Table Of ContentCAMBRIDGE STUDIES IN ADVANCED MATHEMATICS 204
EditorialBoard
J. BERTOIN, B. BOLLOBÁS, W. FULTON, B. KRA, I. MOERDIJK,
C. PRAEGER, P. SARNAK, B. SIMON, B. TOTARO
GEOMETRICINVERSEPROBLEMS
This up-to-date treatment of recent developments in geometric inverse problems
introduces graduate students and researchers to an exciting area of research. With
an emphasis on the two-dimensional case, topics covered include geodesic X-ray
transforms,boundaryrigidity,tensortomography,attenuatedX-raytransforms,andthe
Calderónproblem.
Thepresentationisself-containedandbeginswiththeRadontransformandradial
soundspeedsasmotivatingexamples.Therequiredgeometricbackgroundisdeveloped
in detail in the context of simple manifolds with boundary. An in-depth analysis of
various geodesic X-ray transforms is carried out together with related uniqueness,
stability,reconstruction,andrangecharacterizationresults.Highlightsincludeaproof
ofboundaryrigidityforsimplesurfacesaswellasscatteringrigidityforconnections.
The concluding chapter discusses current open problems and related topics. The
numerousexercisesandexamplesmakethisbookanexcellentself-studyresourceor
textforaone-semestercourseorseminar.
Gabriel P. Paternain is Professor of Mathematics at the Department of Pure Math-
ematics and Mathematical Statistics at the University of Cambridge and a Fellow of
TrinityCollege.Hisresearchhascoveredanamplemathematicallandscape,including
Hamiltonian dynamics, symplectic geometry, and geometric inverse problems. He is
theauthorofthemonographGeodesicFlows(1999),andwasawardedthePilkington
Teaching Prize at Cambridge for his ability to explain analysis and geometry with a
claritythathaswonhimtheadmirationandrespectofhisstudents.
Mikko Salo is Professor of Mathematics at the University of Jyväskylä, Finland.
Hehasreceivedseveralawardsforhisworkoninverseproblemsinpartialdifferential
equations and geometry, including the Calderón prize, the Väisälä prize, an ERC
StartingGrant,andanERCConsolidatorGrant.
GuntherUhlmannistheWalkerFamilyEndowedProfessorattheUniversityofWash-
ington and the Si Yuan Professor at the Hong Kong University of Science and
Technology. He has worked on microlocal analysis and a broad spectrum of inverse
problems. He was awarded the AMS Bocher Prize, the Kleinman Prize from SIAM,
theSolomonLefschetzMedalfromtheMathematicalCounciloftheAmericas,andthe
BirkhoffPrize,awardedjointlybySIAMandtheAMS.
Published online by Cambridge University Press
CAMBRIDGE STUDIES IN ADVANCED MATHEMATICS
EditorialBoard
J.Bertoin,B.Bollobás,W.Fulton,B.Kra,I.Moerdijk,C.Praeger,P.Sarnak,B.Simon,B.Totaro
AllthetitleslistedbelowcanbeobtainedfromgoodbooksellersorfromCambridgeUniversityPress.
Foracompleteserieslisting,visitwww.cambridge.org/mathematics.
AlreadyPublished
164 P.SchneiderGaloisRepresentationsand(φ,(cid:3))-Modules
165 P.Gille&T.SzamuelyCentralSimpleAlgebrasandGaloisCohomology(2ndEdition)
166 D.Li&H.QueffelecIntroductiontoBanachSpaces,I
167 D.Li&H.QueffelecIntroductiontoBanachSpaces,II
168 J.Carlson,S.Müller-Stach&C.PetersPeriodMappingsandPeriodDomains(2ndEdition)
169 J.M.LandsbergGeometryandComplexityTheory
170 J.S.MilneAlgebraicGroups
171 J.Gough&J.KupschQuantumFieldsandProcesses
172 T.Ceccherini-Silberstein,F.Scarabotti&F.TolliDiscreteHarmonicAnalysis
173 P.GarrettModernAnalysisofAutomorphicFormsbyExample,I
174 P.GarrettModernAnalysisofAutomorphicFormsbyExample,II
175 G.NavarroCharacterTheoryandtheMcKayConjecture
176 P.Fleig,H.P.A.Gustafsson,A.Kleinschmidt&D.PerssonEisensteinSeriesandAutomorphic
Representations
177 E.PetersonFormalGeometryandBordismOperators
178 A.OgusLecturesonLogarithmicAlgebraicGeometry
179 N.NikolskiHardySpaces
180 D.-C.CisinskiHigherCategoriesandHomotopicalAlgebra
181 A.Agrachev,D.Barilari&U.BoscainAComprehensiveIntroductiontoSub-RiemannianGeometry
182 N.NikolskiToeplitzMatricesandOperators
183 A.YekutieliDerivedCategories
184 C.DemeterFourierRestriction,DecouplingandApplications
185 D.Barnes&C.RoitzheimFoundationsofStableHomotopyTheory
186 V.Vasyunin&A.VolbergTheBellmanFunctionTechniqueinHarmonicAnalysis
187 M.Geck&G.MalleTheCharacterTheoryofFiniteGroupsofLieType
188 B.RichterCategoryTheoryforHomotopyTheory
189 R.Willett&G.YuHigherIndexTheory
190 A.BobrowskiGeneratorsofMarkovChains
191 D.Cao,S.Peng&S.YanSingularlyPerturbedMethodsforNonlinearEllipticProblems
192 E.KowalskiAnIntroductiontoProbabilisticNumberTheory
193 V.GorinLecturesonRandomLozengeTilings
194 E.Riehl&D.VerityElementsof∞-CategoryTheory
195 H.KrauseHomologicalTheoryofRepresentations
196 F.Durand&D.PerrinDimensionGroupsandDynamicalSystems
197 A.ShefferPolynomialMethodsandIncidenceTheory
198 T.Dobson,A.Malnicˇ&D.MarušicˇSymmetryinGraphs
199 K.S.Kedlayap-adicDifferentialEquations
200 R.L.Frank,A.Laptev&T.WeidlSchrödingerOperators:EigenvaluesandLieb–ThirringInequalities
201 J.vanNeervenFunctionalAnalysis
202 A.SchmedingAnIntroductiontoInfinite-DimensionalDifferentialGeometry
203 F.CabelloSánchez&J.M.F.CastilloHomologicalMethodsinBanachSpaceTheory
Published online by Cambridge University Press
Geometric Inverse Problems
With Emphasis on Two Dimensions
GABRIEL P. PATERNAIN
UniversityofCambridge
MIKKO SALO
UniversityofJyväskylä,Finland
GUNTHER UHLMANN
UniversityofWashington
Published online by Cambridge University Press
ShaftesburyRoad,CambridgeCB28EA,UnitedKingdom
OneLibertyPlaza,20thFloor,NewYork,NY10006,USA
477WilliamstownRoad,PortMelbourne,VIC3207,Australia
314–321,3rdFloor,Plot3,SplendorForum,JasolaDistrictCentre,
NewDelhi–110025,India
103PenangRoad,#05–06/07,VisioncrestCommercial,Singapore238467
CambridgeUniversityPressispartofCambridgeUniversityPress&Assessment,
adepartmentoftheUniversityofCambridge.
WesharetheUniversity’smissiontocontributetosocietythroughthepursuitof
education,learningandresearchatthehighestinternationallevelsofexcellence.
www.cambridge.org
Informationonthistitle:www.cambridge.org/9781316510872
DOI:10.1017/9781009039901
©GabrielP.Paternain,MikkoSaloandGuntherUhlmann2023
Thispublicationisincopyright.Subjecttostatutoryexception
andtotheprovisionsofrelevantcollectivelicensingagreements,
noreproductionofanypartmaytakeplacewithoutthewritten
permissionofCambridgeUniversityPress&Assessment.
Firstpublished2023
AcataloguerecordforthispublicationisavailablefromtheBritishLibrary.
LibraryofCongressCataloging-in-PublicationData
Names:Paternain,GabrielP.(GabrielPedro),1964–author.|
Salo,Mikko,author.|Uhlmann,Gunther,1952–author.
Title:Geometricinverseproblems:withemphasisontwodimensions/
GabrielP.Paternain,MikkoSalo,GuntherUhlmann.
Description:Cambridge;NewYork,NY:CambridgeUniversityPress,2023.|
Series:Cambridgestudiesinadvancedmathematics,0950-6330;204|
Includesbibliographicalreferencesandindex.
Identifiers:LCCN2022030681(print)|LCCN2022030682(ebook)|
ISBN9781316510872(hardback)|ISBN9781009039901(epub)
Subjects:LCSH:Inverseproblems(Differentialequations)|
Inversions(Geometry)|Geometry,Differential.
Classification:LCCQA378.5.P382023(print)|LCCQA378.5(ebook)|
DDC515/.357–dc23/eng20221013
LCrecordavailableathttps://lccn.loc.gov/2022030681
LCebookrecordavailableathttps://lccn.loc.gov/2022030682
ISBN978-1-316-51087-2Hardback
CambridgeUniversityPress&Assessmenthasnoresponsibilityforthepersistence
oraccuracyofURLsforexternalorthird-partyinternetwebsitesreferredtointhis
publicationanddoesnotguaranteethatanycontentonsuchwebsitesis,orwill
remain,accurateorappropriate.
Published online by Cambridge University Press
Toourfamiliesandallwhohavesupportedus
Published online by Cambridge University Press
Published online by Cambridge University Press
Contents
ForewordbyAndrásVasy pagexi
Preface xv
Acknowledgements xxiii
1 TheRadonTransforminthePlane 1
1.1 UniquenessandStability 1
1.2 RangeandSupportTheorems 5
1.3 TheNormalOperatorandSingularities 8
1.4 TheFunkTransform 19
2 RadialSoundSpeeds 25
2.1 GeodesicsofaRadialSoundSpeed 25
2.2 TravelTimeTomography 31
2.3 GeodesicsofaRotationallySymmetricMetric 36
2.4 GeodesicX-rayTransform 39
2.5 ExamplesandCounterexamples 45
3 GeometricPreliminaries 52
3.1 Non-trappingandStrictConvexity 52
3.2 RegularityoftheExitTime 58
3.3 TheGeodesicFlowandtheScatteringRelation 63
3.4 ComplexStructure 65
3.5 TheUnitCircleBundleofaSurface 75
3.6 TheUnitSphereBundleinHigherDimensions 81
3.7 ConjugatePointsandMorseTheory 87
3.8 SimpleManifolds 99
4 TheGeodesicX-rayTransform 107
4.1 TheGeodesicX-rayTransform 107
4.2 TransportEquations 110
vii
Published online by Cambridge University Press
viii Contents
4.3 PestovIdentity 112
4.4 InjectivityoftheGeodesicX-rayTransform 114
4.5 StabilityEstimateinNon-positiveCurvature 117
4.6 StabilityEstimateintheSimpleCase 123
4.7 TheHigherDimensionalCase 126
5 RegularityResultsfortheTransportEquation 130
5.1 SmoothFirstIntegrals 130
5.2 FoldsandtheScatteringRelation 132
5.3 AGeneralRegularityResult 135
∗
5.4 TheAdjointI 138
A
6 VerticalFourierAnalysis 142
6.1 VerticalFourierExpansions 142
6.2 TheFibrewiseHilbertTransform 149
6.3 SymmetricTensorsasFunctionsonSM 151
6.4 TheX-rayTransformonTensors 157
6.5 Guillemin–KazhdanIdentity 162
6.6 TheHigherDimensionalCase 166
7 TheX-rayTransforminNon-positiveCurvature 171
7.1 TensorTomography 171
7.2 StabilityforFunctions 173
7.3 StabilityforTensors 179
7.4 CarlemanEstimates 182
7.5 TheHigherDimensionalCase 186
∗
8 MicrolocalAspects,SurjectivityofI 189
0
8.1 TheNormalOperator 189
∗
8.2 SurjectivityofI 196
0
8.3 StabilityEstimatesBasedontheNormalOperator 201
8.4 TheNormalOperatorwithaMatrixWeight 203
9 InversionFormulasandRange 208
9.1 Motivation 208
9.2 PropertiesofSolutionsoftheJacobiEquation 211
9.3 TheSmoothingOperatorW 213
9.4 FredholmInversionFormulas 218
9.5 RevisitingtheEuclideanCase 224
9.6 Range 227
9.7 NumericalImplementation 229
Published online by Cambridge University Press
Contents ix
10 TensorTomography 233
10.1 HolomorphicIntegratingFactors 233
10.2 TensorTomography 236
10.3 RangeforTensors 238
11 BoundaryRigidity 241
11.1 TheBoundaryRigidityProblem 241
11.2 BoundaryDetermination 244
11.3 DeterminingtheLensDataandVolume 253
11.4 RigidityinaGivenConformalClass 256
11.5 DeterminingtheDirichlet-to-NeumannMap 257
11.6 CalderónProblem 260
11.7 BoundaryRigidityforSimpleSurfaces 267
12 TheAttenuatedGeodesicX-rayTransform 269
12.1 TheAttenuatedX-rayTransforminthePlane 269
12.2 InjectivityResultsforScalarAttenuations 271
∗
12.3 SurjectivityofI 275
⊥
12.4 DiscussiononGeneralWeights 276
13 Non-AbelianX-rayTransforms 277
13.1 ScatteringData 277
13.2 Pseudo-linearizationIdentity 280
13.3 ElementaryBackgroundonConnections 281
13.4 StructureEquationsIncludingaConnection 283
13.5 ScatteringRigidityandInjectivityforConnections 286
13.6 AnAlternativeProofofTensorTomography 290
13.7 GeneralSkew-HermitianAttenuations 293
13.8 InjectivityforConnectionsandHiggsFields 295
13.9 ScatteringRigidityforConnectionsandHiggsFields 298
13.10 MatrixHolomorphicIntegratingFactors 299
13.11 StabilityEstimate 302
14 Non-AbelianX-rayTransformsII 304
14.1 ScatteringRigidityandInjectivityResultsforgl(n,C) 304
14.2 AFactorizationTheoremfromLoopGroups 308
14.3 ProofofTheorems14.1.1and14.1.2 310
14.4 GeneralLieGroups 314
14.5 RangeofIA,0andIA,⊥ 317
∗ ∗
14.6 SurjectivityofI andI 321
A,0 A,⊥
14.7 AddingaMatrixField 324
Published online by Cambridge University Press
