黎曼-芬斯勒几何导论

图书标准信息

• 作者 鲍 著
• 出版社 世界图书出版公司
• 出版时间 2009-08
• 版次 4
• ISBN 9787510005053
• 定价 50.00元
• 装帧 平装
• 开本 24开
• 纸张 胶版纸
• 页数 425页
• 正文语种 英语
• 原版书名 An Introduction to Riemann-Finsler Geometry

【内容简介】

　　ThesubjectmatterofthisbookhaditsgenesisinRiemanns1854"habil-itation"address："UberdieHypothesen，welchederGeometriezuGrundeliegen"（OntheHypotheses，whichlieattheFoundationsofGeometry）.VolumeIIofSpivaksDifferentialGeometrycontainsanEnglishtranslationofthisinfluentiallecture，withacommentarybySpivakhimself.Riemann，undoubtedlythegreatestmathematicianofthe19thcentury，aimedatintroducingthenotionofamanifoldanditsstructures.Theprob-leminvolvedgreatdifficulties.But，withhypothesesonthesmoothnessofthefunctionsinquestion，theissuescanbesettledsatisfactorilyandthereisnowacompletetreatment.Traditionally，thestructurebeingfocusedonistheRiemannianmetric，whichisaquadraticdifferentialform.Putanotherway，itisasmoothlyvaryingfamilyofinnerproducts，oneoneachtangentspace.Theresultinggeometry——Riemanniangeometry——hasundergonetremendousdevelop-mentinthiscentury.AreasinwhichithashadsignificantimpactincludeEinsteinstheoryofgeneralrelativity，andglobaldifferentialgeometry.InthecontextofRiemannslecture，thisrestrictiontoaquadraticdif-ferentialform

【目录】

Preface
Acknowledgments
PARTONE
FinslerManifoldsandTheirCurvature
CHAPTER1
FinslerManifoldsand
theFundamentalsofMinkowskiNorms
1.0PhysicalMotivations
1.1FinslerStructures：DefinitionsandConventions
1.2TwoBasicPropertiesofMinkowskiNorms
1.2A.EnlersTheorem
1.2B.AFundamentalInequality
1.2C.InterpretationsoftheFundamentalInequality.
1.3ExplicitExamplesofFinslerManifolds
1.3A.MinkowskiandLocallyMinkowskiSpaces
1.3B.RiemannianManifolds
1.3C.RandersSpaces
1.3D.BerwaldSpaces
1.3E.FinslerSpacesofConstantFlagCurvature
1.4TheFundamentalTensorandtheCartanTensor
RefereneesforChapter1

CHAPTER2
TheChernConnection
2.0Prologue
2.1TheVectorBundleandRelatedObjects
2.2CoordinateBasesVersusSpecialOrthonormalBases
2.3TheNonlinearConnectionontheManifoldTM\O
2.4TheChernConnectionon
2.5IndexGymnastics
2.5A.TheSlash(..-)sandtheSemicolon(...);s
2.5B.CovariantDerivativesoftheFundamentalTensorg
2.5C.CovariantDerivativesoftheDistinguished
ReferencesforChapter2

CHAPTER3
CurvatureandSchursLemma
3.1Conventionsandthehh-，hv-，w-curvatures
3.2FirstBianchiIdentitiesfromTorsionFreeness
3.3FormulasforRandPinNaturalCoordinates
3.4FirstBianchiIdentitiesfrom"Almost"g-compatibility
3.4A.Consequencesfromthe
3.4B.Consequencesfromthe
3.4C.Consequencesfromthe
3.5SecondBianchiIdentities
3.6InterchangeFormulasorRicciIdentities
3.7LieBracketsamongthe
Oy
3.8DerivativesoftheGeodesicSprayCoefficients
3.9TheFlagCurvature
3.9A.ItsDefinitionandItsPredecessor
3.9B.AnInterestingFamilyofExamplesofNumataType
3.10SchursLemma
ReferencesforChapter3

CHAPTER4
FinslerSurfacesand
aGeneralizedGauss-BonnetTheorem
4.0Prologue
4.1MinkowskiPlanesandaUsefulBasis
4.1A.RundsDifferentialEquationandItsConsequence
4.1B.ACriterionforCheckingStrongConvexity
4.2TheEquivalenceProblemforMinkowskiPlanes
4.3TheBerwaldFrameandOurGeometricalSetuponSM
4.4TheChernConnectionandtheInvariantsI，J，K
4.5TheRiemannianArcLengthoftheIndicatrix
4.6AGauss-BonnetTheoremforLandsbergSurfaces
ReferencesforChapter4

PARTTWO
CalculusofVariationsandComparisonTheorems
CHAPTER5
VariationsofArcLength，
JacobiFields，theEffectofCurvature
5.1TheFirstVariationofArcLength
5.2TheSecondVariationofArcLength
5.3GeodesicsandtheExponentialMap
5.4JacobiFields
5.5HowtheFlagCurvaturesSignInfluencesGeodesicRays
ReferencesforChapter5

CHAPTER6
TheGaussLemmaandtheHopf-RinowTheorem
6.1TheGaussLemma
6.1A.TheGaussLemmaProper
6.1B.AnAlternativeFormoftheLemma
6.1C.IstheExponentialMapEveraLocalIsometry?
6.2FinslerManifoldsandMetricSpaces
6.2A.AUsefulTechnicalLemma
6.2B.ForwardMetricBallsandMetricSpheres
6.2C.TheManifoldTopologyVersustheMetricTopology.
6.2D.ForwardCauchySequences，ForwardCompleteness.
6.3ShortGeodesicsAreMinimizing
6.4TheSmoothnessofDistanceFunctions
6.4A.OnMinkowskiSpaces
6.4B.OnFinslerManifolds
6.5LongMinimizingGeodesics
6.6TheHopf-RinowTheorem
ReferencesforChapter6

CHAPTER7
TheIndexFormandtheBonnet-MyersTheorem
7.1ConjugatePoints
7.2TheIndexForm"
7.3WhatHappensintheAbsenceofConjugatePoints?
7.3A.GeodesicsAreShortestAmong"Nearby"Curves...
7.3B.ABasicIndexLemma
7.4WhatHappensIfConjugatePointsArePresent?
7.5TheCutPointVersustheFirstConjugatePoint
7.6RicciCurvatures
7.6A.TheRicciScalarRicandtheRicciTensorRicij
7.6B.TheInterplaybetweenRicandRicij
7.7TheBonnet-MyersTheorem
ReferencesforChapter7

CHAPTER8
TheCutandConjugateLoci，andSyngesTheorem
8.1Definitions
8.2TheCutPointandtheFirstConjugatePoint
8.3SomeConsequencesoftheInverseFunctionTheorem
8.4TheMannerinWhichandDependony
8.5GenericPropertiesoftheCutLocus
8.6AdditionalPropertiesofCutxWhenMIsCompact
8.7ShortestGeodesicswithinHomotopyClasses
8.8SyngesTheorem
ReferencesforChapter8

CHAPTER9
TheCartan-HadamardTheoremand
RauchsFirstTheorem
9.1EstimatingtheGrowthofJacobiFields
9.2WhenDoLocalDiffeomorphismsBecomeCoveringMaps?.
9.3SomeConsequencesoftheCoveringHomotopyTheorem...
9.4TheCartan-HadamardTheorem
9.5PreludetoRauchsTheorem
9.5A.TransplantingVectorFields
9.5B.ASecondBasicPropertyoftheIndexForm
9.5C.FlagCurvatureVersusConjugatePoints
9.6RauchsFirstComparisonTheorem
9.7JacobiFieldsonSpaceForms
9.8ApplicationsofRauchsTheorem
ReferencesforChapter9