黎曼几何

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