Gabriel P.PATERNAINMikko SALOGunther UHLMANN

1 Introduction

This paper surveys recent results on the integral geometry problem of recovering a tensor field from its integrals along geodesics.The most basic example of the kinds of transforms studied in this paper is the X-ray(or Radon)transform in the plane,which encodes the integrals of a functionfin R2over straight lines:

Hereω⊥is the rotation ofωby 90 degrees counterclockwise.The properties of this transform are classical and well studied(see[20]).The X-ray transform forms the basis for many imaging methods such as CT and PET in medical imaging.

A number ofimaging methods involve generalizations of this transform.In seismic and ultrasound imaging one encounters ray transforms where the measurements are given by integrals over more general families of curves,often modeled as the geodesics of a Riemannian metric.Moreover,integrals of vector fields or other tensor fields instead of just integrals offunctions over geodesics may arise,and these transforms are also useful in rigidity questions in differential geometry.We will give more specific examples after having defined the relevant transforms precisely.

The geodesic ray transform acts on tensor fields on a compact,oriented Riemannian manifold(M,g)with boundary of dimension dim(M)=n≥2.We denote by⟨·,·⟩theg-inner product of tangent vectors or other tensors,and by|·|theg-norm.Letνdenote the unit outer normal to∂M.We denote bySM→Mthe unit-sphere bundle overM:

The setSMis a(2n−1)-dimensional compact manifold with boundary which can be written as the union∂(SM)=∂+(SM)∪∂−(SM),

The standard volume forms onSMand∂(SM)that we will use are defined by

where dVn(resp.dVn−1)is the volume form ofM(resp.∂M),andwhere dExis the Euclidean volume form ofSxinTxM.For(x,v)∈∂(SM),letµ(x,v)=|⟨ν(x),v⟩|and letbe the space offunctions on∂+(SM)with inner product

Without loss of generality,we may assume that(M,g)is embedded in(N,g),whereNis a compactn-dimensional manifold without boundary.Letφtbe the geodesic flow onNandbe the geodesic vector field.If(x,v)∈SM,letγ(t,x,v)be the unit speedN-geodesic starting fromxin the direction ofv.Then

Define the travel timeτ:SM→[0,∞]by

We say that(M,g)is non-trapping ifτ(x,v)<∞for all(x,v)∈SM.

Definition 1.1The geodesic ray transform of a function f∈C∞(SM)is the function

Note that if the manifold(M,g)is non-trapping and has strictly convex boundary,thenI:C∞(SM)→C(∂+(SM)),and Santal´o’s formula(see[10])implies thatIis also a bounded mapThe general problem in tensor tomography is to determine properties of a functionffrom its integrals over geodesics as encoded by the transformIf.

Question 1.1Given f∈C∞(SM),what properties offmay be determined from the knowledge ofif?

Clearly a general functionfonSMis not determined by its geodesic ray transform alone,sincefdepends on more variables thanIf.In applications one often encounters the transformIacting on special functions onSMthat arise from symmetric tensor fields,and we will now consider this case.

Letbe a smooth symmetricm-tensor field onM.Such a tensor field induces a smooth functionfm(x,v)onSMby

The operatorIm,de fined by

is called the geodesic ray transform of the symmetric tensor fieldf.If the manifold(M,g)is non-trapping and the boundary∂Mis strictly convex,then

whereSm(M)denotes the bundle of symmetricm-tensor fields over(M,g).We will frequently identify the tensor fieldfonMwith the functionfmonSM(see[32]for more details).

It is known that any symmetric smooth enough tensor fieldfmay be decomposed in a potential and solenoidal part(see[42]):

wherepis a smooth symmetric(m−1)-tensor field onM,the inner derivatived=σ∇is the symmetric part of the covariant derivative∇,andδis the divergence(the adjoint of−din the naturalL2inner product).Iffis a 1-tensor,identified with a vector fieldW,this generalizes the usual Helmholtz decomposition of a vector field,

It is easy to see,using the fact thatpvanishes on∂M,that the geodesic ray transform of the potential partdpis zero.We denote bythe space of smooth solenoidalm-symmetric tensor fields.The remark above means that we can only expect to recover the solenoidal part of a tensor field from its ray transform.This leads to the following definition of solenoidal injectivity,ors-injectivity for short.

Definition 1.2The ray transform on symmetric m-tensors,m≥1,is said to be s-injective ifimf=0implies fs=0for any f∈C∞(M,Sm(M)).In the case offunctions on M(m=0),I0is said to be s-injective ifi0f=0implies f=0for any f∈C∞(M).

The transformsImarise in several applications as well as in the boundary rigidity problem.The latter consists in determining the Riemannian metric of a compact Riemannian manifold with boundary,modulo isometries fixing the boundary,from the distance functiondg|∂M×∂Mbetween boundary points(see[26]).The case ofI0when the metric is Euclidean is the standard X-ray transform that integrates a function along lines.Radon found in 1917 an inversion formula to determine a function knowing the X-ray transform.Inversion formulas of this type have been implemented numerically using the filtered backprojection algorithm which is used today in CT scans.

Another important transform in medical imaging and other applications is the Doppler transform which integrates a vector field along lines.This corresponds to the case ofI1for the case of the Euclidean metric.The motivation is ultrasound Doppler tomography.It is known that blood flow is irregular and faster around tumor tissue than in normal tissue and Doppler tomography attempts to reconstruct the blood flow pattern.Mathematically the problem is to what extent a vector field is determined from its integrals along lines.

The case ofintegration along more general geodesics arises in geophysical imaging in determining the inner structure of the Earth since the speed of elastic waves generally increases with depth,thus curving the rays back to the Earth surface.It also arises in ultrasound imaging.The geodesic ray transformI0,that is,the integration of a function along geodesics,arises as the linearization of the boundary rigidity problem in a conformal class of metrics.The linearization of the boundary rigidity problem itself leads toI2,i.e.,the integration of tensors of order two along geodesics.The case ofintegration of tensors of order 4 along geodesics arises in certain inverse problems in elasticity(see[42]).

Many of the results in this survey are valid in the case when(M,g)is simple,a notion that naturally arises in the context of the boundary rigidity problem(see[26]).We recall that a Riemannian manifold with boundary is said to be simple if the boundary is strictly convex and if any two points are connected by a unique geodesic depending smoothly on the endpoints.In particular,a simple manifold is non-trapping and has no conjugate points.

One of the main results we review in this paper is thes-injectivity ofImfor allmfor simple two-dimensional manifolds that was proved recently in[32].

Theorem 1.1If(M,g)is a simple two-dimensional manifold,then Imis s-injective for any m≥0.

This result was known earlier form=0(see[27]),m=1(see[3])andm=2(see[46]).A key point in proving the result for generalmis the efficient use of surjectivity properties of,the adjoint ofI0.In fact,[32]gave the following more general result.

Theorem 1.2If(M,g)is a compact non-trapping two-dimensional manifold with strictly convex boundary,and ifi0and I1are s-injective and is surjective,then Imis s-injective for any m≥0.

To describe in detail the adjoint,for any functionwon∂+(SM),we define the function

Then the solution of the boundary value problem for the transport equation

is equal tou=wψ.

Recall thatIis a bounded mapThe adjointI∗is boundedand it is easy to compute explicitly.In the case ofI0,forf∈C∞(M)andw∈C∞(∂+(SM)),we have

The second equality used Santal´o’s formula(see[10]).From this computation we conclude that

Similarly,the adjoint ofImis the operatorwhich is given by

Definition 1.3We say thatis surjective iffor any f∈C∞(M),there exists a functionw∈C∞(∂+(SM))with

The surjectivity ofin the above sense was proved in[39]on simple manifolds of any dimension.We will show below how this result is used in the uniqueness proof of tensor tomography in two dimensions.

In this paper,we also review results in higher dimensions.Here is a summary of what is known abouts-injectivity on simple manifolds of dimensionn≥2:

(i)I0is injective(see[27]).

(ii)I1iss-injective(see[3]).

(iii)Imiss-injective for allmifn=2(see[32]).

(iv)Imiss-injective for allmfor manifolds of negative sectional curvature(see[37]),or under certain other curvature restrictions(see[9,36,42]).

(v)I2iss-injective for generic simple metrics including real-analytic ones(see[52]).

See[9,43,45,53,56]for uniqueness results on certain non-simple manifolds.We will also review results on the stability and range forIm,and moreover we propose several open problems.

A brief summary of the contents of this paper is as follows.Section 2 contains preliminaries and notation used in the paper.In Section 3 we review the two proofs of Theorem 1.1 given in[32].In Section 4 we explain a natural approach to the proof of the so-called Pestov identities used in Section 3.This energy estimate approach resembles Carleman estimates.In Section 5 we review a microlocal approach to the study of the geodesic ray transform that gives in particular stability estimates which are summarized in Section 6.In Section 7 we consider the scattering relation which is used in the characterization of the range and is ofindependent interest.In Section 8 we state the result of[35]on the range of the geodesic ray transform.In Section 9 we summarize several results for the attenuated ray transform for unitary connections proved in[33].In Section 10 we survey the result of[34]ons-injectivity of the ray transform on 2-tensors on closed Anosov surfaces.Finally in Section 11 we state several open problems.

2 Facts About the Unit Circle Bundle

This section contains some facts needed for explaining the uniqueness prooffor tensor tomography on surfaces,and we will restrict our attention to two-dimensional manifolds.Let(M,g)be a compact oriented two-dimensional Riemannian manifold with smooth boundary∂M.As usualSMwill denote the unit circle bundle which is a compact 3-manifold with boundary given by∂(SM)={(x,v)∈SM:x∈∂M}.

LetXdenote the vector field associated with the geodesic flowφt.SinceMis assumed oriented,there is a circle action on the fibers ofSMwith infinitesimal generatorVcalled the vertical vector field.It is possible to complete the pairX,Vto a global frame ofT(SM)by considering the vector fieldX⊥defined as the commutatorX⊥:=[X,V].There are two additional structure equations given byX=[V,X⊥]and[X,X⊥]=−KV,whereKis the Gaussian curvature of the surface.Using this frame we can define a Riemannian metric onSMby declaring{X,X⊥,V}to be an orthonormal basis.This metric coincides with the Sasaki metric onSM,and the volume form of this metric will be denoted by dΣ3.The fact that{X,X⊥,V}are orthonormal together with the commutator formulas implies that the Lie derivative of dΣ3along the three vector fields vanishes,in other words,the three vector fields preserve the volume form dΣ3.See[48]for more details on these facts.

It will be useful to have explicit forms of the three vector fields in local coordinates.Since(M,g)is two-dimensional,we can always choose isothermal coordinates(x1,x2)so that the metric can be written aswhereλis a smooth real-valued function ofx=(x1,x2).This gives coordinates(x1,x2,θ)onSM,whereθis the angle between a unit vectorvandIn these coordinates the vertical vector field is just

and the other vector fields are given by

Given functionsu,v:SM→C,we consider theL2inner product and norm

SinceX,X⊥,Vare volume preserving,we have

and if additionally

then also

The spaceL2(SM)decomposes orthogonally as a direct sum

whereHkis the eigenspace of−iVcorresponding to the eigenvaluek.A functionu∈L2(SM)has a Fourier series expansion

whereuk∈Hk.Also

whereThe even and odd parts ofuwith respect to velocity are given by

In the(x,θ)-coordinates previously introduced we may write

Observe that fork≥0,ukmay be identified with a section of thek-th tensor power of the canonical line bundle;the identification takesukintowherez=x1+ix2.

The next definition introduces holomorphic and antiholomorphic functions with respect to theθvariable.

Definition 2.1A function u:SM→Cis said to be holomorphic if uk=0for all k<0.Similarly,u is said to be antiholomorphic if uk=0for all k>0.

Let Ωk:=Hk∩C∞(SM).As in[19]we introduce the following first order elliptic operators

given by

ClearlyX=η++η−.From the structure equations for the frame{X,X⊥,V},one easily derives

We will also employ the fiberwise Hilbert transformH:C∞(SM)→C∞(SM),defined in terms offourier coefficients as

Here sgn(k)is the sign ofk,with the convention sgn(0)=0.Thus,uis holomorphic if and only if(Id−iH)u=u0and antiholomorphic if and only if(Id+iH)u=u0.

The following commutator formula for the Hilbert transform and the geodesic vector field,proved in[39],has been a crucial component for many results reviewed in this paper.

Proposition 2.1Let(M,g)be a two-dimensional Riemannian manifold.For any smooth function u on SM,we have the identity

where

is the average value.

ProofIt suffices to show that

SinceX=η++η−,we need to compute[Id+iH,η±],so let us find[Id+iH,η+]u,whereRecall that(Id+iH)We find

Thus

Similarly we find

Therefore using that iX⊥=η+−η−,we obtain

as desired.

3 Tensor Tomography on Surfaces

The paper[32]gave two proofs for uniqueness in tensor tomography on a simple surface(M,g).In this section we will give an outline of both proofs.They are based on Pestov identities,which are energy estimates for operators related to the ray transform,and which will be discussed in more detail in Section 4.Below we will make use of the concepts introduced in Sections 1 and 2.

First ProofTo explain the idea behind the first proof ofs-injectivity,let us first assume thatfis a 0-tensor,that is,f∈C∞(M).Assuming thatI0f=0,it is required to show thatf=0.The first step is a reduction from the integral operatorI0into a PDE question involving a transport equation.The function

solves the transport equation

It is enough to show thatu=0,since then alsof=0.

Isothermal coordinates allow to identify

The vertical vector field onSMisWe want to show that

Iffis a 0-tensor,f=f(x),thenV f=0.Thus it is enough to show that

This calls for a uniqueness result for the operatorP=V X.In isothermal coordinates,this operator has the form

whereh(x,θ)is a certain smooth function.It turns out that the operatorPis rather exotic and there do not seem to be general results on uniqueness properties of such operators in the literature.Here are some facts about the operatorP:

(i)It is a second-order operator on 3D manifoldSM.

(ii)It has multiple characteristics.

(iii)P+Whas compactly supported solutions for some first order perturbationW.

(iv)It enjoys a subelliptic type estimateforu∈C∞(SM)withu|∂(SM)=0.

However,we can still prove a global uniqueness result forPby using energy estimates.This involves the Pestov identity inL2(SM)inner product,whenu|∂(SM)=0,

whereP=A+iB,A∗=A,B∗=B.

We will compute the commutator below,and this gives(see Proposition 4.2)

It is known(see[33])that on simple manifolds,

Note that in the case of non-positive curvature,i.e.,K≤0,one always has(KV u,V u)≥0.ThusPu=0 impliesu=0,showing injectivity ofI0.

We now return to tensor tomography.LetXu=−finSM,u|∂(SM)=0,wherefis the function onSMcorresponding to a symmetricm-tensor field.It will be convenient to switch to a slightly different setup and think ofuandf(which are functionsSM→C)as sections of the trivial bundleE=SM×C.The transport equation then becomes an equation for sections ofE,

whereis the flat connection on the trivial bundleE.

One benefit of this(trivial)change of point of view is that from the equation on sections,one sees that the transport equation has a natural gauge group acting via multiplication by smooth functionsc∈C∞(M).This action preservesm-tensors,and leads to gauge equivalent equations

whereDA=d+Ais a gauge equivalent connection onEandA=−c−1dcis the 1-form determining the connection.

Now we try to use an energy identity for the connectionsDA.This Pestov identity with a connection is proved in the same way as the usual Pestov identity(see Proposition 4.3),and reads inL2(SM)norms

Here∗is Hodge star and

FA=dA+A∧A

is the curvature of the connectionDA=d+A.We observe that if the curvature∗FAand the expression(V u,u)have suitable signs,we gain a positive term in the energy estimate.

This observation does not immediately lead to anything new since curvature is preserved under gauge transformation.Thus,ifDAis gauge equivalent toD0,thenFA=F0=0.However,we can use a generalized gauge transformation that arranges a sign forFA.This involves gauge transformations via functionscthat may depend on thevvariable.Such transformations break them-tensor structure of the equation,but turn out to be manageable if the gauge transforms are holomorphic in a suitable sense.

Recall from Section 2 that a functionu∈L2(SM)is called holomorphic ifuk=0 fork<0.The main point is the following theorem guaranteeing that holomorphic gauge transformations always exist.This is related to injectivity of the attenuated ray transform on simple surfaces(see[41]),and in the form below it is proved in[32–33].The proofis based on the surjectivity of

Theorem 3.1(Holomorphic Gauge Transformation)If A is a1-form on a simple surface,there exists a holomorphic w∈C∞(SM)such that X+A=ew◦X◦e−w.

ProofSinceMis simply connected,there is a Hodge decompositionAjdxj=da+⋆dbfor somea,b∈C∞(M)(⋆is the Hodge star operator).In terms of the corresponding functions onSM,we haveA=Xa+X⊥b.Replacingwbyw−a,it is enough to consider the case whereA=X⊥b.

Let us try a solution of the formw=(Id+iH),where∈C∞(SM)is even with respect tov.By Proposition 2.1 ,

Now it is sufficient to findeven withX=0 andUsing the surjectivity of(see[39]),there exists someh∈C∞(∂+(SM))withBut ifw′∈C∞(SM)is the function withXw′=0 inSMandw′|∂+(SM)=h,we haveIt is enough to taketo be the even part ofw′with respect tov.

We can now explain the end of the proof of the uniqueness result for tensor tomography on simple surfaces.Letbe anm-tensor written in terms ofits Fourier components,and let

Choose a primitiveφof the volume formωgof(M,g),so thatdφ=ωg.Lets>0 be large,letAs=−isφ,and choose a holomorphicwwithX+As=esw◦X◦e−sw.The transport equation becomes

Here the curvature ofAshas a sign and one has information on Fourier coefficients of eswf.The Pestov identity with connection allows to control Fourier coefficients of eswu,eventually provings-injectivity ofIm.

Heuristically,the proof above involves“twisting”the trivial bundleEby a holomorphic gauge transformation to make it positively curved,using the Pestov identity with a large positive term coming from the connection to absorb error terms,and then undoing the gauge transformation(this is possible because of holomorphicity)to get uniqueness.This idea of twisting to impose positivity to prove a vanishing theorem is of course well known in complex geometry and it is the way one proves results like the Kodaira vanishing theorem(see[17]).Our setting is more complicated since the relevant PDE is the transport equation which is harder to handle than the Cauchy-Riemann equation.However this analogy is important and permeates all our work;in particular the injectivity results on the attenuated ray transform for unitary connections,to be discussed later on,are also proved in this fashion.

There is an interesting connection between the Pestov identity with connectionAsabove and with Carleman estimates.In fact,the Pestov identity withAsimplies the estimate

Here we use the norms

Formally this looks very much like a Carleman estimate with exponential weights,but it involves some slightly exotic spaces and one can see that the positivity comes from Im(w)(not Re(w)as is usual in Carleman estimates)!We finally remark that such an estimate is sufficient for

(i)absorbing large attenuation(even for systems,see Section 9),

(ii)absorbing error terms coming fromm-tensors.

However,it seems that the estimate may not be enough to

(i)localize in space,

(ii)absorb error terms coming from curvature ofM.

Second ProofNext we explain a very short alternative proof to a key step in the injectivity result.

Suppose thatuis a smooth solution ofXu=−finSM,wherefk=0 fork≤−m−1 andu|∂(SM)=0.We wish to show thatuk=0 fork≤−m.This,together with the analogous result for positive Fourier coefficients,implies thatf=Xh,where the Fourier expansion ofhhas degreem−1 andh|∂(SM)=0,thus provings-injectivity.

We choose a nonvanishing functionh∈Ωm.In fact,in isothermal coordinates,we can set

Define the 1-form

Thenhusolves the problem

Note thathfis a holomorphic function.Next we employ a holomorphic integrating factor,as above:By Theorem 3.1 there exists a holomorphicw∈C∞(SM)withXw=A.The function ewhuthen satisfies

The right-hand side ewhfis holomorphic.It is known that the solution ewhu,which vanishes on∂(SM)also has to be holomorphic and further(ewhu)0=0.This follows from thes-injectivity ofI0andI1(see[32,41]).Looking at Fourier coefficients shows that(hu)k=0 fork≤0,and thereforeuk=0 fork≤−mas required.

4 Pestov Identity

In this section,we consider the Pestov identity,which is the basic energy identity that has been used since the work of Mukhometov[27]in most injectivity proofs of ray transforms in the absence of real-analyticity or special symmetries.Pestov type identities were also used in[3]to proves-injectivity forI1on simple manifolds and in[37]to proves-injectivity for anymin any dimensions if the sectional curvatures are negative.See[9,36,42]for further results.Pestov identities have often appeared in a somewhat ad hoc way,but here we follow[32]which gave a new point of view making the derivation of these identities more transparent.We will only consider two-dimensional manifolds in this section.

The easiest way to motivate the Pestov identity is to consider the injectivity of the ray transform on functions.The first step,as discussed in Section 3,is to recast the injectivity problem as a uniqueness question for the partial differential operatorPonSM,where

This involves a standard reduction to the transport equation.

Proposition 4.1Let(M,g)be a compact oriented nontrapping surface with strictly convex smooth boundary.The following statements are equivalent:

(a)The ray transform I:C∞(M)→C(∂+(SM))is injective.

(b)Any smooth solution of Pu=0in SM with u|∂(SM)=0is identically zero.

ProofAssume that the ray transform is injective,and letu∈C∞(SM)solvePu=0 inSMwithu|∂(SM)=0.This implies thatXu=−finSMfor some smoothfonly depending onx,and we have 0=u|∂+(SM)=If.SinceIis injective,one hasf=0 and thusXu=0,which impliesu=0 by the boundary condition.

Conversely,assume that the only smooth solution ofPu=0 inSMwhich vanishes on∂(SM)is zero.Letf∈C∞(M)be a function withIf=0,and define the function

This function satisfies the transport equationXu=−finSMandu|∂(SM)=0 sinceIf=0,and alsou∈C∞(SM)(see[33]).Sincefonly depends onx,we haveV f=0,and consequentlyPu=0 inSMandu|∂(SM)=0.It follows thatu=0 and alsof=−Xu=0.

We now focus on proving a uniqueness statement for solutions ofPu=0 inSM.For this it is convenient to expressPin terms ofits self-adjoint and skew-adjoint parts in theL2(SM)inner product as

Here the formal adjointP∗ofPis given by

P∗:=XV.

In fact,ifu∈C∞(SM)withu|∂(SM)=0,then

This computation suggests to study the commutator i[A,B].We note that the argument just presented is typical in the proof ofL2Carleman estimates(see[21]).

By the definition ofAandBit easily follows thatBy the commutation formulas forX,X⊥andV,this commutator may be expressed as

Consequently

If the curvatureKis nonpositive,then[P∗,P]is positive semidefinite.More generally,one can try to use the other positive terms in(4.1).Note that

The identity(4.1)may then be expressed as

(Note that we could have just started from the last identity,but expressing matters viaAandBhighlights the similarity to Carleman estimates.)Moving the termto the other side,we have proved the version of the Pestov identity which is most suited for our purposes.The main point in this proof was that the Pestov identity boils down to a standardL2estimate based on separating the self-adjoint and skew-adjoint parts ofPand on computing one commutator,[P∗,P].

Proposition 4.2If(M,g)is a compact oriented surface with smooth boundary,then

for any u∈C∞(SM)with u|∂(SM)=0.

It is known(see[13,33])that on a simple surface,one has

Also,ifXu=−f,wheref=f0+f1+f−1is the sum of a 0-form and a 1-form,we have

These two facts together with the Pestov identity give the standard proof ofs-injectivity of the ray transform for 0-forms and 1-forms on simple surfaces.It is easy to see where this proof breaks down ifm≥2:The Fourier expansionimplies

This term may be negative,and the Pestov identity may not give useful information unless there is some extra positivity like a curvature bound.

Finally,we consider the Pestov identity in the presence of attenuation given byA(x,v)=Aj(x)vj,whereAjdxjis a purely imaginary 1-form onM.We writeAboth for the 1-form and the function onSM.The geometric interpretation is thatd+Ais a unitary connection on the trivial bundleM×C,and its curvature is the 2-form

Then⋆FAis a function onM,where⋆is the Hodge star.We consider the operator

Since=−A,the formal adjoint ofPin theL2(SM)inner product is

The same argument leading to Proposition 4.2,based on computing the commutator[P∗,P],gives the following Pestov identity proved also in[33,Lemma 6.1].

Proposition 4.3If(M,g)is a compact oriented surface with smooth boundary and if A is a purely imaginary1-form on M,then

for any u∈C∞(SM)with u|∂(SM)=0.

Using the Fourier expansion ofu,the last term in the identity is given by

This shows that ifuis holomorphic and i⋆FA>0,or ifuis antiholomorphic and i⋆FA<0,one gains an additional positive term in the Pestov identity.This is crucial in absorbing negative contributions from the termwhen provings-injectivity on tensor fields.

5 Microlocal Approach

A different approach that is useful to proves-injectivity ofImin some cases and gives stability estimates as well as reconstruction formulas in some cases was started in[50]and developed further in[51–54].We describe the method in more detail forI0.Let(M,g)be a simple manifold embedded in a closed manifold(N,g)and letUbe a simple neighborhood ofMinN.

Theorem 5.1is an elliptic pseudodifferential operator on U of order−1.

ProofIt is easy to see that

Before we continue we make a remark concerning notation.We have used up to now the notationγ(t,x,v)for a geodesic.But it is known,that a geodesic depends smoothly on the pointxand the vectorξt∈Tx(M).Therefore in what follows we will also use sometimes the notationγ(x,vt)for a geodesic.Since the manifoldMis simple,any small enough neighborhoodU(in(N,g))is also simple(an open domain is simple ifits closure is simple).For any pointx∈U,there exists an open domainsuch that exponential map expx:U,expxη=γ(x,η)is a diffeomorphism ontoU.LetDx,x∈Mbe the inverse image ofM.Then expx(Dx)=Mand expx|Dx:Dx→Mis a diffeomorphism.

Now we change variables in(5.1),y=γ(x,vt).Thent=dg(x,y)and

where

Notice that since

it follows that the Jacobian of the exponential map is 1 at 0,and then det=[det(expx)′(x,0)]−1=1.From(5.2)we also conclude that

Therefore the kernel ofcan be written in the form

Thus the kernelKhas at the diagonalx=ya singularity of type|x−y|−n+1.The kernel

has the same singularity.Clearly,the differenceK−K0has a singularity of type|x−y|−n+2.Therefore the principal symbols of both operators coincide.The principal symbol of the integral operator,corresponding to the kernelK0,coincides with its full symbol and is easily calculated.As a result,

Letgbe a simple metric inM.ExtendgnearMand letM1be a simple manifold with boundary so thatMis a compact subset ofM1.We will work withfsupported inM.We assume thatfis extended as 0 outsideM.Choose a smooth functionχsupported inM1such thatχ=1 nearM.

It was shown in[51]that the normal operatoris a pseudodifferential operator of order−1,form=0,1,2 which is elliptic acting on solenoidal tensor fields.We have the following theorem.

Theorem 5.2Let g be a simple metric in M and let χ be as before.Then one can construct a pseudodifferential operator aijkl(x,D)of order1so that for any symmetric2-tensor f∈L2(M),we have

where K:L2(M)→H1(M1)is bounded.Heredenotes the solenoidal part offon M1.

This result was extended to tensor fields of any order in[47].

Whengis a real-analytic simple metric it was shown in[52]thatI2iss-injective.The proof constructs a parametrix as in the previous result withKanalytic regularizing,that is,Kfis real-analytic onM1forf∈L2(M).The idea of the proofforI0is that ifI0f=0,f∈L2(M),thenf=−Kf.SinceKfis real-analytic onM1and supported onM,it must be zero.For the details of the proofforI2see[52].

6 Stability Estimates

It was shown in[51]and[47]that for a simple manifolds-injectivity ofImimplies stability estimates.This is based on the fact thatis an elliptic pseudodifferential operator acting on solenoidal tensor fields.We have the following stability estimate forI0(see[51]).

Theorem 6.1Let g be a simple metric in M and assume that g is extended smoothly as a simple metric near the simple manifold M1⊃⊃M.Then for any function f∈L2(M),

Similarlys-injectivity ofI1implies the stability estimate.

Theorem 6.2Assume that g is simple metric in M and extend g as a simple metric in M1⊃⊃M.Then for any1-form f=fidxiin L2(M),we have

A sharp stability estimate forI2,assuming thatI2is known to bes-injective,was proved in[49].

Theorem 6.3Let g be a simple metric in M and assume that g is extended smoothly as a simple metric near the simple manifold M1⊃⊃M.Also assume that I2is s-injective.Then for any symmetric2-tensor field fin L2(M),

In order to describe possible stability estimates forIm,we describe an earlier result forI2.In order to state the result,we first take boundary normal coordinatesx1,···,xnwithxn=0 the defining function of∂M.Introduce the space(M1)with norm equal to theL2norm outside a neighborhood of∂Mand near∂M(but outsideM)having the following form in normal local coordinates:

HereUis a small neighborhood of a point on∂Mand the norm in(M1)is defined by using a partition of unity.

Next we define the norm

The earlier stability result forI2is the following theorem.

Theorem 6.4Assume that g is simple metric in M and extend g as a simple metric in M1⊃⊃M.

(a)The following estimate holds for each symmetric2-tensor fin H1(M):

(b)KerI2∩SL2(M)is finite dimensional and included in C∞(M).Here S stands for solenoidal.

(c)Assume that I2is s-injective in M,i.e.,thatKerIg∩SL2(M)={0}.Then for any symmetric2-tensor fin H1(M),we have

This result was proved in[51]for the casem=2.Using the results of[47],stability estimates of this type can be shown to be valid for anym.

7 The Scattering Relation

To state the results for the range ofImfor simple surfaces,we need to recall the definition of the scattering relation which is a subject ofinterest in its own right.

Suppose that we have a Riemannian metric in Euclidean space which is the Euclidean metric outside a compact set.The inverse scattering problem for metrics is to determine the Riemannian metric by measuring the scattering operator(see[18]).A similar obstruction to the boundary rigidity problem occurs in this case with the diffeomorphismψequal to the identity outside a compact set.It was proved in[18]that from the wave front set of the scattering operator,one can determine,under some non-trapping assumptions on the metric,the scattering relation on the boundary of a large ball.This uses high frequency information of the scattering operator.In the semiclassical setting Alexandrova has shown for a large class of operators that the scattering operator associated to potential and metric perturbations of the Euclidean Laplacian is a semiclassical Fourier integral operator quantized by the scattering relation(see[2]).The scattering relation maps the point and direction of a geodesic entering the manifold to the point and direction of exit of the geodesic.

We proceed to define in more detail the scattering relation.To do this,letτ0=τ|∂(SM)and note that this function is equal to zero on∂−(SM)and is smooth on∂+(SM).Its odd part with respect tov,

is a smooth function on∂(SM)(see for instance[10]).

Definition 7.1Let(M,g)be non-trapping with strictly convex boundary.The scattering relation α:∂(SM)→∂(SM)is defined by

The scattering relation is a diffeomorphism∂(SM)→∂(SM).Notice thatα|∂+(SM):∂+(SM)→∂−(SM),α|∂−SM:∂−(SM)→∂+(SM)are diffeomorphisms as well.The manifolds ofinner vectors∂+(SM)and outer vectors∂−(SM)intersect at the set of tangent vectors

Obviously,αis an involution,α2=id and∂0(SM)is the hypersurface ofits fixed points,α(x,v)=(x,v),(x,v)∈∂0(SM).

A natural inverse problem is whether the scattering relation determines the metricgup to an isometry which is the identity on the boundary.This information takes into account all the travel times,not just the first arrivals like the boundary distance function.

We remark that in the case that(M,g)is a simple manifold,and we know the metric at the boundary(and this is determined ifdgis known),knowing the scattering relation is equivalent to knowing the boundary distance function(see[26]).

We introduce the operators of even and odd continuation with respect toα:

We will examine next the boundedness properties ofA−,A+.

Lemma 7.1A±:are bounded.

Proof

whereα:∂+(SM)→∂−(SM)is a diffeomorphism.Thus it is enough to show that

Letw∈C∞(∂+(SM)).Then

Varyingwshows thatα∗(−µdΣ2n−2)=µdΣ2n−2on∂+(SM)∂0SM.

The adjointsatisfies

so

In[39]the following characterization of the space of smooth solutions of the transport equation was given.Here we define

Lemma 7.2

Thenwheneverw

We conclude this section by defining certain operators which combine the operatorsA±introduced above with the fibrewise Hilbert transformH.These operators will be essential to determine the range of the ray transform in the next section.SetH±u=Hu±,whereu+(resp.u−)denotes the even(resp.odd)part ofu∈C∞(SM).

We define

8 Range of the Geodesic Ray Transform

We now give the characterization of the range ofI0andI1in terms of the scattering relation only.We have that these are the projections of the operatorsP−,P+,respectively(defined in(7.1)).For the details see[38].

Theorem 8.1Let(M,g)be simple two-dimensional compact Riemannian manifold with boundary.Then

(1)A function u∈C∞(∂+(SM))belongs to the range ofi0if and only if u=P−w,where

(2)A function u∈C∞(∂+(SM))belongs to the range ofi1if and only if u=P+w,where

We now move on to describe the range of the geodesic ray transform for tensors of order≥2.For this we apply the ideas of the second proof of Theorem 1.1 described in Section 3.For the details see[35].

Let(M,g)be a simple surface.The metricginduces a complex structure onMand letκbe the canonical line bundle(which we may identify withT∗M).Recall thatHm(m∈Z)is the set offunctions inf∈L2(SM,C)such thatV f=imf.The set Ωm=Hm∩C∞(SM,C)can be identified with the set Γ(M,κ⊗m)of smooth sections ofm-th tensor power of the canonical line bundleκ.This identification depends on the metric and is explained in detail in[34,Section 2],but let us give a brief description ofit.Given a sectionξ∈Γ(M,κ⊗m)we can obtain a function on Ωmsimply by restriction toSM:ξdetermines the functionSM∋(x,v)7→ξx(v⊗m)and this gives a 1-1 correspondence.

SinceMis a disk,there existsξ∈Γ(M,κ)which is nowhere vanishing.Having picked this section we may define a functionh:SM→S1by settingBy construction,h∈Ω1.Our description of the range will be based on this choice ofh.Define

Observe that sinceAlsoXh=η+h+η−h∈Ω2⊕Ω0which implies thatA∈Ω1⊕Ω−1.It follows thatAis the restriction toSMof a purely imaginary 1-form onM,hence we have a unitary connection(see Section 9).

First we describe the range of the geodesic ray transformIrestricted to Ωm:

Observe that ifusolves the transport equationXu=−fwithu|∂−(SM)=0,thenh−musolves(X−mA)(h−mu)=−h−mfandh−mu|∂−(SM)=0.Also note thath−mf∈Ω0.Thus

where the left-hand side is the attenuated ray transform of the unitary connection−mA.Attenuated transforms will be described in more detail in the next section,but the upshot is that we can prove a theorem similar to Theorem 8.1 but introducing this time a unitary connection as attenuation.Putting everything together one obtains a description of the range forImas follows.Let

In other words,

We define

Theorem 8.2(see[35])Let(M,g)be a simple surface.Then a function u∈C∞(∂+(SM),C)belongs to the range ofimif and onlywhere this last space denotes the set of all smooth w such that Qmw is smooth.

Suppose that nowFis a complex-valued symmetric tensor of ordermand we denote its restriction toSMbyf.Recall from[32,Section 2]that there is a 1-1 correspondence between complex-valued symmetric tensors of ordermand functions inSMof the formwherefk∈Ωkandfk=0 for allkodd(resp.even)ifmis even(resp.odd).

Since

we deduce directly from Theorem 8.2 the following.

Theorem 8.3Let(M,g)be a simple surface.If m=2l is even,a function u∈C∞(∂+(SM),C)belongs to the range of the ray transform acting on complex-valued symmetricm-tensors if and only if there existsuch that

Similarly,if m=2l+1is odd,a function u∈C∞(∂+(SM),C)belongs to the range of the ray transform acting on complex-valued symmetric m-tensors if and only if there exist w2k+1∈such that

9 Attenuated Ray Transform for Unitary Connections

In this section we describe in detail certain injectivity results for the attenuated ray transform of a unitary connection(see[33]).We saw the appearance of the attenuated ray transform in the last section when we discussed the range of the(unattenuated)ray transform on tensors of any order.We also saw how useful it was for the tensor tomography problem to introduce a connection to gain positivity in the Pestov identity.Here we take a closer and more systematic look.We motivate this section by discussing first another natural inverse problem:Determine a unitary connection from its scattering relation,that is,parallel transport along geodesics between boundary points.Our results are for simple surfaces,but the definitions can be given in the context of non-trapping manifolds(M,g)with strictly convex boundary.

Suppose thatE→Mis a Hermitian vector bundle of ranknoverMand∇is a unitary connection onE.Associated with∇there is the following additional piece of scattering data:Given(x,v)∈∂+(SM),letP(x,v)=P∇(x,v):E(x)→E(π◦α(x,v))denote the parallel transport along the geodesicγ(t,x,v).This map is a linear isometry and the main inverse problem we wish to discuss here is the following question.

QuestionDoesPdetermine∇?

The first observation is that the problem has a natural gauge equivalence.Letψbe a gauge transformation,that is,a smooth section of the bundle of automorphisms AutE.The set of all these sections naturally forms a group(known as the gauge group)which acts on the space of unitary connections by the rule

wheresis any smooth section ofE.Ifin additionψ|∂M=Id,then it is a simple exercise to check that

Thus we can rephrase the question above more precisely as follows.

Question ILet∇1and∇2be two unitary connections withP∇1=P∇2.Does there exist a gauge transformationψwithψ|∂M=Id andψ∗∇1=∇2?

It is easy to see from the definition that a simple manifold must be diffeomorphic to a ball in Rn.Therefore any bundle over suchMis necessarily trivial and from now on we shall assume thatE=M×Cn.

Question I arises naturally when considering the hyperbolic Dirichlet-to-Neumann map associated to the Schrödinger equation with a connection.It was shown in[16]that when the metric is Euclidean,the scattering data for a connection can be determined from the hyperbolic Dirichlet-to-Neumann map.A similar result holds true on simple Riemannian manifolds:a combination of the methods in[16]and[55]shows that the hyperbolic Dirichlet-to-Neumann map for a connection determines the scattering dataP∇.

Elementary background on connectionsConsider the trivial bundleM×Cn.For us a connectionAwill be a complexn×nmatrix whose entries are smooth 1-forms onM.Another way to think ofAis to regard it as a smooth mapA:TM→Cn×nwhich is linear inv∈TxMfor eachx∈M.

Very often in physics and geometry one considers unitary or Hermitian connections.This means that the range ofAis restricted to skew-Hermitian matrices.In other words,if we denote by u(n)the Lie algebra of the unitary groupU(n),we have a smooth mapA:TM→u(n)which is linear in the velocities.There is yet another equivalent way to phrase this.The connectionAinduces a covariant derivativedAon sectionss∈C∞(M,Cn)by settingdAs=ds+As.ThenAbeing Hermitian or unitary is equivalent to requiring compatibility with the standard Hermitian inner product of Cnin the sense that

for any pair offunctionss1,s2.

Given two unitary connectionsAandB,we say thatAandBare gauge equivalent if there exists a smooth mapu:M→U(n)such that

It is easy to check that this definition coincides with the one given in the previous section if we setψ=u−1.

The curvature of the connection is the 2-formFAwith values in u(n)given by

IfAandBare related by(9.1),then

Given a smooth curveγ:[a,b]→M,the parallel transport alongγis obtained by solving the linear differential equation in Cn:

The isometryPA(γ):Cn→Cnis defined asPA(γ)(w):=s(b).We may also consider the fundamental unitary matrix solutionU:[a,b]→U(n)of(9.2).It solves

ClearlyPA(γ)(w)=U(b)w.

The transport equation and the attenuated ray transformConsider now the case of a compact simple Riemannian manifold.We would like to pack the information provided by(9.3)along every geodesic into one PDE inSM.For this we consider the vector fieldXassociated with the geodesic flowϕtand we look at the unique solutionUA:SM→U(n)of

The scattering data of the connectionAis now the mapCA:∂−(SM)→U(n)defined asCA:=UA|∂−(SM).

We can now rephrase Question I as follows.

Question ILetAandBbe two unitary connections withCA=CB.Does there exist a smooth mapU:M→U(n)withU|∂M=Id andB=U−1dU+U−1AU?

SupposeCA=CBand defineU:=UA(UB)−1:SM→U(n).One easily checks thatUsatisfies

If we show thatUis in fact smooth and it only depends on the base pointx∈M,we would have an answer to Question I,since the equation above reduces todU+AU−UB=0 andU|∂M=Id which is exactly gauge equivalence.Showing thatUonly depends onxis not an easy task and it often is the crux of the matter in these types of problems.To tackle this issue we rephrase the problem in terms of an attenuated ray transform.

ConsiderW:=U−Id:SM→Cn×n,where as before Cn×nstands for the set of alln×ncomplex matrices.ClearlyWsatisfies

We introduce a new connectionon the trivial bundleM×Cn×nas follows:Given a matrixR∈Cn×n,we define(R):=AR−RB.One easily checks thatbAis Hermitian ifAandBare.Then equations(9.5)and(9.6)are of the form

whereAis a unitary connection,f:SM→CNis a smooth function linear in the velocities,u:SM→CNis a function that we would like to prove smooth and only dependent onx∈MandN=n×n.As we will see shortly this amounts to understanding which functionsflinear in the velocities are in the kernel of the attenuated ray transform of the connectionA.

First recall that in the scalar case,the attenuated ray transformIafof a functionf∈C∞(SM,C)with attenuation coefficienta∈C∞(SM,C)can be defined as the integral

Alternatively,we may setIaf:=u|∂+(SM),whereuis the unique solution of the transport equation

The last definition generalizes without difficulty to the case of connections.Assume thatAis a unitary connection and letf∈C∞(SM,Cn)be a vector valued function.Consider the following transport equation foru:SM→Cn,

On a fixed geodesic the transport equation becomes a linear ODE with zero initial condition,and therefore this equation has a unique solution denoted byuf.

Definition 9.1The attenuated ray transform off∈C∞(SM,Cn)is given by

We note thatIAacting on sums of 0-forms and 1-forms always has a nontrivial kernel,since

IA(dp+Ap)=0for anyp∈C∞(M,Cn)withp|∂M=0.

Thus from the ray transformIAfone only expects to recoverfup to an element having this form.

The transformIAalso has an integral representation.Consider the unique matrix solutionUA:SM→U(n)from above.Then it is easy to check that

We are now in a position to state the next main question.

Question II(Kernel ofIA)Let(M,g)be a compact simple Riemannian manifold and letAbe a unitary connection.Assume thatf:SM→Cnis a smooth function of the formF(x)+αj(x)vj,whereF:M→Cnis a smooth function andαis a Cn-valued 1-form.IfIA(f)=0,is it true thatF=0 andα=dAp=dp+Ap,wherep:M→Cnis a smooth function withp|∂M=0?

As explained above a positive answer to Question II gives a positive answer to Question I.The next recent result provides a full answer to Question II in the two-dimensional case.

Theorem 9.1(see[33])Let M be a compact simple surface.Assume that f:SM→Cn is a smooth function of the form F(x)+αj(x)vj,where F:M→Cnis a smooth function and α is aCn-valued1-form.Let also A:TM→u(n)be a unitary connection.IfiA(f)=0,then F=0and α=dAp,where p:M→Cnis a smooth function with p|∂M=0.

Let us explicitly state the positive answer to Question I in the case of surfaces.

Theorem 9.2(see[33])Assume that M is a compact simple surface and let A and B be two unitary connections.Then CA=CBimplies that there exists a smooth U:M→U(n)such that U|∂M=Idand B=U−1dU+U−1AU.

The proof of Theorem 9.1 is based on the ideas explained in Section 3.One introduces a suitable additional attenuation(twists with a positive line bundle)which adds positivity to the Pestov identity with a connection and then gauges the twist away via the key Theorem 3.1.

In the case of Euclidean space with the Euclidean metric,the attenuated ray transform is the basis of the medical imaging technology of SPECT and has been extensively studied,see[15]for a review.We remark that in connection with injectivity results for ray transforms,there is great interest in reconstruction procedures and inversion formulas.For the attenuated ray transform in R2with Euclidean metric and scalar attenuation function,an explicit inversion formula was proved by Novikov[28].A related formula also including 1-form attenuations appears in[6],inversion formulas for matrix attenuations in Euclidean space are given in[14,29],and the case of hyperbolic space H2is considered in[5].

Various versions of Theorem 9.2 have been proved in the literature.Sharafutdinov[44]proved the theorem assuming that the connections areC1close to another connection with small curvature(but in any dimension).In the case of domains in the Euclidean plane the theorem was proved by Finch and Uhlmann[16]assuming that the connections have small curvature and by Eskin[14]in general.Novikov[29]considerd the case of connections which are not compactly supported(but with suitable decay conditions at infinity)and established local uniqueness of the trivial connection and gave examples in which global uniqueness fails(existence of“ghosts”).

For more on inverse problems for connections we refer to[30].

10 Anosov Manifolds

There are versions of the ideas in the previous sections in the context of closed manifolds.The first requirement is to have a notion that replaces the concept of simple manifold.It is easy to motivate this as follows.Simple manifolds have two characteristic properties:They have no conjugate points and they are open in theC2-topology of metrics.Recall that a Riemannian manifold is said to have no conjugate points if any two points in the universal covering are joined by a unique geodesic.Hence it seems natural to seek an analogue by requiring that the metric is aC2-interior point among the set of all metric without conjugate points.

Definition 10.1A closed Riemannian manifold(M,g)is said to be Anosov if g belongs to the C2-interior of the set of metrics without conjugate points.

It turns out that the name“Anosov”is completely justified:(M,g)is Anosov if and only if the geodesic flow ofgis Anosov in the sense of dynamical systems(see[40]).We will not give here the definition of an Anosov flow since it will not be explicitly needed and instead we refer the reader to[22].

From our definition it is clear that negatively curved manifolds are Anosov and that there are no Anosov metrics on tori since the only metrics without conjugate points on tori must be flat(see[7]).

The notion of“Imiss-injective”makes sense for closed manifolds as follows.

Definition 10.2We say that Imis s-injective if given any symmetric m-tensor f such that

for every unit speed closed geodesic γ:[0,T]→M,then fis potential,i.e.,there exists an(m−1)-symmetric tensor h such that f=dh.

The tensor tomography problem for an Anosov manifold consists in proving thatImissinjective for anym.There are numerous motivations for this,but perhaps the most notorious one is that of spectral rigidity which involvesI2.In[19],Guillemin and Kazhdan proved that if(M,g)is an Anosov manifold such thatI2iss-injective,then(M,g)is spectrally rigid.This means that if(gs)is a smooth family of Riemannian metrics onMfors∈(−ε,ε)such thatg0=gand the spectra ofcoincide up to multiplicity,

then there exists a family of diffeomorphismsψs:M→Mwithψ0=Id and

Let us summarize what is known about the tensor tomography problem on an Anosov manifold:

(i)I0andI1ares-injective(see[11]);

(ii)I2iss-injective for surfaces(see[34]);

(iii)Imiss-injective for allmfor non-positively curved manifolds(see[8]).

11 Open Problems

In this section we mention some open problems related to tensor tomography.

(1)In the two-dimensional case there is by now,as surveyed in this paper,a rather good understanding of the injectivity and range of the geodesic ray transform on tensor fields for simple manifolds.Important questions remaining are inversion formulas or reconstruction procedures of the solenoidal part of the tensor field from its geodesic ray transform.Certain inversion formulas were given in[24,38]for the case of constant curvature and close to constant curvature.

(2)In the case where dim(M)≥3,it is not known whetherImiss-injective for a general simple manifold.This is known forI0andI1,but even the case ofI2is unknown at present.

(3)Support type theorems for the geodesic ray transform,where a tensor field is determined locally from its line integrals in a certain neighborhood,are known for the case of real analytic simple manifolds forIm,m=0,1,2(see[23,25]).Is it possible to extend these results to all simple manifolds?This has been done forI0in three dimensions or higher(see[56]).

(4)The study ofs-injectivity of the geodesic ray transform for non-simple manifolds is an important problem for which not much is known.Certain results are given in[9,43,45].A microlocal analysis ofI0when the exponential map has fold type singularities was done in[54].Injectivity,stability and reconstruction were proved forI0in the case of three dimensions or higher when the manifold can be foliated by strictly convex hypersurfaces(see[56]).This allows for conjugate points.Thes-injectivity ofI2was analyzed in[53]for a class of non-simple manifolds.However,the question ifI0is injective on a compact non-trapping manifold with strictly convex boundary is open.

(5)The attenuated ray transform for a unitary connection on simple surfaces and Anosov surfaces has been extensively studied in[31,33,41].It would be interesting to extend the results to the case of a non-unitary connection.

(6)For closed Anosov surfaces it is known thatImiss-injective form=0,1,2.Is it true for allm?Also,isI2s-injective for Anosov manifolds of dimension≥3?

(7)Finally,it would be natural to extend all this theory to more general classes of curves.By this we mean replacing geodesics by other natural set of curves like magnetic geodesics or geodesics of affine connections with torsion(thermostats).Concerning magnetic geodesics,the tensor tomography problem in 2D is solved in[1]using the ideas presented here and the results in[10],see also[4].

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