Inverting the local geodesic X-ray transform on tensors
We prove the local invertibility, up to potential fields, and stability of the geodesic X-ray transform on tensor fields of order and near a strictly convex boundary point, on manifolds with boundary of dimension . We also present an inversion formula. Under the condition that the manifold can be foliated with a continuous family of strictly convex surfaces, we prove a global result which also implies a lens rigidity result near such a metric. The class of manifolds satisfying the foliation condition includes manifolds with no focal points, and does not exclude existence of conjugate points.
1991 Mathematics Subject Classification:53C65, 35R30, 35S05, 53C21
Let be a compact Riemannian manifold with boundary. The X-ray transform of symmetric covector fields of order is given by
where, in local coordinates, , and runs over all (finite length) geodesics with endpoints on . When , we integrate functions; when , is a covector field, in local coordinates, ; when , is a symmetric 2-tensor field , etc. The problem is of interest by itself but it also appears as a linearization of boundary and lens rigidity problems, see, e.g., [19, 18, 23, 24, 26, 6, 5, 4]. Indeed, when , can be interpreted as the infinitesimal difference of two conformal factors, and when , can be thought of as an infinitesimal difference of two metrics. The problem arises as a linearization of recovery a velocity fields from the time of fly. The problem appears in linearized elasticity.
The problem we study is the invertibility of . It is well known that potential vector fields, i.e., which are a symmetric differential of a symmetric field of order vanishing on (when ), are in the kernel of . When , there are no potential fields; when , potential fields are just ordinary differentials of functions vanishing at the boundary; for , potential fields are given by , with one form, on ; etc. The natural invertibility question is then whether implies that is potential; we call that property s-injectivity below.
This problem has been studied extensively for simple manifolds, i.e., when is strictly convex and any two points are connected by a unique minimizing geodesic smoothly depending on the endpoints. For simple metrics, in case of functions (), uniqueness and a non-sharp stability estimate was established in [13, 12, 2] using the energy method initiated by Mukhometov, and for , in . Sharp stability follows from . The case is harder with less complete results and the one already contains all the difficulties. In two dimensions, uniqueness for simple metrics and has been proven in  following the boundary rigidity proof in . For any , this was done in .
In dimensions , the problem still remains open for . Under an explicit upper bound of the curvature, uniqueness and a non-sharp stability was proved by Sharafutdinov, see [18, 19] and the references there, using a suitable version of the energy method developed in . Convexity of is not essential for those kind of results and the curvature assumption can be replaced by an assumption stronger than requiring no conjugate points, see [20, 7]. This still does not answer the uniqueness question for metrics without conjugate points however. The first and the second author proved in [23, 24], using microlocal and analytic microlocal techniques, that for simple metrics, the problem is Fredholm (modulo potential fields) with a finitely dimensional smooth kernel. For analytic simple metrics, there is uniqueness; and in fact, the uniqueness extends to an open and dense set of simple metrics in , . Moreover, there is a sharp stability estimate for , where is some extension of , see . We study the case there for simplicity of the exposition but the methods extend to any .
The reason why is harder than the and the cases can be seen from the analysis in [23, 24]. When , the presence of the boundary is not essential — we can extend to a complete and just restrict to functions supported in a fixed compact set. When is an one-form (), we have to deal with non-uniqueness due to exact one-forms but then the symmetric differential is just the ordinary one . When , is an elliptic operator but recovery of from is not a local operator. One way to deal with the non-uniqueness due to potential fields is to project on solenoidal ones (orthogonal to the potential fields). This involves solving an elliptic boundary value problem and the presence of the boundary becomes an essential factor. The standard pseudo-differential calculus is not suited naturally to work on manifolds with boundary.
In , the first two authors study manifolds with possible conjugate points of dimension . The geodesic manifold (when it is a smooth manifold) has dimension which exceeds when . We restrict there to an open set of geodesics. Assuming that consists of geodesics without conjugate points so that the conormal bundle covers , we show uniqueness and stability for analytic metrics, and moreover for an open and dense set of such metrics. In this case, even though conjugate points are allowed, the analysis is done on the geodesics in assumed to have no such points.
A significant progress is done in the recent work , where the second and the third author prove the following local result: if is strictly convex at and , then , acting on functions (), known for all geodesics close enough to the tangent ones to at , determine near in a stable way. The new idea in  was to introduce an artificial boundary near cutting off a small part of including and to apply the scattering calculus in the new domain , treating the artificial boundary as infinity, see Figure 1. Then is small enough, then a suitable “filtered” backprojection operator is not only Fredholm, but also invertible. We use this idea in the present work, as well. The authors used this linear results in a recent work  to prove local boundary and lens rigidity near a convex boundary point.
The purpose of this paper is to invert the geodesic X-ray transform on one forms and symmetric 2-tensors ( and ) for near a strictly convex boundary point. We give a local recovery procedure for on suitable open sets from the knowledge of for -local geodesics , i.e. contained in with endpoints on . More precisely, there is an obstacle to the inversion explained above: one-forms or tensors which are potential, i.e. of the form , where is scalar or a one-form, vanishing at , have vanishing integrals along all the geodesics with endpoints there, so one may always add a potential (exact) form or a potential two-tensor to and obtain the same localized transform . Our result is thus the local recovery of from up to this gauge freedom; in a stable way. Further, under an additional global convex foliation assumption we also give a global counterpart to this result.
We now state our main results more concretely. Let be a local boundary defining function, so that in . It is convenient to also consider a manifold without boundary extending . First, as in , the main local result is obtained for sufficiently small regions , ; see Figure 1. Here is an ‘artificial boundary’ which is strictly concave as viewed from the region between it and the actual boundary ; this (rather than ) is the boundary that plays a role in the analysis below.
We set this up in the same way as in  by considering a function with strictly concave level sets from the super-level set side for levels , , and letting
(A convenient normalization is that there is a point such that and such that ; then one can take e.g. for small , which localizes in a lens shaped region near , or indeed which only localizes near .) Here the requirement on is, if we assume that is compact, that there is a continuous function such that and such that
i.e. as , is a thinner and thinner shell in terms of . As in , our constructions are uniform in for . We drop the subscript from , i.e. simply write , again as in , to avoid overburdening the notation.
A weaker version, in terms of function spaces, of the main local theorem, presented in Corollaries 4.17-4.18, is then the following. The notation here is that local spaces mean that the condition is satisfied on compact subsets of , i.e. the conclusions are not stated uniformly up to the artificial boundary (but are uniform up to the original boundary); this is due to our efforts to minimize the analytic and geometric background in the introduction. The dot denotes supported distributions in the sense of Hörmander relative to the actual boundary , i.e. distributions in (within the extension ) whose support lies in , i.e. for , this is the space.
(See Corollaries 4.17-4.18.) With as above, there is such that for , if then , where , while can be stably determined from restricted to -local geodesics in the following sense. There is a continuous map , where for , in , the norm of restricted to any compact subset of is controlled by the norm of restricted to the set of -local geodesics.
Replacing by , can be taken uniform in for in a compact set on which the strict concavity assumption on level sets of holds.
The uniqueness part of the theorem generalizes Helgason’s type of support theorems for tensors fields for analytic metrics [9, 10, 3]. In those works however, analyticity plays a crucial role and the proof is a form of a microlocal analytic continuation. In contrast, no analyticity is assumed here.
As in , this theorem can be applied in a manner to obtain a global conclusion. To state this, assume that is a globally defined function with level sets which are strictly concave from the super-level set for , with on the manifold with boundary . Then we have:
(See Theorem 4.19.) Suppose is compact. Then the geodesic X-ray transform is injective and stable modulo potentials on the restriction of one-forms and symmetric 2-tensors to in the following sense. For all there is such that can be stably recovered from in the sense that for and locally on , the norm of restricted to compact subsets of is controlled by the norm of on local geodesics.
This theorem, combined with Theorem 2 in  (with a minor change — the no-conjugate condition there is only needed to guarantee a stability estimate, and we have it in our situation), implies a local, in terms of a perturbation of the metric, lens rigidity uniqueness result near metric satisfying the foliation condition.
Manifolds satisfying the foliation condition include manifolds without focal points . Subdomains of with the metric , satisfying the Herglotz  and Wiechert and Zoeppritz  condition on satisfy it as well since then the Euclidean spheres form a strictly convex foliation. Conjugate points in that case may exist, and small perturbations of such metrics satisfy the condition, as well. We can also formulate semi-global results: if we can foliate with compact, then we can recover up to a potential field there in a stable way, with stability degenerating near . This can be considered as a linearized model of the seismology problem for anisotropic speeds of propagation. One such example is metrics (and close to them) for which holds for and . Then can be stably recovered for up to a potential field.
Similarly to our work , this paper, and its methods, will have applications to the boundary rigidity problem; in this case without the conformal class restriction. This paper is forthcoming.
The plan of the paper is the following. In Section 2 we sketch the idea of the proof, and state the main technical result. In Section 3 we show the ellipticity of the modified version of , modified by the addition of gauge terms. This essentially proves the main result if one can satisfy the gauge condition. In Section 4 we analyze the gauge condition and complete the proof of our main results.
2. The idea of the proof and the scattering algebra
We now explain the basic ideas of the paper.
The usual approach in dealing with the gauge freedom is to add a gauge condition, which typically, see e.g. the work of the first two authors , is of the solenoidal gauge condition form, , where is the adjoint of with respect to the Riemannian metric on . Notice that actually the particular choice of the adjoint is irrelevant; once one recovers in one gauge, one could always express it in terms of another gauge, e.g. in this case relative to a different Riemannian metric.
In order to motivate our gauge condition, we need to recall the method introduced by the last two authors in  to analyze the geodesic X-ray transform on functions: the underlying analysis strongly suggests the form the gauge condition should take.
As in  we consider an operator that integrates over geodesics in a small cone at each point, now multiplying with a one form or symmetric 2-tensor, in the direction of the geodesic, mapping (locally defined) functions on the space of geodesics to (locally defined) one forms or tensors. The choice of the operator, or more concretely the angle, plays a big role; we choose it to be comparable to the distance to the artificial boundary, . In this case ends up being in Melrose’s scattering pseudodifferential algebra, at least once conjugated by an exponential weight. (The effect of this weight is that we get exponentially weak estimates as we approach the artificial boundary.) The main analytic problem one faces then is that, corresponding to the gauge freedom mentioned above, is not elliptic, unlike in the scalar (function) setting.
Concretely is defined as follows. Near , one can use coordinates , with as before, coordinates on . Correspondingly, elements of can be written as . The unit speed geodesics which are close to being tangential to level sets of (with the tangential ones being given by ) through a point can be parameterized by say (with the actual unit speed being a positive multiple of this) where is unit length with respect to say a Euclidean metric. The concavity of the level sets of , as viewed from the super-level sets, means that is bounded below by a positive constant along geodesics in , as long as is small, which in turn means that, for sufficiently small , geodesics with indeed remain in (as long as they are in ). Thus, if is known along -local geodesics, it is known for geodesics in this range. As in  we use a smaller range because of analytic advantages, namely the ability work in the well-behaved scattering algebra. Thus, for smooth, even, non-negative, of compact support, to be specified, in the function case  considered the operator
where is a (locally, i.e. on , defined) function on the space of geodesics, here parameterized by . (In fact, had a factor only in , with another placed elsewhere; here we simply combine these, as was also done in [27, Section 3]. Also, the particular measure is irrelevant; any smooth positive multiple would work equally well.) In this paper, with still a locally defined function on the space of geodesics, for one-forms we consider the map
while for 2-tensors
so in the two cases maps into one-forms, resp. symmetric 2-cotensors, where is a scattering metric used to convert vectors into covectors — this is discussed in detail below.
Since it plays a crucial role even in the setup, by giving the bundles of which our tensors are sections of, as well as the gauge condition, we need to discuss scattering geometry and the scattering pseudodifferential algebra, introduced by Melrose in , at least briefly. There is a more thorough discussion in [28, Section 2], though the cotangent bundle, which is crucial here, is suppressed there. Briefly, the scattering pseudodifferential algebra on a manifold with boundary is the generalization of the standard pseudodifferential algebra given by quantizations of symbols , i.e. satisfying
for all multiindices in the same way that on a compact manifold without boundary , arises from (localized) pseudodifferential operators on via considering coordinate charts. More precisely, can be compactified to a ball , by gluing a sphere at infinity, with the gluing done via ‘reciprocal polar coordinates’; see [28, Section 2]. One then writes for the quantizations of the symbols (2.3). Then is defined by requiring that locally in coordinate charts, including charts intersecting with , the algebra arises from . (One also has to allow smooth Schwartz kernels on which are vanishing to infinite order at , in analogy with the smooth Schwartz kernels on .) Thus, while the compactification is extremely useful to package information, the reader should keep in mind that ultimately almost all of the analysis reduces to uniform analysis on . Since we are working with bundles, we also mention that scattering pseudodifferential operators acting on sections of vector bundles are defined via local trivializations, in which these operators are given by matrices of scalar scattering pseudodifferential operators (i.e. are given by the definition above if in addition these trivializations are made to be coordinate charts), up to the same smooth, infinite order vanishing at Schwartz kernels as in the scalar case.
Concretely, the compactification , away from , is just , where the identification with is just the ‘inverse polar coordinate’ map , with the standard radial variable. Then a straightforward computation shows that translation invariant vector fields on lift to the compactification (via this identification) to generate, over , the Lie algebra of vector fields, where on a manifold with boundary is the Lie algebra of smooth vector fields tangent to the boundary of . In general, if is a boundary defining function of , we let . Then contains , corresponding to the analogous inclusion on Euclidean space, and the vector fields in are essentially the elements of , after a slight generalization of coefficients (since above does not have an asymptotic expansion at infinity in , only symbolic estimates; the expansion would correspond to smoothness of the coefficients).
Now, a local basis for , in a coordinate chart , is
directly from the definition, i.e. means exactly that locally, on
This gives that elements of are exactly smooth sections of a vector bundle, , with local basis . In the case of , this simply means that one is using the local basis , , where the are local coordinates on the sphere. An equivalent global basis is just , , i.e. is a trivial bundle with this identification.
The dual bundle of correspondingly has a local basis , which in case of becomes , with local coordinates on the sphere. A global version is given by using the basis , with covectors written as ; thus ; this is exactly the same notation as in the description of the symbol class (2.3), i.e. one should think of this class as living on . Thus, smooth scattering one-forms on , i.e. sections of , are simply smooth one-forms on with an expansion at infinity. Similar statements apply to natural bundles, such as the higher degree differential forms , as well as symmetric tensors, such as . The latter give rise to scattering metrics , which are positive definite inner products on the fibers of (i.e. positive definite sections of ) of the form , a standard smooth 2-cotensor on (i.e. a section of ). For instance, one can take, in a product decomposition near , , a metric on the level sets of .
The principal symbol of a pseudodifferential operator is the equivalence class of as in (2.3) modulo , i.e. modulo additional decay both in and in on . In particular, full ellipticity is ellipticity in this sense, modulo , i.e. for a scalar operator lower bounds for , where is suitably large. This contrasts with (uniform) ellipticity in the standard sense, which is a similar lower bound, but only for . Fully elliptic operators are Fredholm between the appropriate Sobolev spaces corresponding to the scattering structure, see [28, Section 2]; full ellipticity is needed for this (as shown e.g. by taking on , the flat positive Laplacian). If is matrix valued, ellipticity can be stated as invertibility for large , together with upper bounds for the inverse: ; this coincides with the above definition for scalars.
We mention also that the exterior derivative for all . Explicitly, for , in local coordinates, this is the statement that
with , while are smooth sections of (locally, where this formula makes sense). Such a computation also shows that the principal symbol, in both senses, of , at any point , is wedge product with . A similar computation shows that the gradient with respect to a scattering metric is a scattering differential operator (on any of the natural bundles), with principal symbol given by tensor product with , hence so is the symmetric gradient on one forms, with principal symbol given by the symmetrized tensor product with . Note that all of these principal symbols are actually independent of the metric , and itself is completely independent of any choice of a metric (scattering or otherwise).
If we instead consider the symmetric differential with respect to a smooth metric on , as we are obliged to use in our problem since its image is what is annihilated by the (-geodesic) X-ray transform , it is a first order differential operator between sections of bundles and . Writing , resp., , and for the corresponding bases, this means that we have a matrix of first order differential operators. Now, as the standard principal symbol of is just tensoring with the covector at which the principal symbol is evaluated, the first order terms are the same, modulo zeroth order terms, as when one considers , and in particular they correspond to a scattering differential operator acting between section of and . (This can also be checked explicitly using the calculation done below for zeroth order term, but the above is the conceptual reason for this.) On the other hand, with , , etc., these zeroth order terms form a matrix with smooth coefficients in the local basis
of the homomorphism bundle . In terms of the local basis
of , these are all smooth, and vanish at to order respectively, showing that , and that the only non-trivial contribution of these zeroth order terms to the principal symbol is via the entry corresponding to , which however is rather arbitrary.
Returning to the choice of gauge, in our case the solenoidal gauge relative to would not be a good idea: the metric on is an incomplete metric as viewed at the artificial boundary, and does not interact well with . We circumvent this difficulty by considering instead the adjoint relative to a scattering metric, i.e. one of the form , a metric on the level sets of . While are then scattering differential operators, unfortunately on functions, or one forms, is not fully elliptic in the scattering sense (full ellipticity is needed to guarantee Fredholm properties on Sobolev spaces in a compact setting), with the problem being at finite points of , . For instance, in the case of being the radial compactification of , we would be trying to invert the Laplacian on functions or one-forms, which has issues at the 0-section. However, if we instead use an exponential weight, which already arose when was discussed, we can make the resulting operator fully elliptic, and indeed invertible for suitable weights.
Thus, we introduce a Witten-type (in the sense of the Witten Laplacian) solenoidal gauge on the scattering cotangent bundle, or its second symmetric power, . Fixing , our gauge is
or the -solenoidal gauge. (Keep in mind here that is the adjoint of relative to a scattering metric.) We are actually working with
throughout; in terms of this the gauge is
(See Theorem 4.15 for the proof and the formula.) There exists such that for the following holds.
For , small, the geodesic X-ray transform on -solenoidal one-forms and symmetric 2-tensors , i.e. ones satisfying , is injective, with a stability estimate and a reconstruction formula.
In addition, replacing by , can be taken uniform in for in a compact set on which the strict concavity assumption on level sets of holds.
3. Ellipticity up to gauge
With defined in (2.1)-(2.2), the main analytic points are that, first, is (after a suitable exponential conjugation) a scattering pseudodifferential operator of order , and second, by choosing an additional appropriate gauge-related summand, this operator is elliptic (again, after the exponential conjugation). These results are stated in the next two propositions, with the intermediate Lemma 3.2 describing the gauge related summand.
On one forms, resp. symmetric 2-cotensors, the operators , lie in
The proof of this proposition follows that of the scalar case given in [28, Proposition 3.3] and in a modified version of the scalar case in [27, Proposition 3.2]. For convenience of the reader, we follow the latter proof very closely, except that we do not emphasize the continuity statements in terms of the underlying metric itself, indicating the modifications.
Thus, recall that the map
is a local diffeomorphism, and similarly for in which takes the place of ; see the discussion around [28, Equation (3.2)-(3.3)]; indeed this is true for more general curve families. Here is the blow-up of at the diagonal , which essentially means the introduction of spherical/polar coordinates, or often more conveniently projective coordinates, about it. Concretely, writing the (local) coordinates from the two factors of as ,
give (local) coordinates on this space. Since the statement regarding the pseudodifferential property of is standard away from , we concentrate on the latter region. Correspondingly, in our coordinates , we write
for the lifted geodesic .
Recall from [28, Section 2] that coordinates on Melrose’s scattering double space, on which the Schwartz kernels of elements of are conormal to the diagonal, near the lifted scattering diagonal, are (with )
Note that here are as in  around Equation (3.10), not as in [28, Section 2] (where the signs are switched), which means that we need to replace by in the Fourier transform when computing principal symbols. Further, it is convenient to write coordinates on in the region of interest (see the beginning of the paragraph of Equation (3.10) in ), namely (the lift of) , as
with the norms being Euclidean norms, instead of (3.2); we write in terms of these. Note that these are . Moreover, by [28, Equation(3.10)] and the subsequent equations, combined also with Equations (3.14)-(3.15) there, are given in terms of as
with smooth and
Thus, a smooth metric applied to this yields
while so applied to this yields
Notice that on the right hand side of (3.4) the singular factor of in front of disappears due to the factor in , while on the right hand side of (3.3) correspondingly has a vanishing factor . This means, as we see below, that the component behaves trivially at the level of the boundary principal symbol of the operator defined like but with in place of , so in fact one can never have full ellipticity in this case; this is the reason we must use in the definition of .
One also needs to have evaluated at , since this is the tangent vector with which our tensors are contracted as they are being integrated along the geodesic. In order to compute this efficiently, we recall from [28, Equation (3.14)] that
with the , resp. terms having smooth coefficients in terms of . Correspondingly,
This gives that in terms of , is given by
with smooth in terms of . Substituting these in yields