ExpositorybooksonthetheoryofLiegroupsgenerallyconfinethemselvestothelocalaspectofthetheory.Thislimitationwasprobablynecessaryaslongasgeneraltopologywasnotyetsufficientlywellelaboratedtoprovideasolidbaseforatheoryinthelarge.Thesedaysarenowpassed，andwehavethoughtthatitwouldbe'usefultohaveasystematictreatmentofthetheoryfromaglobalpointofview.ThepresentvolumeintroducesthemainbasicprincipleswhichgovernthetheoryofLiegroups.　　ALiegroupisatthesametimeagroup，atopologicalspaceandamanifold:ithasthereforethreekindsof"structures，"whichareinterrelatedwitheachother.Theelem，entarypropertiesofabstractgroupsarebynowsufficientlywellknowntothegeneralmathematicalpublictomakeitunnecessaryforsuchabookasthisonetocontainapurelygroup-theoreticchapter.Thetheoryoftopologicalgroups，however，hasbeenincludedandistreatedinChapterII.Thegreat-estpartofthischapterisconcernedwiththetheoryofcoveringspacesandgroups，whichisdevelopedindependentlyfromthetheoryofpaths.ChapterIIIisconcernedwiththetheoryof（analytic）mani-folds（independentlyofthenotionofgroup）.OurdefinitionofamanifoldisinspiredbythedefinitionofaRiemannsurfacegivenbyH.Weylinhisbook。‘DieIdeederRiemannschenFlache";ithas，comparedwiththedefinitionbyoverlappingsystemofcoordinates，theadvantageofbeingintrinsic.Thetheoryofinvolutivesystemsofdifferentialequationsonamanifoldistreatednotonlyfromthelocalpointofviewbutalsointhelarge.Inordertoachievethis，adefini-tionofthesubmanifoldsofamanifoldisgivenaccordingtowhichasubmanifoldisnotnecessarilyatopologicalsubspaceofthemanifoldinwhichitisimbedded.