Aspects of Differential Geometry

156 Pages·2015·3.398 MB·English

by  Peter Gilkey, JeongHyeong Park, Ramón Vázquez-Lorenzo

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Aspects of Differential Geometry
I
Synthesis Lectures on
Mathematics and Statistics
Editor
StevenG.Krantz,WashingtonUniversity,St.Louis
AspectsofDifferentialGeometryI
PeterGilkey,JeongHyeongPark,andRamónVázquez-Lorenzo
2015
AnEasyPathtoConvexAnalysisandApplications
BorisS.MordukhovichandNguyenMauNam
2013
ApplicationsofAffineandWeylGeometry
EduardoGarcía-Río,PeterGilkey,StanaNikčević,andRamónVázquez-Lorenzo
2013
EssentialsofAppliedMathematicsforEngineersandScientists,SecondEdition
RobertG.Watts
2012
ChaoticMaps:Dynamics,Fractals,andRapidFluctuations
GoongChenandYuHuang
2011
MatricesinEngineeringProblems
MarvinJ.Tobias
2011
eIntegral:ACruxforAnalysis
StevenG.Krantz
2011
StatisticsisEasy!SecondEdition
DennisShashaandMandaWilson
2010
iii
LecturesonFinancialMathematics:DiscreteAssetPricing
GregAndersonandAlecN.Kercheval
2010
JordanCanonicalForm:eoryandPractice
StevenH.Weintraub
2009
eGeometryofWalkerManifolds
MiguelBrozos-Vázquez,EduardoGarcía-Río,PeterGilkey,StanaNikčević,andRamón
Vázquez-Lorenzo
2009
AnIntroductiontoMultivariableMathematics
LeonSimon
2008
JordanCanonicalForm:ApplicationtoDifferentialEquations
StevenH.Weintraub
2008
StatisticsisEasy!
DennisShashaandMandaWilson
2008
AGyrovectorSpaceApproachtoHyperbolicGeometry
AbrahamAlbertUngar
2008
Copyright©2015byMorgan&Claypool
Allrightsreserved.Nopartofthispublicationmaybereproduced,storedinaretrievalsystem,ortransmittedin
anyformorbyanymeans—electronic,mechanical,photocopy,recording,oranyotherexceptforbriefquotations
inprintedreviews,withoutthepriorpermissionofthepublisher.
AspectsofDifferentialGeometryI
PeterGilkey,JeongHyeongPark,andRamónVázquez-Lorenzo
www.morganclaypool.com
ISBN:9781627056625 paperback
ISBN:9781627056632 ebook
DOI10.2200/S00632ED1V01Y201502MAS015
APublicationintheMorgan&ClaypoolPublishersseries
SYNTHESISLECTURESONMATHEMATICSANDSTATISTICS
Lecture#15
SeriesEditor:StevenG.Krantz,WashingtonUniversity,St.Louis
SeriesISSN
Print1938-1743 Electronic1938-1751
Aspects of Differential Geometry
I
Peter Gilkey
UniversityofOregon,Eugene,OR
JeongHyeong Park
SungkyunkwanUniversity,Suwon,Korea
InstituteforAdvancedStudy,Seoul,Korea
Ramón Vázquez-Lorenzo
UniversityofSantiagodeCompostela,SantiagodeCompostela,Spain
SYNTHESISLECTURESONMATHEMATICSANDSTATISTICS#15
M
&C Morgan &cLaypool publishers
ABSTRACT
DifferentialGeometryisawidefield.Wehavechosentoconcentrateuponcertainaspectsthatare
appropriateforanintroductiontothesubject;wehavenotattemptedanencyclopedictreatment.
InBookI,wefocusonpreliminaries.Chapter1providesanintroductiontomultivariable
calculusandtreatstheInverseFunctioneorem,ImplicitFunctioneorem,thetheoryofthe
Riemann Integral, and the Change of Variable eorem. Chapter 2 treats smooth manifolds,
the tangent and cotangent bundles, and Stokes’ eorem. Chapter 3 is an introduction to Rie-
mannian geometry. e Levi–Civita connection is presented, geodesics introduced, the Jacobi
operatorisdiscussed,andtheGauss–Bonneteoremisproved.ematerialisappropriatefor
anundergraduatecourseinthesubject.
Wehavegivensomedifferentproofsthanthosethatareclassicallygivenandthereissome
newmaterialinthesevolumes.Forexample,thetreatmentoftheChern–Gauss–Bonneteorem
forpseudo-Riemannianmanifoldswithboundaryisnew.
KEYWORDS
ChangeofVariableeorem,derivativeasbestlinearapproximation,Fubini’seo-
rem,Gauss–Bonneteorem,Gauss’seorem,geodesic,Green’seorem,Implicit
Functioneorem,improperintegrals,InverseFunctioneorem,Levi–Civitacon-
nection, partitions of unity, pseudo-Riemannian geometry, Riemann integral, Rie-
manniangeometry,Stokes’eorem
vii
is book is dedicated to
Alison, Arnie, Carmen, Junmin,
Junpyo, Manuel, Montse, Rosalía, and Susana.