黎曼几何

Prefacetothefirstedition

Prefacetothesecondedition

PrefacetotheEnglishedition

Howtousethisbook

CHAPTER0-DIFFERENTIABLEMANIFOLDS

1.Introduction

2.Differentiablemanifolds；tangentspace

3.Immersionsandembeddings；examples

4.Otherexamplesofmanifolds，Orientation

5.Vectorfields；brackets，Topologyofmanifolds

CHAPTER1-RIEMANNIANMETRICS

1.Introduction

2.RiemannianMetrics

CHAPTER2-AFFINECONNECTIONS；RIEMANNIANCONNECTIONS

1.Introduction

2.Affineconnections

3.Riemannianconnections

CHAPTER3-GEODESICS；CONVEXNEIGHBORHOODS

1．Introduction

2．Thegeodesicflow

3．Minimizingpropertiesofgeodesics

4．Convexneighborhoods

CHAPTER4-CURVATURE

1．Introduction

2．Curvature

3．Sectionalcurvature

4．Riccicurvatureand8calarcurvature

4．Riccicurvatureand8calarcurvature

5．Tensors0nRiemannianmanifoids

CHAPTER5-JACOBIFIELDS

1．Introduction

2．TheJacobiequation

3．Conjugatepoints

CHAPTER6-ISOMETRICIMMERSl0NS

1．Introduction．

2．Thesecondfundamentalform

3．Thefundarnentalequations

CHAPTER7-COMPLETEMANIFoLDS；HOPF-RINOWANDHADAMARDTHEOREMS

1．Introduction．

2．Completemanifolds；Hopf-RinowTheorem．

3．TheTheoremofHadamazd．

CHAPTER8-SPACES0FCONSTANTCURVATURE

1．Introduction

2．TheoremofCartanonthedeterminationofthemetricbymebnsofthecurvature．

3．Hyperbolicspace

4．Spaceforms

5．Isometriesofthehyperbolicspace；TheoremofLiouville

CHAPTER9一VARIATl0NS0FENERGY

1．Introduction．

2．Formulasforthefirstandsecondvariationsofenezgy

3．ThetheoremsofBonnetMyersandofSynge-WeipJtein

CHAPTER10-THERAUCHCOMPARISONTHEOREM

1．Introduction

2．TtleTheoremofRauch．

3．ApplicationsoftheIndexLemmatoimmersions

4．FocalpointsandanextensionofRauch’sTheorem

CHAPTER11THEMORSElNDEXTHEOREM

1．Introduction

2.TheIndexTheorem

CHAPTER12-THEFUNDAMENTALGROUPOFMANIFOLDS0FNEGATIVECURVATURE

1．Introduction

2．Existenceofclosedgeodesics

3.PreissmansTheorem

CHAPTER13-THESPHERETHEOREM

1.Introduction

2.Thecutlocus

3.Theestimateoftheinjectivityradius

4.TheSphereTheorem

5.Somefurtherdevelopments

References

Index

　　TheobjectofthisbookistofamiliarizethereaderwiththebasiclanguageofandsomefundamentaltheoremsinRiemannianGeometry.Toavoidreferringtopreviousknowledgeofdifferentiablemanifolds,weincludeChapter0,whichcontainsthoseconceptsandresultsondifferentiablemanifoldswhichareusedinanessentialwayintherestofthebook.　　ThefirstfourchaptersofthebookpresentthebasicconceptsofRiemannianGeometry(Riemannianmetrics,Riemannianconnections,geodesicsandcurvature).AgoodpartofthestudyofRiemannianGeometryconsistsofunderstandingtherelationshipbetweengeodesicsandcurvature.Jacobifields,anessentialtoolforthisunderstanding,areintroducedinChapter5.InChapter6weintroducethesecondfundamentalformassociatedwithanisometricimmersion,andproveageneralizationoftheTheoremEgregiumofGauss.ThisallowsustorelatethenotionofcurvatureinRiemannianmanifoldstotheclassicalconceptofGaussiancurvatureforsurfaces.