数学物理中的整体分析（影印版）

　　The first edition of this book entitled Analysis on Riemannian Manifolds and Some Problems of Mathematical Physics was published by Voronezh University Press in 1989. For its English edition， the book has been substantially revised and expanded. In particular， new material has been added to Sections 19 and 20. I am grateful to Viktor L. Ginzburg for his hard work on the translation and for writing Appendix F， and to Tomasz Zastawniak for his numerous suggestions. My special thanks go to the referee for his valuable remarks on the theory of stochastic processes. Finally， I would like to acknowledge the support of the AMS fSU Aid Fund and the International Science Foundation （Grant NZB000）， which made possible my work on some of the new results included in the English edition of the book.此书为英文版。

暂缺《数学物理中的整体分析（影印版）》作者简介

PrefacetotheEnglishEdition
PrefacetotheRussianEdition
PartI.Finite-DimensionalDifferentialGeometryandMechanics
Chapter1
SomeGeometricConstructionsinCalculusonManifolds
1.CompleteRiemannianMetricsandtheCompletenessof
VectorFields
1.AANecessaryandSufficientConditionfortheCompletenessofaVectorField
1.BAWaytoConstructCompleteRiemannianMetrics
2.RiemannianManifolds
PossessingaUniformRiemannianAtlas
3.IntegralOperatorswithParallelTranslation
3.ATheOperatorS
3.BTheOperator
3.CIntegralOperators
Chapter2
GeometricFormalismofNewtonianMechanics
4.GeometricMechanics:Introductionand
ReviewofStandardExamples
4.ABasicNotions
4.BSomeSpecialClassesofForceFields
4.CMechanicalSystemsonGroups
5.GeometricMechanicswithLinearConstraints
5.ALinearMechanicalConstraints
5.BReducedConnections
5.CLengthMinimizingandLeast-ConstrainedNonholonomicGeodesics
6.MechanicalSystemswithDiscontinuousForcesandSystemswithControl:DifferentialInclusions
7.IntegralEquationsofGeometricMechanics:TheVelocityHodograph
7.AGeneralConstructions
7.BIntegralFormalismofGeometricMechanicswithConstraints
8.MechanicalInterpretationofParallelTranslationandSystemswithDelayedControlForce
Chapter3
AccessiblePointsofMechanicalSystems
9.ExamplesofPointsthatCannotBeConnectedbyaTrajectory
10.TheMainResultonAccessiblePoints
11.GeneralizationstoSystemswithConstraints
PartII.StochasticDifferentialGeometryanditsApplicationstoPhysics
Chapter4
StochasticDifferentialEquationsonRiemannianManifolds
12.ReviewoftheTheoryofStochasticEquationsandIntegralsonFinite-DimensionalLinearSpaces
12.AWienerProcesses
12.BTheIt5Integral
12.CTheBackwardIntegralandtheStratonovichIntegral
12.DTheIt5andStratonovichStochasticDifferentialEquations
12.ESolutionsofSDEs
12.FApproximationbySolutionsofOrdinaryDifferentialEquations
12.GARelationshipBetweenSDEsandPDEs
13.StochasticDifferentialEquationsonManifolds
14.StochasticParallelTranslationandtheIntegralFormalismfortheIt5Equations
15.WienerProcessesonRiemannianManifoldsandRelatedStochasticDifferentialEquations
15.AWienerProcessesonRiemannianManifolds
15.BStochasticEquations
15.CEquationswithIdentityastheDiffusionCoefficient
16.StochasticDifferentialEquationswithConstraints
Chapter5
TheLangevinEquation
17.TheLangevinEquationofGeometricMechanics
18.StrongSolutionsoftheLangevinEquation,Ornstein-UhlenbeckProcesses
Chapter6
MeanDerivatives,Nelson'sStochasticMechanics,andQuantization
19.MoreonStochasticEquationsandStochasticMechanicsinIRn
19.APreliminaries
19.BForwardMeanDerivatives
19.CBackwardMeanDerivativesandBackwardEquations
19.DSymmetricandAntisymmetricDerivatives
19.ETheDerivativesofaVectorFieldAlong(t)andtheAccelerationof(t)
19.FStochasticMechanics
20.MeanDerivativesandStochasticMechanicsonRiemannianManifolds
20.AMeanDerivativesonManifoldsandRelatedEquations
20.BGeometricStochasticMechanics
20.CTheExistenceofSolutionsinStochasticMechanics
21.RelativisticStochasticMechanics
PartIII.Infinite-DimensionalDifferentialGeometryandHydrodynamics
Chapter7
GeometryofManifoldsofDiffeomorphisms
22.ManifoldsofMappingsandGroupsofDiffeomorphisms
22.AManifoldsofMappings
22.BTheGroupofHs-Diffeomorphisms
22.CDiffeomorphismsofaManifoldwithBoundary
22.DSomeSmoothOperatorsandVectorBundlesoverDs(M)
23.WeakRiemannianMetricsandConnectionsonManifoldsofDiffeomorphisms
23.ATheCaseofaClosedManifold
23.BTheCaseofaManifoldwithBoundary
23.CTheStrongRiemannianMetric
24.LagrangianFormalismofHydrodynamicsofanIdealBarotropicFluid
24.ADiffuseMatter
24.BABarotropicFluid
Chapter8
LagrangianFormalismofHydrodynamicsofanIdealIncompressibleFluid
25.GeometryoftheManifoldofVolume-PreservingDiffeomorphismsandLHSsofanIdealIncompressibleFluid
25.AVolume-PreservingDiffeomorphismsofaClosedManifold
25.BVolume-PreservingDiffeomorphismsofaManifoldwithBoundary
25.CLHS'sofanIdealIncompressibleFluid
26.TheFlowofanIdealIncompressibleFluidonaManifoldwithBoundaryasanLHSwithanInfinite-DimensionalConstraintontheGroupofDiffeomorphismsofaClosedManifold
27.TheRegularityTheoremandaReviewofResultsontheExistenceofSolutions
Chapter9
HydrodynamicsofaViscousIncompressibleFluidandStochasticDifferentialGeometryofGroupsofDiffeomorphisms
28.StochasticDifferentialGeometryontheGroupsofDiffeomorphismsofthen-DimensionalTorus
29.AViscousIncompressibleFluid
Appendices
A.IntroductiontotheTheoryofConnections
ConnectionsonPrincipalBundles
ConnectionsontheTangentBundle
CovariantDerivatives
ConnectionCoefficientsandChristoffelSymbols
Second-OrderDifferentialEquationsandtheSpray
TheExponentialMapandNormalCharts
B.IntroductiontotheTheoryofSet-ValuedMaps
C.BasicDefinitionsofProbabilityTheoryandtheTheoryofStochasticProcesses
StochasticProcessesandCylinderSets
TheConditionalExpectation
MarkovianProcesses
MartingalesandSemimartingales
D.TheIt8GroupandthePrincipalIt8Bundle
E.SobolevSpaces
F.AccessiblePointsandClosedTrajectoriesofMechanicalSystems(byViktorL.Ginzburg)
GrowthoftheForceFieldandAccessiblePoints
AccessiblePointsinSystemswithConstraints
ClosedTrajectoriesofMechanicalSystems
References
Index