Table Of Contentℝ
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Series ISSN: 1938-1743 C
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Series Editor: Steven G. Krantz, Washington University in St. Louis O
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Aspects of Differential
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Aspects of Differential Geometry IV A
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Esteban Calviño-Louzao, University of Santiago de Compostela
Eduardo García-Río, University of Santiago de Compostela Geometry IV
Peter Gilkey, University of Oregon
JeongHyeong Park, SungKyunkan University A
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Ramón Vázquez-Lorenzo, University of Santiago de Compostela P
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Book IV continues the discussion begun in the first three volumes. Although it is aimed at first-year S
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graduate students, it is also intended to serve as a basic reference for people working in affine differential
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geometry. It also should be accessible to undergraduates interested in affine differential geometry. We D
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are primarily concerned with the study of affine surfaces which are locally homogeneous. We discuss F
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affine gradient Ricci solitons, affine Killing vector fields, and geodesic completeness. Opozda has E
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classified the affine surface geometries which are locally homogeneous; we follow her classification. E
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Up to isomorphism, there are two simply connected Lie groups of dimension 2.The translation group T
ℝ αx b I
² is Abelian and the + group is non-Abelian. The first chapter presents foundational material. A
A ℝ L
The second chapter deals with Type surfaces.These are the left-invariant affine geometries on ². G Esteban Calviño-Louzao
Associating to each Type A surface the space of solutions to the quasi-Einstein equation corresponding E
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to the eigenvalue µ = -1 turns out to be a very powerful technique and plays a central role in our study as M
Eduardo García-Río
it links an analytic invariant with the underlying geometry of the surface. The third chapter deals with E
B αx b T
Type surfaces; these are the left-invariant affine geometries on the + group. These geometries R
Y Peter Gilkey
form a very rich family which is only partially understood. The only remaining homogeneous geometry I
V
is that of the sphere S². The fourth chapter presents relations between the geometry of an affine surface
and the geometry of the cotangent bundle equipped with the neutral signature metric of the modified JeongHyeong Park
Riemannian extension.
Ramón Vázquez-Lorenzo
About SYNTHESIS
This volume is a printed version of a work that appears in the Synthesis M
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Digital Library of Engineering and Computer Science. Synthesis
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books provide concise, original presentations of important research and G
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development topics, published quickly, in digital and print formats. N
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store.morganclaypool.com O
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Aspects of Differential Geometry
IV
Synthesis Lectures on
Mathematics and Statistics
Editor
StevenG.Krantz,WashingtonUniversity,St.Louis
AspectsofDifferentialGeometryIV
EstebanCalviño-Louzao,EduardoGarcía-Río,PeterGilkey,JeongHyeongPark,andRamón
Vázquez-Lorenzo
2019
AnalyticalTechniquesforSolvingNonlinearPartialDifferentialEquations
DanielJ.Arrigo
2019
IntroductiontoStaticsUsingR
MustaphaAkinkunmi
2019
SymmetryProblems.TheNavie–StokesProblem.
AlexanderG.Ramm
2019
PDEModelsforAtherosclerosisComputerImplementationinR
WilliamE.Schiesser
2018
AnIntroductiontoPartialDifferentialEquations
DanielJ.Arrigo
2017
NumericalIntegrationofSpaceFractionalPartialDifferentialEquations:Vol2–
ApplicatonsfromClassicalIntegerPDEs
YounesSalehiandWilliamE.Schiesser
2017
iv
NumericalIntegrationofSpaceFractionalPartialDifferentialEquations:Vol1–
IntroductiontoAlgorithmsandComputerCodinginR
YounesSalehiandWilliamE.Schiesser
2017
AspectsofDifferentialGeometryIII
EstebanCalviño-Louzao,EduardoGarcía-Río,PeterGilkey,JeongHyeongPark,andRamón
Vázquez-Lorenzo
2017
TheFundamentalsofAnalysisforTalentedFreshmen
PeterM.Luthy,GuidoL.Weiss,andStevenS.Xiao
2016
AspectsofDifferentialGeometryII
PeterGilkey,JeongHyeongPark,RamónVázquez-Lorenzo
2015
AspectsofDifferentialGeometryI
PeterGilkey,JeongHyeongPark,RamónVázquez-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
TheIntegral:ACruxforAnalysis
StevenG.Krantz
2011
v
StatisticsisEasy!SecondEdition
DennisShashaandMandaWilson
2010
LecturesonFinancialMathematics:DiscreteAssetPricing
GregAndersonandAlecN.Kercheval
2010
JordanCanonicalForm:TheoryandPractice
StevenH.Weintraub
2009
TheGeometryofWalkerManifolds
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©2019byMorgan&Claypool
Allrightsreserved.Nopartofthispublicationmaybereproduced,storedinaretrievalsystem,ortransmittedin
anyformorbyanymeans—electronic,mechanical,photocopy,recording,oranyotherexceptforbriefquotations
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AspectsofDifferentialGeometryIV
EstebanCalviño-Louzao,EduardoGarcía-Río,PeterGilkey,JeongHyeongPark,
andRamónVázquez-Lorenzo
www.morganclaypool.com
ISBN:9781681735634 paperback
ISBN:9781681735641 ebook
ISBN:9781681735658 hardcover
DOI10.2200/S00917ED1V04Y201904MAS026
APublicationintheMorgan&ClaypoolPublishersseries
SYNTHESISLECTURESONMATHEMATICSANDSTATISTICS
Lecture#26
SeriesEditor:StevenG.Krantz,WashingtonUniversity,St.Louis
SeriesISSN
Print1938-1743 Electronic1938-1751
Aspects of Differential Geometry
IV
Esteban Calviño-Louzao
UniversityofSantiagodeCompostela,Spain
Eduardo García-Río
UniversityofSantiagodeCompostela,Spain
Peter Gilkey
UniversityofOregon
JeongHyeong Park
SungKyunkanUniversity,Korea
Ramón Vázquez-Lorenzo
UniversityofSantiagodeCompostela,Spain
SYNTHESISLECTURESONMATHEMATICSANDSTATISTICS#26
M
&C Morgan &cLaypool publishers
ABSTRACT
BookIVcontinuesthediscussionbeguninthefirstthreevolumes.Althoughitisaimedatfirst-
year graduate students, it is also intended to serve as a basic reference for people working in
affine differential geometry. It also should be accessible to undergraduates interested in affine
differential geometry. We are primarily concerned with the study of affine surfaces which are
locallyhomogeneous.WediscussaffinegradientRiccisolitons,affineKillingvectorfields,and
geodesic completeness. Opozda has classified the affine surface geometries which are locally
homogeneous;wefollowherclassification.Uptoisomorphism,therearetwosimplyconnected
Lie groups of dimension 2. The translation group R2 is Abelian and the ax b group is non-
C
Abelian.Thefirstchapterpresentsfoundationalmaterial.ThesecondchapterdealswithTypeA
surfaces.Thesearetheleft-invariantaffinegeometriesonR2.AssociatingtoeachTypeAsurface
the space of solutions to the quasi-Einstein equation corresponding to the eigenvalue (cid:22) 1
D(cid:0)
turnsouttobeaverypowerfultechniqueandplaysacentralroleinourstudyasitlinksananalytic
invariant with the underlying geometry of the surface. The third chapter deals with Type B
surfaces; these are the left-invariant affine geometries on the ax b group. These geometries
C
form a very rich family which is only partially understood. The only remaining homogeneous
geometryisthatofthesphereS2.Thefourthchapterpresentsrelationsbetweenthegeometryof
anaffinesurfaceandthegeometryofthecotangentbundleequippedwiththeneutralsignature
metricofthemodifiedRiemannianextension.
KEYWORDS
affine gradient Ricci solitons, affine Killing vector fields, geodesic completeness,
locally homogeneous affine surfaces, locally symmetric affine surfaces, projectively
flat,quasi-Einsteinequation
