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