may differ from the original in pagination and typographic details. Author(s): Title: Year: Version: Copyright: Rights: Rights url: Please cite the original version: CC BY 4.0 https://creativecommons.org/licenses/by/4.0/ The Geodesic Ray Transform on Spherically Symmetric Reversible Finsler Manifolds © The Author(s) 2023 Published version Ilmavirta, Joonas; Mönkkönen, Keijo Ilmavirta, J., & Mönkkönen, K. (2023). The Geodesic Ray Transform on Spherically Symmetric Reversible Finsler Manifolds. Journal of Geometric Analysis, 33(4), Article 137. https://doi.org/10.1007/s12220-022-01182-w 2023
The Journal of Geometric Analysis (2023) 33:137 https://doi.org/10.1007/s12220-022-01182-w The Geodesic Ray Transform on Spherically Symmetric Reversible Finsler Manifolds Joonas Ilmavirta1·Keijo Mönkkönen1 Received: 31 March 2022 / Accepted: 21 December 2022 © The Author(s) 2023 Abstract We show that the geodesic ray transform is injective on scalar functions on spherically symmetric reversible Finsler manifolds where the Finsler norm satisfies a Herglotz condition. We use angular Fourier series to reduce the injectivity problem to the invertibility of generalized Abel transforms and by Taylor expansions of geodesics we show that these Abel transforms are injective. Our result has applications in linearized boundary rigidity problem on Finsler manifolds and especially in linearized elastic travel time tomography. Keywords Inverse problems ·Geodesic ray transform ·Integral geometry Mathematics Subject Classification 44A12 ·53A99 ·86A22 1 Introduction In this paper, we study the following mathematical inverse problem arising in integral geometry: If we know the integrals of a scalar function fover all geodesics of a Finsler manifold (M,F), can we determine f? Since the problem is linear, we can formulate it in terms of the kernel of the geodesic ray transform I:IfIf(γ ) =0 for all geodesics γ, does it follow that f=0? In other words, is the geodesic ray transform Iinjective on scalar fields? This inverse problem (and its generalization to tensor fields) has been usually studied on Riemannian manifolds and a variety of results under different types of assumptions is known in the Riemannian setting [28, 42,47]. We show that in the case of spherical symmetry, reversibility and a Herglotz BJoonas Ilmavirta [email protected] Keijo Mönkkönen [email protected] 1Department of Mathematics and Statistics, University of Jyväskylä, P.O. Box 35 (MaD), 40014 Jyväskylä, Finland 0123456789().: V,-vol 123
137 Page 2 of 27 J. Ilmavirta, K. Mönkkönen condition the answer is positive for Finsler manifolds as well: Iis injective on scalar fields. A Finsler norm Fis a non-negative function F:TM →[0,∞)such that for every x∈Mthe map y→ F(x,y)defines a Minkowski norm in TxM(see Sect. 2.1 for details). We focus on spherically symmetric and reversible Finsler norms. We show that if M⊂Rnis an annulus centered at the origin and Fis a spherically symmetric reversible Finsler norm on Mwhich satisfies the Herglotz condition [see Eq. (1) and Sect. 2.2], then the geodesic ray transform Iis injective on L2-functions (see Sect. 1.1 and Theorem 1.1). This generalizes earlier Riemannian results in [14, 46] to the Finslerian case and our theorem can be seen as a Helgason-type support theorem on Finsler manifolds (see, e.g. [14,22]). An example of a non-Riemannian geometrywhereourmaintheoremappliesis a Finsler norm arising from an anisotropic sound speed which is reversible and spherically symmetric and satisfies the Herglotz condition (see Sect. 2.2). We use angular Fourier series to reduce the inverse problem to the invertibility of certain Abel-type integral transforms. This approach was used in [14] where the authorsprovedvariousinjectivityresultsofgeneralized Abeltransformswhich wealso use in the proof of our main result. By a careful treatment of the Taylor expansions of geodesics near their lowest point to the origin we show that the Abel transforms we encounter are indeed injective. Our result is related to the travel time tomography or the boundary rigidity problem. Travel time tomography is an imaging method used in seismology where one wants to determine the speed of sound inside the Earth by measuring travel times of seismic waves on the surface of the Earth [55]. The ray paths correspond to geodesics and travel times to lengths of geodesics. The boundary rigidity problem is a more general geometric inverse problem where one wants to determine a Riemannian metric or more generally a Finsler norm from the distances between boundary points [55]. The travel time tomography problem was already solved in the 1900s for radial sound speeds satisfying the Herglotz condition [23,57]. However, it is observed that the Earth exhibits more complicated and especially anisotropic behavior with respect to the sound speed [11,20,51]. In the anisotropic case, seismic rays propagate along geodesics of a Finsler norm [4,58] and Riemannian geometry is not enough to describe the most general types of anisotropies. The boundary rigidity problem is already a difficult non-linear inverse problem in the Riemannian case and anisotropies complicate things even more. It is known that Finsler norms arising in elasticity are reversible [13] (see also Sect. 5.3) which puts some constraints on the geometry. Invariance under rotations is a natural physical requirement for the Finsler norm (or sound speed) since the Earth is (roughly) spherically symmetric. Our Herglotz condition (1) is a natural generalization of the usual Herglotz condition to anisotropic sound speeds [see Eq. (11)] and it implies that certain geodesics behave nicely (see Sect. 2.2). We can further simplify the problem by linearizing it. If the variations of the Riemannian metric or Finsler norm are conformal, then linearization of the boundary rigidity problem leads to the geodesicray transformof scalarfunctionson thebase manifold(see[47] andSect.5.2). This especially holds for a family of conformal Finsler norms induced by a conformal family of stiffness tensors. 123
The Geodesic Ray Transform on Reversible Finsler Manifolds Page 3 of 27 137 Our main theorem implies boundary rigidity up to first order for a conformal family of spherically symmetric reversible Finsler norms satisfying the Herglotz condition. In terms of elasticity, if we have a conformal family of stiffness tensors (a family of factorized anisotropic inhomogeneous media [9,58]) such that the induced family of Finsler norms give the same distances between boundary points and satisfy the assumptions of Theorem 1.1, then the stiffness tensors are equal up to first order (see Sect. 5.3). 1.1 The Main Theorem Let us first quickly introduce the key definitions and notation. More details can be found in Sect. 2. Let Mbe a smooth manifold. A Finsler norm F:TM →[0,∞)is a non-negative function on the tangent bundle TM so that the map y→ F(x,y)is a positively homogeneous (but not necessarily homogeneous) norm in the tangent space TxMfor each x∈M. Finsler norm Fis reversible if F(x,−y)=F(x,y)for all x∈Mand y∈TxM. A reversible Finsler norm defines a homogeneous norm in TxM. The length of a curve γ:[a,b]→Mis defined as L(γ ) =b aF(γ (t), ˙γ(t))dt. The geodesics of a Finsler norm are critical points of the length functional γ→ L(γ ), or equivalently they satisfy the geodesic equation [see Eq, (4)]. Our manifold will eventually be an annulus M⊂R2with outer boundary centered at the origin. The inner boundary of the annulus is not included in Mso ∂Monly consists of the outer boundary. We say that a Finsler norm Fon Mis spherically symmetric, if U∗F=Ffor all U∈SO(2).Let(x1,x2)=(r,θ) be the polar coordinateson M.Thesecoordinatesinduceacoordinatebasis{∂r,∂ θ}ineverytangent space T(r,θ) M.Ify=y1∂r+y2∂θ∈T(r,θ) M, then we denote its coordinates by (y1,y2)=(ρ, φ); see Fig. 1. We can then equivalently say that the Finsler norm F on Mis spherically symmetric if it is independent of the angular variable θ, i.e. F=F(r,ρ,φ). We say that a spherically symmetric reversible Finsler norm F=F(r,ρ,φ)on M satisfies the Herglotz condition, if ∂rF2(r,0,φ)>0(1) for all r∈(R,1]and φ= 0. Here R∈(0,1)is the inner radius of the annulus. The Herglotz condition implies that (M,F)admits a strictly convex foliation and geodesics, which are initially tangential to circles, reach the outer boundary in finite time (see Lemmas 2.1 and 2.2). More generally, we say that a spherically symmetric reversible Finsler norm Fon an n-dimensional annulus M⊂Rnsatisfies the Herglotz condition if Fsatisfies the two-dimensional Herglotz condition (1) on all slices M∩P where P⊂Rnis a two-dimensional subspace. The geodesic ray transform Itakes a sufficiently regular scalar field fon M and integrates it over geodesics, i.e. If(γ ) =γfdswhere γis a geodesic of the Finsler norm F. The Herglotz condition guarantees that geodesics which are initially tangential to circles have a unique closest point to the origin and the integrals exist for such geodesics. 123
137 Page 4 of 27 J. Ilmavirta, K. Mönkkönen Fig. 1 Our manifold Mand the coordinate system (r,θ,ρ,φ) on TM. The coordinate vector fields ∂r and ∂θform a basis in each tangent space T(r,θ) Mand the coordinates of y∈T(r,θ) Mwith respect to this basis are (ρ, φ). The components are given by ρ=dr(y)and φ=dθ(y). We use polar coordinates only on M; the induced coordinates on T(r,θ) Mare Euclidean. The inner boundary (dashed) is not included in M Our main theorem is the following injectivity result. The proof of the theorem can be found in Sect. 3. Theorem 1.1 Let n ≥2,M=¯ B(0,1)\¯ B(0,R)⊂Rnwhere R ∈(0,1)and equip M with a smooth spherically symmetric reversible Finsler norm F which satisfies the Herglotz condition. Then the geodesic ray transform Iis injective on L2(M). Remark 1.2 It is enough to prove Theorem 1.1 in two dimensions. Namely, if we intersect a higher-dimensional annulus M⊂Rnwith any two-dimensional linear subspace P⊂Rn, we get a totally geodesic submanifold M∩P⊂Msince F is reversible and spherically symmetric. Also, by [14, Lemma 17] it holds that if f∈L2(M), then f|M∩P∈L2(M∩P)for almost every two-dimensional plane P. Hence if Theorem 1.1 is true for n=2, then it is also true for all n≥2. Remark 1.3 We assume that the Finsler norm Fin Theorem 1.1 is smooth. This regularity assumption could be weakened: From the proof of Theorem 1.1 one sees that a finite number of derivatives with respect to the variables x∈Mand y∈TxM is enough. However, we are not going to quantify or optimize the needed regularity assumptions in this paper. In Theorem 1.1 we assume that the Finsler norm is reversible since it simplifies the proof and Finsler norms arising in elasticity are reversible. Our main application 123
The Geodesic Ray Transform on Reversible Finsler Manifolds Page 5 of 27 137 of Theorem 1.1 is the seismic imaging of the Earth and therefore we let Mto be an annulus and Fto be spherically symmetric. We could formulate Theorem 1.1 in terms of a general family of curves satisfying certain properties (see Remark 1.4). In fact, in the proof of Theorem 1.1 we only use integrals of fover geodesics which have a unique lowest point to the origin. Due to the Herglotz condition geodesics cannot have more than one point where the radial speed ˙rvanishes. However, we do not know whether our manifold is non-trapping, i.e. we do not know if all geodesics reach the boundary in finite time or if there exists trapped geodesics. Theorem 1.1 is proved in the following way (see Sects. 2,3and 4for more details). Since our manifold is annulus we can express any L2-function fas an angular Fourier series. Using this and the reversibility of Fthe geodesic ray transform of fcan be written as a sum of generalized Abel transforms acting on the Fourier components of f. By the Taylor expansions of geodesics and careful treatment of the error terms we show that these Abel transforms are injective. From this, it follows that the Fourier components of fall vanish giving the claim. Theorem 1.1 can be seen as a generalization of the corresponding Riemannian result in [14](seealso[46]) and the proof is similar in spirit. In fact, we use the theory of Abel transforms introduced in [14] to prove our result. However, many formulas which were explicit in [14] become implicit and less tractable in our Finslerian case. For this reason,we use the Taylor expansions of geodesics near their lowest point to show the needed regularity properties of the integral kernels of the Abel transforms (see Sect. 4). Remark 1.4 We could express Theorem 1.1 in terms of a more general family of curves than geodesics. From the proof of our main theorem, one sees that the curves only needtobe“sufficientlysmooth” with respect to the Taylor expansionsand“sufficiently symmetric” with respect to the Finsler norm. The family of curves can be characterized by the following properties (compare to the assumptions in [5]): (A1) All the curves in the family are smooth with unit speed. (A2) For every x∈Mand y∈TxMthere is unique curve going through xto the direction y. (A3) The curves depend smoothly on the initial conditions xand y. (A4) Every curve reaches the boundary in finite time and has unique closest point to the origin where ˙r0=0 and ¨r0>0. (A5) The curves are symmetric with respect to the lowest point and they consist of two parts where ˙r>0 and ˙r<0. (A6) The curves satisfy the weak reversibility condition (18). The assumptions (A1)–(A6) allow the existence of conjugate points on M:ifFis for example induced by the Riemannian metric g=c−2(r)ewhere c=c(r)is smooth and satisfies the Herglotz condition and eis the Euclidean metric, then the (non-radial) geodesics of gsatisfy conditions (A1)–(A6) (see, e.g. [32,34,43]). We also note that the regularity assumptions for the admissible curves could be weakened (finite number of derivatives is enough, see remark 1.3). Remark 1.5 Theorem 1.1 can also be seen as a generalization of the famous Helgason support theorem in Euclidean geometry [22](seealso[14, Remark 31]). According 123
137 Page 6 of 27 J. Ilmavirta, K. Mönkkönen to Helgason’s theorem, if a function integrates to zero on all lines not intersecting a given convex and compact set, then the function has to vanish outside that set. Since the Herglotz condition allows the presence of conjugate points (see [32]) and on Riemannian manifolds the existence of conjugate points implies instability for the geodesic ray transform [33], we do not expect stability for our injectivity or uniqueness result. Remark 1.6 By combining our approach with the ideas and methods of the proof of Theorem 29 in [14] we could also prove (with minor changes in the proof of Theorem 1.1) that the attenuated geodesic ray transform is injective on (sufficiently smooth) scalar fields on our manifold (M,F)when the attenuation is a sufficiently regular radial function. See [25,40] for results on attenuated transforms on manifolds. 1.2 Related Results Thegeodesicraytransform has been widely studied butmost of the results areobtained in the Riemannian setting. If (M,g)is a compact simple Riemannian manifold with boundary (and smooth metric), then the geodesic ray transform is known to be injective [37]. Recently it was proved in [26] that injectivity holds also when the simple Riemannian metric is only C1,1-regular. Injectivity is known in the presence of conjugate points as well: If the Riemannian metric is of the form g=c−2(r)ewhere eis the Euclidean metric and the radial sound speed c=c(r)satisfies the Herglotz condition, then Iis injective on scalar fields [14,32,46,56](seealso[43] and the generalization to tensor fields in [49]). In this case, the Herglotz condition is equivalent to that the manifold has a foliation with strictly convex hypersurfaces (see also Lemma 2.1). Our main theorem is also related to the Helgason support theorem in Euclidean space [22] (see [14, Remark 31]). Whenthegeodesicraytransformoperatesontensorfields,theuniquenessresultsare known as solenoidal injectivity since one can only uniquely determine the solenoidal part of the tensor field [42,47]. Solenoidal injectivity is known for example on twodimensional compact simple manifolds [41], on simply connected compact manifolds with strictly convex boundary and non-positive curvature [39,44,47], on certain noncompact Cartan–Hadamard manifolds [31] and on manifolds which admit strictly convex foliation [16,49,54,56]. A more comprehensive treatment of the geodesic ray transform on Riemannian manifolds can be found in [28,42,47]. Therearesome injectivityresultsintheFinslerian case. It isknown thatthegeodesic ray transform is injective on scalar fields on simple Finsler manifolds [29,48]. The geodesic ray transform is also injective on a certain family of curves on general Finsler surfaces [5]. This result extends to one forms as well when uniqueness is understood modulo potential fields. Compared to the results in [5,29,48] our theorem allows the existence of conjugate points (see, e.g. [32]). We also note that we could express our main theorem in terms of a family of general geodesic-like curves satisfying certain assumptions (see Remark 1.4 and compare to the assumptions in [5]). Other injectivity results for a general family of curves can be found in [21,24,36,50]. The geodesic ray transform arises naturally in the linearization of the travel time tomography or the boundary rigidity problem where one wants to uniquely determine 123
The Geodesic Ray Transform on Reversible Finsler Manifolds Page 7 of 27 137 (up to a gauge) the Riemannian metric (more generally a Finsler norm) from the distances between boundary points [47]. When we have conformal variations then the linearized problem reduces to the injectivity of the geodesic ray transform in the background geometry (see Sect. 5.2). The travel time tomography problem was solved over a century ago for radial sound speeds satisfying the Herglotz condition [23,57] (see also [43]). In this case, the solution of the problem reduces to the inversion of an Abel transform [38,51]. There are also recent spectral rigidity results for radial sound speeds which satisfy the Herglotz condition [19]. In the more general setting boundary rigidity is known for two-dimensional compactsimpleRiemanniansurfaces[45], formanifoldsadmittingstrictlyconvex foliation [53,55] and for compact simple Riemannian manifolds which are in the same conformal class [12,37,55]. There are some Finslerian results as well including Randers metrics [35], reversible Finsler norms which satisfy a strictly convex foliation [17] and projectively flat Finsler norms in the plane [2,3,30]. Our main result can be seen as a boundary rigidity result up to first order for a conformal family of spherically symmetric reversible Finsler norms satisfying the Herglotz condition (see Sect. 5.2). A survey of the boundary rigidity or the travel time tomography problem can be found in [55]. 1.3 Organization of the Paper In Sect. 2we go through basic definitions and properties of Finsler manifolds and Abel transforms and we study the Herglotz condition. We prove our main theorem in Sect. 3. In Sect. 4we prove the regularity properties of the integral kernel of the Abel transforms. Finally, in Sect. 5we discuss the linearization of the boundary rigidity problem on Finsler manifolds and the application of our result to linearized elastic travel time tomography. 2 Preliminaries In this section, we go through definitions, notation and lemmas which are needed in the proof of our main theorem. The basic theory of Finsler geometry can be found in [1,6,10,52] and the geodesic ray transform is treated in detail in [47]. Generalized Abel transforms are studied for example in [14,27]. 2.1 Finsler Manifolds Let Mbe a smooth manifold with or without a boundary. We use x∈Mto denote the base point and y∈TxMto denote the direction in the tangent space. A non-negative function F:TM →[0,∞)of the tangent bundle is called a Finsler norm if it satisfies the following conditions: (i) Fis smooth in TM\{0} (ii) F(x,y)=0 if and only if y=0 123
137 Page 8 of 27 J. Ilmavirta, K. Mönkkönen (iii) F(x,λy)=λF(x,y)for every λ≥0 (iv) 1 2∂2F2(x,y) ∂yi∂yjis positive definite for all y= 0. The pair (M,F)is called a Finsler manifold. In other words, the map y→ F(x,y) defines a Minkowski norm in TxMfor every x∈M. The length of a piecewise smooth curve γ:[a,b]→Mis defined as L(γ ) =b aF(γ (t), ˙γ(t))dt. In this way, a Finsler norm Fdefines a (not necessarily symmetric) distance function on M. Finsler norm Fis reversible, if F(x,−y)=F(x,y)for all x∈Mand y∈TxM. Riemannian metrics are a special case of reversible Finsler norms: If gis a Riemannian metric, then Fg(x,y)=gij(x)yiyjdefines a reversible Finsler norm where we have used the Einstein summation convention under the square root. A distance function inducedbyareversibleFinsler norm is symmetric. Not allFinsler norms are reversible: Examples include Randers metrics F=Fg+βwhere gis a Riemannian metric and β is a one-form. On the other hand, there are reversible Finsler norms which are not induced by any Riemannian metric. Using convexity property (iv) we can define the Finslerian metric tensor gij(x,y)=1 2 ∂2F2(x,y) ∂yi∂yj.(2) If F=Fgis induced by a Riemannian metric, then gij(x,y)=gij(x)is independent of y∈TxM. Using the Finslerian metric tensor one can define the Legendre transformation L:TM →T∗Mwhich in the Riemannian case corresponds to the musical isomorphisms. Legendre transformation allows us to define the co-Finsler norm (or dual norm) F∗:T∗M→[0,∞)so that for every ω∈T∗ xMwe have F∗(x,ω)=sup y∈TxM F(x,y)=1 ω(y). (3) Let γ:[a,b]→Mbe a smooth curve on M. We call γa geodesic if it is a critical point of the length functional γ→ L(γ ). Equivalently, we say that γis geodesic if it satisfies the geodesic equation ¨γi(t)+2Gi(γ (t), ˙γ(t)) =0(4) where Gi=Gi(x,y)are the spray coefficients defined as Gi(x,y)=1 4gil(x,y)yk∂2F2(x,y) ∂xk∂yl−∂F2(x,y) ∂xl.(5) Here gij(x,y)is the inverse matrix of gij(x,y)and we have used the Einstein summation convention. Geodesics correspond to straightest possible paths on a Finsler manifold and they minimize distances locally. It follows thatif Fis a reversible Finsler norm and γis a geodesic of F, then the reversed reparametrization ←− γ(t)=γ(−t)is also a geodesic of F. 123
The Geodesic Ray Transform on Reversible Finsler Manifolds Page 15 of 27 137 also implies that r(r0,t)=r(r0,−t)and θ(r0,t)=θ0−θ(r0,−t)(see Lemma 2.2). Differentiating this with respect to twe obtain ... r(r0,0)=0 ¨ θ(r0,0)=0.(18) Condition (18) can be called “weak reversibility” of a Finsler norm since it does not necessarily require reversibility. Using weak reversibility we write the Taylor expansion for the second derivative of the radial coordinate ¨r(r0,t)=a(r0)+E1(r0,t), E1(r0,t)=O(t2), where a(r0)=¨r(r0,0)>0 for all r0∈(R,1)by the Herglotz condition. Since ˙r(r0,0)=0 we can integrate the expansion for ¨r(r0,t)to obtain that ˙r(r0,t)=a(r0)t+E2(r0,t)with E2(r0,t)=O(t3)and r(r0,t)−r0=a(r0)t2 2+E3(r0,t)with E3(r0,t)=O(t4). Here f(t)=O(h(t)) means that |f(t)|≤M|h(t)|for small twhere M>0is constant. Note that the maps (r0,t)→ r(r0,t)and (r0,t)→ θ(r0,t)are smooth because of smooth dependence on initial conditions. Therefore a=a(r0)is bounded both from above and below by a positive constant. 4.2 Changing Between Time and Radial Coordinate From the Taylor expansion of r=r(r0,t)we get an important relation t2≈r−r0 for small t, i.e. there is constant C>0 such that 1 Ct2≤r−r0≤Ct2 when t(or equivalently r−r0) is small enough. Therefore when we write estimates using the Taylor expansions, it does not matter whether we express them in terms of t or r−r0. When we change between tand rwe need to know how the derivatives transform. We introduce the coordinates z=(x,t)=(r0,t) ˜z=(y,r)=(r0,r). Coordinate transformation z→˜zis well-defined because the Herglotz condition implies that ∂r/∂t>0 on the rising part of the geodesic and ∂r/∂t<0onthe 123
137 Page 16 of 27 J. Ilmavirta, K. Mönkkönen descending part (see Lemma 2.2). Using the chain rule we see that ∂ ∂y=∂ ∂˜z1=∂ ∂x+∂t ∂y ∂ ∂t ∂ ∂r=∂ ∂˜z2=∂t ∂r ∂ ∂t. Notice that ∂t/∂r=(∂r/∂t)−1which can be seen for example by looking at the Jacobian matrices of the transformations z=(˜z)and ˜z=−1(z). 4.3 Proofs of the Lemmas The strategy to prove Lemmas 3.1 and 3.2 is the following. We use the Taylor expansions for the coordinate functions of geodesics to calculate the derivatives with respect to the variables r0and r. By a careful treatment of the error terms we show that both of the derivatives are bounded when t(or equivalently r−r0) is small. Using the mean value theorem we obtain that K=K(r0,r)and (r0,r)→ ω2(r0,r)are Lipschitz in a small neighborhood of the diagonal of R={(u1,u2)∈R2:R≤u1≤u2≤1}. We start by proving Lemma 3.1. Proof of Lemma 3.1 We write ˙r−1(r0,r):= (˙r(r0,r))−1etc. The leading order behavior of the kernel Kcan be seen by writing the expansion K(r0,r)=(r−r0)1/2˙r−1(r0,r)=a(r0) 2t+O(t3) 1 a(r0)t+O(t) =1 √2a(r0)+O(t2). From this, we easily see that Kis non-zero on the diagonal and bounded everywhere in R. We then focus on the derivative ∂rK(r0,r). Using the chain rule we get ∂rK(r0,r)=∂ ∂r(r−r0)1/2˙r−1=˙r−3(r−r0)−1/21 2˙r2−(r−r0)¨r. The Taylor expansions from Sect. 4.1 imply that 1 2˙r2=a2(r0) 2t2+O(t4) (r−r0)¨r=a2(r0) 2t2+O(t4) ˙r3=a3(r0)t3+O(t5). 123
The Geodesic Ray Transform on Reversible Finsler Manifolds Page 17 of 27 137 From the expression for ˙r3we obtain 1 ˙r3=1 a3(r0)t3+O(t3)=1 a3(r0)t3+O(t−1)=O(t−3). Thus finally we have ∂rK(r0,r)=O(t−3)·O(t−1)·O(t4)=O(1) whichimpliesthatthederivative∂rK(r0,r)is bounded when tis small,orequivalently when r−r0is small. The estimate for the derivative ∂r0K(r0,r)is a little bit trickier. We use the coordinates (x,t)=(r0,t)and (y,r)=(r0,r)introduced in Sect. 4.2. First we obtain that ∂yK(y,r)=∂ ∂y(r−y)1/2˙r−1=−(r−y)−1/2 2˙r−1−(r−y)1/2˙r−2∂˙r ∂y. We use the derivative transformations from Sect. 4.2 to see that ∂˙r ∂y=∂˙r ∂x+∂t ∂y ∂˙r ∂t=∂˙r ∂x+∂t ∂y¨r where ∂˙r ∂x=a(r0)t+∂xE2. The Taylor expansion for rallows us to write t(y,r)=2 a(y)((r−y)−E3(y,r)) which can be differentiated with respect to y ∂t ∂y=−a(y)(1+∂yE3)+a(y)((r−y)−E3) a2(y)t.(19) Therefore the derivative becomes ∂yK(y,r)=−(r−y)−1/2 2˙r−1−(r−y)1/2˙r−2(a(y)t+∂xE2) −a(y)(1+∂yE3)+a(y)((r−y)−E3) a2(y)t(a(y)+E1). We estimate the different terms separately. 123
137 Page 18 of 27 J. Ilmavirta, K. Mönkkönen First of all (r−y)1/2˙r−2=O(t)·O(t−2)=O(t−1). Now E2(x,0)=0 for all xand therefore ∂xE2(x,0)=0. Compactness and smooth dependence on initial conditions give us that ∂xE2is Lipschitz with respect to t. Thus ∂xE2(x,t)=O(t)and (r−y)1/2˙r−2·(a(y)t+∂xE2)=O(t−1)·O(t)=O(1). Similarly (r−y)=O(t2)and E3=O(t4)=O(t2)so (r−y)1/2˙r−2·a(y)((r−y)−E3) a2(y)t=O(t−1)·O(t)=O(1). Additionally E1=O(t2)and (r−y)1/2˙r−2·a(y)E1 a2(y)t=O(t−1)·O(t)=O(1). Next, we take a look at the term ∂yE3. From the transformation law ∂ ∂y=∂ ∂x+∂t ∂y ∂ ∂t we get ∂yE3=∂xE3−∂tE3a(y)+a(y)∂yE3+a(y)((r−y)−E3) a2(y)t ⇔∂yE3=a2(y)t∂xE3−∂tE3(a(y)+a(y)((r−y)−E3)) a2(y)t+a(y)∂tE3. Now ∂xE2=O(t)and Ei(x,t)=t 0 Ei−1(x,s)ds,i∈{2,3}. By smoothness we have ∂xE3(x,t)=t 0 ∂xE2(x,s)ds so ∂xE3=O(t2).Also∂tE3=E2=O(t3)and we have an estimate for the derivative ∂yE3=O(t3) a2(y)t+a(y)∂tE3=O(t2)(1+O(t2)) =O(t2). Thus (r−y)1/2˙r−2·a(y)∂yE3 a2(y)t=O(t−1)·O(t)=O(1). 123
The Geodesic Ray Transform on Reversible Finsler Manifolds Page 19 of 27 137 Finally, the y-derivative of the integral kernel is ∂yK(y,r)=−(r−y)−1/2 2˙r−1+(r−y)1/2˙r−21 t+O(1) =˙r−2(r−y)−1/2t−1−˙rt 2+(r−y)+O(1) =˙r−2(r−y)−1/2t−1−a(y)t2 2−tE2 2+a(y)t2 2+E3+O(1) =O(t−4)·O(t4)+O(1)=O(1). This implies that ∂r0K(r0,r)is bounded for small t, or equivalently for small r−r0. Let >0 be small enough so that both ∂rK(r0,r)and ∂r0K(r0,r)are bounded when r−r0<.Nowif(r0,r), (r0,r)∈ R={(u1,u2)∈R:u2−u1<}, then the mean value theorem implies that |K(r0,r)−K(r0,r)|≤|∇K(r0,r)||(r0,r)−(r0,r)| ≤M(∂r0K(r0,r)+|∂rK(r0,r)|)|(r0,r)−(r0,r)| ≤ M|(r0,r)−(r0,r)|. Here we used the fact that the point (r0,r)belongs to the segment connecting (r0,r) and (r0,r)sor−r0<since Ris a convex set. Therefore K=K(r0,r)is Lipschitz in a small neighborhood of the diagonal of R. Next, we prove Lemma 3.2. The proof is similar to the proof of Lemma 3.1. Proof of Lemma 3.2 Using fundamental theorem of calculus we can write ω(r0,t)=θ(r0,t)−θ0=t 0˙ θ(r0,s)ds=r r0˙ θ(r0,u)˙r−1(r0,u)du. Since ˙r−1(r0,t)=O(t−1)and ˙ θ(r0,t)=O(1)(by compactness and smooth dependence on initial conditions) we have by the chain rule ∂rω(r0,r)=˙ θ(r0,r)˙r−1(r0,t)=O(1)·O(t−1)=O(t−1). Using ˙r−1=O(t−1)and t−1≈(r−r0)−1/2for small twe obtain |ω(r0,r)|≤Mr r0|˙r−1(r0,u)|du≤ Mr r0 (u−r0)−1/2du= M(r−r0)1/2 for small r−r0which implies ω(r0,r)=O(t). Hence ∂rω2(r0,r)=2ω(r0,r)∂rω(r0,r)=O(t)·O(t−1)=O(1). Thus the derivative ∂rω2(r0,r)is bounded for small t(or for small r−r0). 123
137 Page 20 of 27 J. Ilmavirta, K. Mönkkönen For the derivative ∂r0ω(r0,r)we write ω(r0,r)=r r0˙ θ(r0,u)(u−r0)−1/2K(r0,u)du=r r0 (u2−r2 0)−1/2ϕ(r0,u)du where ϕ(r0,r)=˙ θ(r0,r)K(r0,r)(r+r0)1/2. The weak reversibility condition ¨ θ(r0,0)=0 implies that ¨ θ(r0,t)=t 0 ... θ(r0,s)ds so ¨ θ(r0,t)=O(t)since ... θ(r0,t)=O(1)by compactness and smooth dependence on initial conditions. The transformation rules for the derivatives imply that ∂r˙ θ(y,r)=˙r−1(y,r)¨ θ(y,r)=O(t−1)·O(t)=O(1) and ∂y˙ θ(y,r)=∂x˙ θ(x,t)+∂t ∂y¨ θ(y,r)=O(1)+O(t−1)·O(t)=O(1). Hereweusedthecalculationsfromtheproof of lemma 3.1 to deduce that ∂t ∂y=O(t−1) [(see Eq. (19)] and ∂x˙ θ(x,t)=O(1)by compactness and smooth dependence on initial conditions. Thus ˙ θ=˙ θ(r0,r)is Lipschitz for small r−r0. Because K= K(r0,r)and (r0,r)→ (r+r0)1/2are Lipschitz for small r−r0also the map ϕ=ϕ(r0,r)is Lipschitz for small r−r0(the terms in ϕare bounded). Therefore the derivatives ∂r0ϕand ∂rϕare bounded for small r−r0and ϕ=O(1). The derivative ∂r0ω(r0,r)becomes (see, e.g. [14, Proposition 15]) ∂r0ω(r0,r)=r r0 (u2−r2 0)−1/2∂r0ϕ(r0,u)−r0 u2ϕ(r0,u)+r0 u∂uϕ(r0,u)du −r0(r2−r2 0)−1/2ϕ(r0,r). The term inside the big parenthesis is O(1)because ϕand its derivatives are bounded and u≥r0>R>0. Hence the term coming from the integral is O(t). The latter term is O(t−1)and thus ∂r0ω(r0,r)=O(t−1). Finally, we have ∂r0ω2(r0,r)=2ω(r0,r)∂r0ω(r0,r)=O(t)·O(t−1)=O(1) whichimpliesthatthe derivative ∂r0ω2(r0,r)isboundedfor small t,orequivalently for small r−r0. The Lipschitz continuity of (r0,r)→ ω2(r0,r)in a small neighborhood of the diagonal of Rthen follows from the mean value theorem as in the proof of Lemma 3.1. 123
The Geodesic Ray Transform on Reversible Finsler Manifolds Page 21 of 27 137 5 Linearized Travel Time Tomography on Finsler Manifolds In this section, we consider the linearization of the boundary rigidity problem on Finsler manifolds: If two Finsler norms give the same distances between boundary points, are they equal up to a gauge? We show that the linearization of boundary distances for a general family of Finsler norms leads to the geodesic ray transform of a function on the sphere bundle SM. We also show that if the family of Finsler norms arises from conformal variations, then linearization leads to the geodesic ray transform of scalar fields on M. This implies that if the geodesic ray transform is injective on scalar fields, then we have boundary rigidity in the first-order approximation. The linearization of the boundary rigidity problem has been done earlier for Riemannian metrics for example in [50, Section 3.1]. We show that the Riemannian linearization result follows from the linearization of Finsler norms as a special case. 5.1 Linearization for General Finsler Norms Let Mbe a smooth compact manifold with boundary ∂M.Letx,x∈∂M,>0 and s∈(−, ). Assume that we have a family of curves γs:[0,T]→Msmoothly depending on sand connecting xto xsuch that each γsis a unit speed geodesic of a Finsler norm Fs. We denote by ˙γsthe derivative of γs=γs(t)with respect to t.Let dFs(x,x)be the length of the geodesic γswith respect to Fs, i.e. we assume that γs minimizes the distance from xto x. The derivative of dFs(x,x)with respect to the parameter sat zero is ∂dFs(x,x) ∂ss=0=T 0 ∂ ∂s(Fs(γs(t), ˙γs(t)))s=0 dt.(20) A calculation shows that ∂ ∂s(Fs(γs(t), ˙γs(t)))s=0=∂Fs(γ0(t), ˙γ0(t)) ∂ss=0+∂ ∂s(F0(γs(t), ˙γs(t)))s=0 and we obtain ∂dFs(x,x) ∂ss=0=T 0 ∂Fs(γ0(t), ˙γ0(t)) ∂ss=0 dt+∂ ∂sT 0 F0(γs(t), ˙γs(t))dts=0 . The second term vanishes since γ0is a geodesic of F0and hence a critical point of the length functional. Thus, we obtain ∂dFs(x,x) ∂ss=0=ISMh(γ0)(21) 123
137 Page 22 of 27 J. Ilmavirta, K. Mönkkönen where the function h:SM →Ris defined as h(x,y)=∂Fs(x,y) ∂ss=0 (22) and ISM is the geodesic ray transform on the sphere bundle SM. If Fs(x,y)=gs ij(x)yiyjwhere gs=gs ij(x)is a family of Riemannian metrics, then ∂Fs(γ0(t), ˙γ0(t)) ∂ss=0=1 2g0 ij(γ0(t)) ˙γi 0(t)˙γj 0(t) ∂gs ij(γ0(t)) ∂ss=0˙γi 0(t)˙γj 0(t) =1 2 ∂gs ij(γ0(t)) ∂ss=0˙γi 0(t)˙γj 0(t) where we used the fact that γ0is a unit speed geodesic of g0. Hence ∂dgs(x,x) ∂ss=0=I2h(γ0)(23) where the components of the 2-tensor field h=hij(x)are hij(x)=1 2 ∂gs ij(x) ∂ss=0 (24) and I2is the geodesic ray transform of 2-tensor fields. If there are no constraints on the family Fsof Finsler geometries, then any smooth function h:SM →Rcan be realized as a variation in the sense of Eq. (22). Therefore the linearized problem in general Finsler geometry is that of finding the kernel of ISM. This kernel characterization in the Finsler setting is surprisingly simple: ISMh=0if and only if h=Xu for a smooth function u:SM →Rwith u|∂SM =0, where X is the geodesic vector field. The claim can be proved by defining uto be the integral of hover forward geodesics. Constraints on hinduce constraints on the potential uas is the case in Riemannian linearizations and tomography of 2-tensors. If one studies the Riemannian version of linearized travel time tomography, then the deformation has the special form h(x,y)=hij(x)yiyjwith coefficients as in (24). This structure of h:SM →R(rank two tensor field) implies a special structure for u:SM →R(rank one tensor field), and it is proving this structural implication that makes the Riemannian problem hard in comparison to the general Finslerian one. See [28,42,47] for Riemannian results. If one studies the linearized problem in the family of Finsler metrics arising from elasticity (see Sect. 5.3), the difficulty returns: One needs a structural characterization of possible variation fields hand the corresponding structure for the potential u. Neither of these structures is known in the general elastic setting. 123
The Geodesic Ray Transform on Reversible Finsler Manifolds Page 23 of 27 137 5.2 Linearization for Conformal Variations Let us consider the case Fs(x,y)=cs(x)F0(x,y)where cs=cs(x)is a family of positive functions on Msuch that c0≡1 and F0is some fixed Finsler norm. Now ∂Fs(γ0(t), ˙γ0(t)) ∂ss=0=∂cs(γ0(t)) ∂ss=0 (25) where we used the fact that F0(γ0(t), ˙γ0(t)) =1 since γ0is a unit speed geodesic of F0. We obtain ∂dFs(x,x) ∂ss=0=If(γ0)(26) where the function f:M→Ris defined as f(x)=∂cs(x) ∂ss=0 .(27) Hence the linearization of boundary distances of conformal family of Finsler norms leads to the geodesic ray transform of scalar fields on the Finsler manifold (M,F0). If all the geodesics γsgive the same boundary distances dFs(x,x), then the derivative with respect to svanishes and If(γ0)=0 (28) for all geodesics γ0of F0connecting two points on the boundary. If the geodesic ray transform is injective on (M,F0), then to first order in swe have Fs≈F0+s·∂Fs ∂ss=0=F0(29) since the derivative satisfies ∂Fs(x,y) ∂ss=0=∂cs(x) ∂ss=0 F0(x,y)=f(x)F0(x,y)=0(30) where we used the fact that f=0 whenever Iis injective on (M,F0). In this case, we have boundary rigidity up to first order in the parameter s. 5.3 Conformally Linearized Elastic Travel Time Tomography Next, we consider the travel time tomography problem in R3arising in elasticity. Basic theory of elasticity can be found for example in [9,51]. The stiffness tensor 123
137 Page 24 of 27 J. Ilmavirta, K. Mönkkönen cijkl =cijkl(x)describes the elastic properties of a given material. The stiffness tensor has the symmetries cijkl =cjikl =ckli j .(31) The density-normalized elastic modulus is aijkl(x)=cijkl(x) ρ(x)(32) where ρ=ρ(x)is the density of the material. If pis the momentum covector, then the Christoffel matrix is il(x,p)= j,kaijkl(x)pjpk. The Christoffel matrix is symmetric and we also assume that it is positive definite so it has three positive eigenvalues λi=λi(x,p)where i∈{1,2,3}. Let us assume that λ1>λ ifor i∈{2,3}.Itwasshownin[13] that √λ1(x,p)defines a co-Finsler norm in T∗R3. Using the Legendre transformation we obtain a Finsler norm in TR3. Assume that the stiffness tensor cijkl =cijkl(x)is fixed and consider the conformal variations cs ijkl(x)=fs(x)cijkl(x)where fs=fs(x)is a smooth family of positive functions such that f0≡1 (i.e. we have a family of “factorized anisotropic inhomogeneous media", see, e.g. [7–9,58]). The density-normalized elastic modulus becomes as ijkl(x)=cs ijkl(x) ρ(x)=fs(x)aijkl(x). (33) Thus, we have a family of Christoffel matrices s il(x,p)=fs(x)il(x,p). Since the eigenvalues only get scaled by fs, the largest eigenvalue corresponds to λs 1(x,p)= fs(x)λ1(x,p). We obtain a family of co-Finsler norms F∗ s(x,p)=√fs(x)F∗(x,p) where F∗is the co-Finsler norm corresponding to the stiffness tensor cijkl.Now as the Legendre transformation acts fiberwise, we obtain a family of Finsler norms Fs(x,y)=√fs(x)F(x,y)where Fis the Legendre transformation of F∗. We have shown that conformal variations of the stiffness tensor leads to conformal variations of the Finsler norm induced by the background stiffness tensor. If we consider the travel time tomography or the boundary rigidity problem for the family of induced Finsler norms Fs, then using the observations done in Sect. 5.2 we obtain that ∂fs(x) ∂ss=0=0(34) whenever the geodesic ray transform is injective on scalar fields on the base manifold (M,F). This in turn implies that the stiffness tensors cs ijkl =fscijkl all agree to first order in s, i.e., cs ijkl ≈cijkl +s·∂cs ijkl ∂ss=0=cijkl.(35) 123