現(xiàn)代幾何學(xué)方法和應(yīng)用（第2卷）

定　價(jià)：￥72.00

作　者：	B.A.DubrovinA.T.FomenkoS.P.Novikov
出版社：	世界圖書(shū)出版公司
叢編項(xiàng)：	Graduate Texts in Mathematics
標(biāo)　簽：	暫缺

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ISBN：	9787506201339	出版時(shí)間：	1999-11-01	包裝：	簡(jiǎn)裝本
開(kāi)本：	24開(kāi)	頁(yè)數(shù)：	430	字?jǐn)?shù)：

內(nèi)容簡(jiǎn)介

　　Up until recently， Riemannian geometry and basic topology were not included， even by departments or faculties of mathematics， as compulsory subjects in a university-level mathematical education. The standard courses in the classical differential geometry of curves and surfaces which were given instead （and still are given in some places） have come gradually to be viewed as anachronisms. However， there has been hitherto no unanimous agreement as to exactly how such courses should be brought up to date， that is to say， which parts of modern geometry should be regarded as absolutely essential to a modern mathematical education， and what might be the appropriate levelof abstractness of their exposition.本書(shū)為英文版。

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圖書(shū)目錄

CHAPTER1
ExamplesofManifolds
1.Theconceptofamanifold
1.1.Definitionofamanifold
1.2.Mappingsofmanifolds;tensorsonmanifolds
1.3.Embeddingsandimmersionsofmanifolds.Manifoldswith
boundary
2.Thesimplestexamplesofmanifolds
2.1.SurfacesinEuclideanspace.Transformationgroupsasmanifolds
2.2.Projectivespaces
2.3.Exercises
3.EssentialfactsfromthetheoryofLiegroups
3.1.ThestructureofaneighbourboodoftheidentityofaLiegroup.
TheLiealgebraofaLiegroup.Semisimplicity
3.2.Theconceptofalinearrepresentation.Anexampleofa
non-matrixLiegroup
4.Complexmanifolds
4.1.Definitionsandexamples
4.2.Riemannsurfacesasmanifolds
5.Thesimplesthomogeneousspaces
5.1.Actionofagrouponamanifold
5.2.Examplesofhomogeneousspaces
5.3.Exercises
6.Spacesofconstantcurvature(symmetricspaces)
6.1.Theconceptofasymmetricspace
6.2.Theisometrygroupofamanifold.PropertiesofitsLiealgebra
6.3.Symmetricspacesofthefirstandsecondtypes
6.4.Liegroupsassymmetricspaces
6.5.Constructingsymmetricspaces.Examples
6.6.Exercises
7.Vectorbundlesonamanifold
7.1.Constructionsinvolvingtangentvectors
7.2.Thenormalvectorbundleonasubmanifold
CHAPTER2
FoundationalQuestions.EssentialFactsConcerningFunctions
onaManifold.TypicalSmoothMappings
8.Partitionsofunityandtheirapplications
8.1.Partitionsofunity
8.2.Thesimplestapplicationsofpartitionsofunity.Integralsovera
manifoldandthegeneralStokesformula
8.3.Invariantmetrics
9.TherealizationofcompactmanifoldsassurfacesinRN
10.Variouspropertiesofsmoothmapsofmanifolds
10.1.Approximationofcontinuousmappingsbysmoothones
10.2.Sard'stheorem
10.3.Transversalregularity
10.4.Morsefunctions
11.ApplicationsofSard'stheorem
11.1.Theexistenceofembeddingsandimmersions
11.2.TheconstructionofMorsefunctionsasheightfunctions
11.3.Focalpoints
CHAPTER3
TheDegreeofaMapping.TheIntersectionIndexofSubmanifolds.
Applications
12.Theconceptofhomotopy
12.1.Definitionofhomotopy.Approximationofcontinuousmaps
andhomotopiesbysmoothones
12.2.Relativehomotopies
13.Thedegreeofamap
13.1.Definitionofdegree
13.2.Generalizationsoftheconceptofdegree
13.3.Classificationofhomotopyclassesofmapsfromanarbitrary
manifoldtoasphere
13.4.Thesimplestexamples
14.Applicationsofthedegreeofamapping
14.1.Therelationshipbetweendegreeandintegral
14.2.Thedegreeofavectorfieldonahypersurface
14.3.TheWhitneynumber.TheGauss-Bonnetformula
14.4.Theindexofasingularpointofavectorfield
14.5.Transversesurfacesofavectorfield.ThePoincare-Bendixson
theorem
15.Theintersectionindexandapplications
15.1.Definitionoftheintersectionindex
15.2.Thetotalindexofavectorfield
15.3.Thesignednumberoffixedpointsofaself-map(theLefschetz
number).TheBrouwerfixed-pointtheorem
15.4.Thelinkingcoefficient
CHAPTER4
OrientabilityofManifolds.TheFundamentalGroup.
CoveringSpaces(FibreBundleswithDiscreteFibre)
16.Orientabilityandhomotopiesofclosedpaths
16.1.Transportinganorientationalongapath
16.2.Examplesofnon-orientablemanifolds
17.Thefundamentalgroup
17.1.Definitionofthefundamentalgroup
17.2.Thedependenceonthebasepoint
17.3.Freehomotopyclassesofmapsofthecircle
17.4.Homotopicequivalence
17.5.Examples
17.6.Thefundamentalgroupandorientability
18.Coveringmapsandcoveringhomotopies
18.1.Thedefinitionandbasicpropertiesofcoveringspaces
18.2.Thesimplestexamples.Theuniversalcovering
18.3.Branchedcoverings.Riemannsurfaces
18.4.Coveringmapsanddiscretegroupsoftransformations
19.Coveringmapsandthefundamentalgroup.Computationofthe
fundamentalgroupofcertainmanifolds
19.1.Monodromy
19.2.Coveringmapsasanaidinthecalculationoffundamental
groups
19.3.Thesimplestofthehomologygroups
19.4.Exercises
20.ThediscretegroupsofmotionsoftheLobachevskianplane
CHAPTER5
HomotopyGroups
21.Definitionoftheabsoluteandrelativehomotopygroups.Examples
21.1.Basicdefinitions
21.2.Relativehomotopygroups.Theexactsequenceofapair
22.Coveringhomotopies.Thehomotopygroupsofcoveringspaces
andloopspaces
22.1.Theconceptofafibrespace
22.2.Thehomotopyexactsequenceofafibrespace
22.3.Thedependenceofthehomotopygroupsonthebasepoint
22.4.ThecaseofLiegroups
22.5.Whiteheadmultiplication
23.Factsconcerningthehomotopygroupsofspheres.Framednormal
bundles.TheHopfinvariant
23.1.Framednormalbundlesandthehomotopygroupsofspheres
23.2.Thesuspensionmap
23.3.Calculationofthegroups
23.4.Thegroups
CHAPTER6
SmoothFibreBundles
24.Thehomotopytheoryoffibrebundles
24.1.Theconceptofasmoothfibrebundle
24.2.Connexions
24.3.Computationofhomotopygroupsbymeansoffibrebundles
24.4.Theclassificationoffibrebundles
24.5.Vectorbundlesandoperationsonthem
24.6.Meromorphicfunctions
24.7.ThePicard-Lefschetzformula
25.Thedifferentialgeometryoffibrebundles
25.1.G-connexionsonprincipalfibrebundles
25.2.G-connexionsonassociatedfibrebundles.Examples
25.3.Curvature
25.4.Characteristicclasses:Constructions
25.5.Characteristicclasses:Enumeration
26.Knotsandlinks.Braids
26.1.Thegroupofaknot
26.2.TheAlexanderpolynomialofaknot
26.3.Thefibrebundleassociatedwithaknot
26.4.Links
26.5.Braids
CHAPTER7
SomeExamplesofDynamicalSystemsandFoliations
onManifolds
27.Thesimplestconceptsofthequalitativetheoryofdynamicalsystems.
Two-dimensionalmanifolds
27.1.Basicdefinitions
27.2.Dynamicalsystemsonthetorus
28.Hamiltoniansystemsonmanifolds.Liouville'stheorem.Examples
28.1.Hamiltoniansystemsoncotangentbundles
28.2.Hamiltoniansystemsonsymplecticmanifolds.Examples
28.3.Geodesicflows
28.4.Liouville'stheorem
28.5.Examples
29.Foliations
29.1.Basicdefinitions
29.2.Examplesoffoliationsofcodimension1
30.Variationalproblemsinvolvinghigherderivatives
30.1.Hamiltonianformalism
30.2.Examples
30.3.Integrationofthecommutativityequations.Theconnexionwith
theKovalevskajaproblem.Finite-zonedperiodicpotentials
30.4.TheKorteweg-deVriesequation.Itsinterpretationasan
infinite-dimensionalHamiltoniansystem
30.5Hamiltonianformalismoffieldsystems
CHAPTER8
TheGlobalStructureofSolutionsofHigher-Dimensional
VariationalProblems
31.Somemanifoldsarisinginthegeneraltheoryofrelativity(GTR)
31.1.Statementoftheproblem
31.2.Sphericallysymmetricsolutions
31.3.Axiallysymmetricsolutions
31.4.Cosmologicalmodels
31.5.Friedman'smodels
31.6.Anisotropicvacuummodels
31.7.Moregeneralmodels
32.SomeexamplesofglobalsolutionsoftheYang-Millsequations.
Chiralfields
32.1.Generalremarks.Solutionsofmonopoletype
32.2.Thedualityequation
32.3.Chiralfields.TheDirichletintegral
33.Theminimalityofcomplexsubmanifotds
Bibliography
Index