[黎曼几何].postnikov.riemannian.geometry

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1、M.M.PostnikovGeometryVIRiemannianGeometrySpringerPrefaceTheoriginalRussianeditionofthisbookisthefifthinmyseries“LecturesonGeometry.”Therefore,tomakethepresentationrelativelyindependentandself-containedintheEnglishtranslation,Ihaveaddedsupplementarychaptersinaspecialaddendum(Chaps.30-36),inwh

2、ichthenecessaryfactsfrommanifoldtheoryandvectorbundletheoryarebrieflysummarizedwithoutproofsasarule.Intheoriginaledition,thebookisdividednotintochaptersbutintolec-tures.ThisisexplainedbyitsoriginasclassroomlecturesthatIgave.Theprincipaldistinctionbetweenchaptersandlecturesisthatthematerialof

3、eachchaptershouldbecompletetoacertainextentandthelengthofchapterscandiffer,while,incontrast,alllecturesshouldbeapproximatelythesameinlengthandthetopicofanylecturecanchangesuddenlyinthemiddle.Fortheseries“EncyclopediaofMathematicalSciences,”theoriginofabookhasnosignificance,andthename“chapter

4、”ismoreusual.Therefore,thenameofsubdivisionswaschangedinthetranslation,althoughnostructuralsurgerywasperformed.Ihavealsoaddedabriefbibliography,whichwasabsentintheoriginaledition.Thefirsttenchaptersaredevotedtothegeometryofaffineconnectionspaces.Inthefirstchapter,Ipresentthemainpropertiesofg

5、eodesicsinthesespaces.Chapter2isdevotedtotheformalismofcovariantderivatives,torsiontensor,andcurvaturetensor.ThemajorpartofChap.3isdevotedtothegeometryofsubmanifoldsofaffineconnectionspaces(Gauss-Weingartenformulas,etc.).InChap.4,Cartanstructuralequationsinpolarcoordinatesarededuced.Thesecon

6、dhalfofthischapterisdevotedtolocallysymmetricaffineconnec-tionspaces.GloballysymmetricspacesareconsideredinChap.5(andthebeginningofChap.6).Inparticular,theircoincidencewithsymmetricspaceintheLoossenseisproved.InthemajorpartofChap.6,thegeneraltheoryisillustratedbyexaminingLiegroups.InChap.7,t

7、helanguageofcategoriesandfunctorsisexplained(thismaterialissetinasmallerfont),andalsothemaintheoremsontherelationbetweenLiegroupsandLiealgebrasarepresentedinessencewithoutproofs.InChaps.8and9,thesetheoremsaregeneralizedtothecaseofsymmetricspaces;in