A Spinorial Approach to Riemannian and Conformal Geometry

462 Pages·2015·2.154 MB·English

by  Jean-Pierre Bourguignon et al.

EMS Monographs in Mathematics
Edited by
Ivar Ekeland (Pacific Institute, Vancouver, Canada)
Gerard van der Geer (University of Amsterdam, The Netherlands)
Helmut Hofer (Institute for Advanced Study, Princeton, USA)
Thomas Kappeler (University of Zürich, Switzerland)
EMS Monographs in Mathematics is a book series aimed at mathematicians and scientists. It publishes
research monographs and graduate level textbooks from all fields of mathematics. The individual volumes are
intended to give a reasonably comprehensive and selfcontained account of their particular subject.
They present mathematical results that are new or have not been accessible previously in the literature.
Previously published in this series:
Richard Arratia, A.D. Barbour and Simon Tavaré, Logarithmic Combinatorial Structures: A Probabilistic Approach
Demetrios Christodoulou, The Formation of Shocks in 3-Dimensional Fluids
Sergei Buyalo and Viktor Schroeder, Elements of Asymptotic Geometry
Demetrios Christodoulou, The Formation of Black Holes in General Relativity
Joachim Krieger and Wilhelm Schlag, Concentration Compactness for Critical Wave Maps
Jean-Pierre Bourguignon
Oussama Hijazi
Jean-Louis Milhorat
Andrei Moroianu
Sergiu Moroianu
A Spinorial Approach
to Riemannian and
Conformal Geometry
Authors:
Jean-Pierre Bourguignon Oussama Hijazi
IHÉS Institut Élie Cartan de Lorraine (IÉCL)
Le Bois-Marie Université de Lorraine, Nancy
35 route de Chartres B. P. 239
91440 Bures-sur-Yvette, France 54506 Vandoeuvre-Lès-Nancy Cedex, France
[email protected] [email protected]
Jean-Louis Milhorat Andrei Moroianu
Département de Mathématiques, Université de Versailles-St Quentin
Université de Nantes Laboratoire de Mathématiques
2, rue de la Houssinière, B.P. 92208 UMR 8100 du CNRS
44322 Nantes, France 45 avenue des États-Unis
78035 Versailles, France
[email protected]
[email protected]
Sergiu Moroianu
Institutul de Matematica˘ al Academiei Române
Calea Grivit‚ei 21
010702 Bucures‚ti, Romania
[email protected]
2010 Mathematics Subject Classification: Primary: 53C27, 53A30, Secondary: 53C26, 53C55, 53C80, 17B10,
34L40, 35S05
Key words: Dirac operator, Penrose operator, Spin geometry, Spinc geometry, conformal geometry, Kähler
manifolds, Quaternion-Kähler manifolds, Weyl geometry, representation theory, Killing spinors, eigenvalues
ISBN 978-3-03719-136-1
The Swiss National Library lists this publication in The Swiss Book, the Swiss national bibliography, and the
detailed bibliographic data are available on the Internet at http://www.helveticat.ch.
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9 8 7 6 5 4 3 2 1
Contents
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
PartI Basicspinorialmaterial . . . . . . . . . . . . . . . . . . . . . . . . 9
1 Algebraicaspects. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11
1.1 Cliffordalgebras . . . . . . . . . . . . . . . . . . . . . . . . . . . 11
1.1.1 Definitions . . . . . . . . . . . . . . . . . . . . . . . . . 11
1.1.2 ClassificationofCliffordalgebras . . . . . . . . . . . . . . 18
1.2 Spingroupsandtheirrepresentations . . . . . . . . . . . . . . . . 20
1.2.1 Spingroups . . . . . . . . . . . . . . . . . . . . . . . . . 20
1.2.2 Representationsofspingroups . . . . . . . . . . . . . . . 28
1.2.3 Realandquaternionicstructures . . . . . . . . . . . . . . 36
2 Geometricalaspects . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39
2.1 Spinorialstructures . . . . . . . . . . . . . . . . . . . . . . . . . 39
2.1.1 Spinstructuresandspinorialmetrics . . . . . . . . . . . . . 39
2.1.2 Spinorialconnectionsandcurvatures . . . . . . . . . . . . 42
2.2 Spincandconformalstructures . . . . . . . . . . . . . . . . . . . . 45
2.2.1 Spincstructures . . . . . . . . . . . . . . . . . . . . . . . 45
2.2.2 Weylstructures . . . . . . . . . . . . . . . . . . . . . . . 49
2.2.3 SpinandSpincconformalmanifolds . . . . . . . . . . . . . 52
2.3 Naturaloperatorsonspinors . . . . . . . . . . . . . . . . . . . . . 56
2.3.1 Generalalgebraicsetting . . . . . . . . . . . . . . . . . . 57
2.3.2 First-orderdifferentialoperators. . . . . . . . . . . . . . . 59
2.3.3 Basicdifferentialoperatorsonspinorfields . . . . . . . . . 61
2.3.4 TheDiracoperator:basicpropertiesandexamples . . . . . 63
2.3.5 ConformalcovarianceoftheDiracandPenroseoperators . . 69
2.3.6 ConformallycovariantpowersoftheDiracoperator. . . . . 72
vi Contents
2.4 Spinorsinclassicalgeometricalcontexts . . . . . . . . . . . . . . . 73
2.4.1 Restrictionsofspinorstohypersurfaces . . . . . . . . . . . 73
2.4.2 Spinorsonwarpedproducts . . . . . . . . . . . . . . . . . 75
2.4.3 SpinorsonRiemanniansubmersions . . . . . . . . . . . . 76
2.5 TheSchrödinger–Lichnerowiczformula . . . . . . . . . . . . . . . 81
3 Topologicalaspects. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85
3.1 Topologicalaspectsofspinstructures . . . . . . . . . . . . . . . . 85
3.1.1 Cˇechcohomologyandprincipalbundles . . . . . . . . . . . 86
3.1.2 Liftingprincipalbundlesviacentralextensions . . . . . . . 88
3.1.3 Stiefel–Whitneyclasses . . . . . . . . . . . . . . . . . . . 90
3.2 TopologicalclassificationofSpincstructures . . . . . . . . . . . . . 91
3.3 Spinstructuresinlowdimensions . . . . . . . . . . . . . . . . . . 93
3.3.1 Dimension1 . . . . . . . . . . . . . . . . . . . . . . . . . 93
3.3.2 Dimension2 . . . . . . . . . . . . . . . . . . . . . . . . . 94
3.3.3 Dimensions3and4 . . . . . . . . . . . . . . . . . . . . . 95
3.4 Examplesofobstructedmanifolds . . . . . . . . . . . . . . . . . . 96
4 Analyticalaspects . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 99
4.1 Fouriertransform . . . . . . . . . . . . . . . . . . . . . . . . . . 99
4.2 Pseudo-differentialcalculus . . . . . . . . . . . . . . . . . . . . . 101
4.2.1 Symbols . . . . . . . . . . . . . . . . . . . . . . . . . . . 101
4.2.2 Asymptoticsummation . . . . . . . . . . . . . . . . . . . 102
4.3 Pseudo-differentialoperators . . . . . . . . . . . . . . . . . . . . 104
4.4 Compositionofpseudo-differentialoperators . . . . . . . . . . . . 107
4.5 Actionofdiffeomorphismsonpseudo-differentialoperators . . . . . 108
4.6 Pseudo-differentialoperatorsonvectorbundles . . . . . . . . . . . 109
4.7 Ellipticoperators. . . . . . . . . . . . . . . . . . . . . . . . . . . 112
4.8 Adjoints . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113
4.9 Sobolevspaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113
4.10 Compactoperators . . . . . . . . . . . . . . . . . . . . . . . . . . 118
4.11 Eigenvaluesofself-adjointellipticoperators . . . . . . . . . . . . . 121
PartII LowesteigenvaluesoftheDiracoperatoronclosedspinmanifolds . . 125
5 LowereigenvalueboundsonRiemannianclosedspinmanifolds . . . . . . 127
5.1 TheLichnerowicztheorem . . . . . . . . . . . . . . . . . . . . . . 127
5.2 TheFriedrichinequality . . . . . . . . . . . . . . . . . . . . . . . 129
5.3 Specialspinorfields . . . . . . . . . . . . . . . . . . . . . . . . . 133
5.4 TheHijaziinequality . . . . . . . . . . . . . . . . . . . . . . . . . 137
5.5 TheactionofharmonicformsonKillingspinors . . . . . . . . . . . 140
Contents vii
5.6 OtherestimatesoftheDiracspectrum . . . . . . . . . . . . . . . . 142
5.6.1 Moroianu–Ornea’sestimate . . . . . . . . . . . . . . . . . 142
5.7 Furtherdevelopments . . . . . . . . . . . . . . . . . . . . . . . . 146
5.7.1 Theenergy–momentumtensor . . . . . . . . . . . . . . . 147
5.7.2 Witten’sproofofthepositivemasstheoremand
applications . . . . . . . . . . . . . . . . . . . . . . . . . 148
5.7.3 Furtherapplications . . . . . . . . . . . . . . . . . . . . . 158
6 LowereigenvalueboundsonKählermanifolds . . . . . . . . . . . . . . . 161
6.1 Kählerianspinorbundledecomposition . . . . . . . . . . . . . . . 161
6.2 Thecanonicallinebundle . . . . . . . . . . . . . . . . . . . . . . 164
6.3 Kähleriantwistoroperators . . . . . . . . . . . . . . . . . . . . . 166
6.4 ProofofKirchberg’sinequalities . . . . . . . . . . . . . . . . . . . 170
6.5 Thelimitingcase . . . . . . . . . . . . . . . . . . . . . . . . . . . 172
7 Lowereigenvalueboundsonquaternion-Kählermanifolds . . . . . . . . . 175
7.1 Thegeometryofquaternion-Kählermanifolds . . . . . . . . . . . . 176
7.2 Quaternion-Kählerspinorbundledecomposition . . . . . . . . . . 179
7.3 Themainestimate . . . . . . . . . . . . . . . . . . . . . . . . . . 182
7.4 Thelimitingcase . . . . . . . . . . . . . . . . . . . . . . . . . . . 190
7.5 Asystematicapproach . . . . . . . . . . . . . . . . . . . . . . . . 202
7.5.1 GeneralWeitzenböckformulas . . . . . . . . . . . . . . . 202
7.5.2 Applicationtoquaternion-Kählermanifolds. . . . . . . . . 206
7.5.3 Proofoftheestimate . . . . . . . . . . . . . . . . . . . . 210
PartIII Specialspinorfieldandgeometries . . . . . . . . . . . . . . . . . . 213
8 SpecialspinorsonRiemannianmanifolds . . . . . . . . . . . . . . . . . 215
8.1 ParallelspinorsonspinandSpincmanifolds . . . . . . . . . . . . . 215
8.1.1 Parallelspinorsonspinmanifolds . . . . . . . . . . . . . . 215
8.1.2 ParallelspinorsonSpincmanifolds . . . . . . . . . . . . . 219
8.2 Specialholonomiesandrelationstowarpedproducts . . . . . . . . 223
8.2.1 Sasakianstructures . . . . . . . . . . . . . . . . . . . . . 223
8.2.2 3-Sasakianstructures . . . . . . . . . . . . . . . . . . . . 225
8.2.3 TheexceptionalgroupG2 . . . . . . . . . . . . . . . . . . 227
8.2.4 NearlyKählermanifolds. . . . . . . . . . . . . . . . . . . 229
8.2.5 ThegroupSpin7 . . . . . . . . . . . . . . . . . . . . . . . 233
8.3 ClassificationofmanifoldsadmittingrealKillingspinors . . . . . . . 236
8.4 DetectingmodelspacesbyKillingspinors . . . . . . . . . . . . . . 239
8.5 GeneralizedKillingspinors . . . . . . . . . . . . . . . . . . . . . 241
8.6 TheCauchyproblemforEinsteinmetrics . . . . . . . . . . . . . . 243
viii Contents
9 Specialspinorsonconformalmanifolds . . . . . . . . . . . . . . . . . . 251
9.1 TheconformalSchrödinger–Lichnerowiczformula . . . . . . . . . . 251
9.2 ParallelspinorswithrespecttoWeylstructures . . . . . . . . . . . . 254
9.2.1 ParallelconformalspinorsonRiemannsurfaces . . . . . . . 254
9.2.2 Thenon-compactcase . . . . . . . . . . . . . . . . . . . . 256
9.2.3 Thecompactcase . . . . . . . . . . . . . . . . . . . . . . 262
9.3 AconformalproofoftheHijaziinequality . . . . . . . . . . . . . . 263
10 SpecialspinorsonKählermanifolds . . . . . . . . . . . . . . . . . . . . 265
10.1 Anintroductiontothetwistorcorrespondence . . . . . . . . . . . . 265
10.1.1 Quaternion-Kählermanifolds . . . . . . . . . . . . . . . . 265
10.1.2 3-Sasakianstructures . . . . . . . . . . . . . . . . . . . . 267
10.1.3 Thetwistorspaceofaquaternion-Kählermanifold . . . . . 268
10.2 KählerianKillingspinors . . . . . . . . . . . . . . . . . . . . . . . 269
10.3 ComplexcontactstructuresonpositiveKähler–Einsteinmanifolds . . 272
10.4 ThelimitingcaseofKirchberg’sinequalities . . . . . . . . . . . . . 275
11 Specialspinorsonquaternion-Kählermanifolds . . . . . . . . . . . . . . 279
11.1 Thecanonical3-SasakianSO3-principalbundle . . . . . . . . . . . 280
11.2 TheDiracoperatoractingonprojectablespinors . . . . . . . . . . . 283
11.3 Characterizationofthelimitingcase . . . . . . . . . . . . . . . . . 288
11.4 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 299
PartIV Diracspectraofmodelspaces . . . . . . . . . . . . . . . . . . . . 301
12 Abriefsurveyonrepresentationtheoryofcompactgroups. . . . . . . . . 305
12.1 Reductionoftheproblemtothestudyofirreduciblerepresentations . 305
12.2 Reductionof theproblemtothestudy of irreduciblerepresentations
ofamaximaltorus . . . . . . . . . . . . . . . . . . . . . . . . . . 315
12.3 Characterizationofirreduciblerepresentationsbymeansofdominant
weights . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 321
12.3.1 Restrictiontosemi-simplesimplyconnectedgroups . . . . . 321
12.3.2 Rootsandtheirproperties . . . . . . . . . . . . . . . . . . 322
12.3.3 IrreduciblerepresentationsofthegroupSU2 . . . . . . . . 325
12.3.4 Proofofthefundamentalpropertiesofroots. . . . . . . . . 327
12.3.5 Dominantweights . . . . . . . . . . . . . . . . . . . . . . 334
12.3.6 TheWeylformulas . . . . . . . . . . . . . . . . . . . . . 339
Contents ix
12.4 Application: irreducible representations of the classical groups SUn,
Spin ,andSp . . . . . . . . . . . . . . . . . . . . . . . . . . . . 343
n n
12.4.1 IrreduciblerepresentationsofthegroupsSUn
andUn,n 3 . . . . . . . . . . . . . . . . . . . . . . . . 343

12.4.2 IrreduciblerepresentationsofthegroupsSpin
n
andSOn,n 3 . . . . . . . . . . . . . . . . . . . . . . . 348

12.4.3 IrreduciblerepresentationsofthegroupSp . . . . . . . . . 361
n
13 Symmetricspacestructureofmodelspaces . . . . . . . . . . . . . . . . . 369
13.1 Symmetricspacestructureofspheres . . . . . . . . . . . . . . . . . 372
13.2 Symmetricspacestructureofthecomplexprojectivespace . . . . . . 374
13.3 Symmetricspacestructureofthequaternionicprojectivespace. . . . 375
14 Riemanniangeometryofmodelspaces . . . . . . . . . . . . . . . . . . . 379
14.1 TheLevi-Civitaconnection . . . . . . . . . . . . . . . . . . . . . 381
14.2 Spinstructures . . . . . . . . . . . . . . . . . . . . . . . . . . . . 388
14.3 Spinorbundlesonsymmetricspaces . . . . . . . . . . . . . . . . . 390
14.4 TheDiracoperatoronsymmetricspaces . . . . . . . . . . . . . . . 392
15 ExplicitcomputationsoftheDiracspectrum . . . . . . . . . . . . . . . . 395
15.1 Thegeneralprocedure . . . . . . . . . . . . . . . . . . . . . . . . 395
15.2 SpectrumoftheDiracoperatoronspheres . . . . . . . . . . . . . . 407
15.3 SpectrumoftheDiracoperatoronthecomplexprojectivespace . . . 409
15.4 SpectrumoftheDiracoperatoronthequaternionicprojectivespace . 412
15.5 Otherexamplesofspectra . . . . . . . . . . . . . . . . . . . . . . 414
Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 421
Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 451