[黎曼几何].postnikov.riemannian.geometry

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1、M.M.PostnikovGeometryVIRiemannianGeometrySpringerPrefaceTheoriginalRussianeditionofthisbookisthefifthinmyseries“LecturesonGeometry.”Therefore,tomakethepresentationrelativelyindependentandself-containedintheEnglishtranslation,Ihaveaddedsupplementarychaptersinaspecialaddendum(C

2、haps.30-36),inwhichthenecessaryfactsfrommanifoldtheoryandvectorbundletheoryarebrieflysummarizedwithoutproofsasarule.Intheoriginaledition,thebookisdividednotintochaptersbutintolec-tures.ThisisexplainedbyitsoriginasclassroomlecturesthatIgave.Theprincipaldistinctionbetweenchapte

3、rsandlecturesisthatthematerialofeachchaptershouldbecompletetoacertainextentandthelengthofchapterscandiffer,while,incontrast,alllecturesshouldbeapproximatelythesameinlengthandthetopicofanylecturecanchangesuddenlyinthemiddle.Fortheseries“EncyclopediaofMathematicalSciences,”theo

4、riginofabookhasnosignificance,andthename“chapter”ismoreusual.Therefore,thenameofsubdivisionswaschangedinthetranslation,althoughnostructuralsurgerywasperformed.Ihavealsoaddedabriefbibliography,whichwasabsentintheoriginaledition.Thefirsttenchaptersaredevotedtothegeometryofaffin

5、econnectionspaces.Inthefirstchapter,Ipresentthemainpropertiesofgeodesicsinthesespaces.Chapter2isdevotedtotheformalismofcovariantderivatives,torsiontensor,andcurvaturetensor.ThemajorpartofChap.3isdevotedtothegeometryofsubmanifoldsofaffineconnectionspaces(Gauss-Weingartenformul

6、as,etc.).InChap.4,Cartanstructuralequationsinpolarcoordinatesarededuced.Thesecondhalfofthischapterisdevotedtolocallysymmetricaffineconnec-tionspaces.GloballysymmetricspacesareconsideredinChap.5(andthebeginningofChap.6).Inparticular,theircoincidencewithsymmetricspaceintheLooss

7、enseisproved.InthemajorpartofChap.6,thegeneraltheoryisillustratedbyexaminingLiegroups.InChap.7,thelanguageofcategoriesandfunctorsisexplained(thismaterialissetinasmallerfont),andalsothemaintheoremsontherelationbetweenLiegroupsandLiealgebrasarepresentedinessencewithoutproofs.In

8、Chaps.8and9,thesetheoremsaregeneralizedtothecaseofsymmetricspaces;in