Differential manifolds微分流形（高等数学）.pdf

Differentialmanifolds

AnintroductiontoRiemanniangeometryandRicciflow

Author:YiLi

Institute:SchoolofMathematicsandShing-TungYauCenter,SoutheastUniversity

Date:March1,2021

Version:2.3;©YiLi2021

Email:yilicms@;yilicms@;yilicms@163.com

Ratherless,butbetter.–CarlFriedrichGauss

MainReferences

Chow,Bennett;Lu,Peng;Ni,Lei.Hamilton’sRicciflow.GSM77,AMS/SciencePress,

2006

Lee,JohnM.Riemannianmanifolds:Anintroductiontocurvature.GTM176,Springer-

Verlag,1997.

Petersen,Peter.Riemanniangeometry.GTM171,Springer-Verlag,NewYork,1998.

Schoen,Richard;Yau,Shing-Tung.Lecturesondifferentialgeometry.ConferencePro-

ceedingsandLEctureNotesinGeometryandTopology,I,InternationalPress,Cambridge,

MA,1994.

Contents

1Differentialmanifolds1

1.1Manifolds.....................................1

1.2Tensorsandforms.................................32

1.3Integrationonmanifolds.............................57

2Riemanniangeometry73

2.1Introduction....................................73

2.2Metrics,connections,curvaturesandcovariantdifferentiation..........73

2.3Basicformulasandidentities...........................89

2.4Examples