国外数学名著系列

图书标准信息

• 作者 [俄罗斯]列舍特尼亚克（Reshetnyak Y.G） 著
• 出版社 科学出版社
• 出版时间 2009-01
• 版次 1
• ISBN 9787030235015
• 定价 60.00元
• 装帧 精装
• 开本 16开
• 纸张 胶版纸
• 页数 250页
• 字数 315千字
• 正文语种 英语

【内容简介】

ThisvolumeoftheEncyclopaediacontainstwoarticleswhichgiveasurveyofmodernresearchintonon-regularRiemanniangeometry，carriedoutmostlybyRussianmathematicians．
ThefirstarticlewrittenbyReshetnyakisdevotedtothetheoryoftwo-dimensionalRiemannianmanifoldsofboundedcurvature．ConceptsofRiemanniangeometrysuchastheareaandintegralcurvatureofasetandthelengthandintegralcurvatureofacurvearealsodefinedforthesemanifolds．SomefundamentalresultsofRiemanniangeometryliketheGauss．Bonnetformulaaretrueinthemoregeneralcaseconsideredinthebook．
ThesecondarticlebyBerestovskijandNikolaevisdevotedtothetheoryofmetricspaceswhosecurvatureliesbetweentwogiyenconstants．ThemainresultiSthatthesespacesareinfactRiemannian．ThisresulthasimportantapplicationsinglobalRiemanniangeometry．
Bothpartscovertopicswhichhavenotyetbeentreatedinmonographform．Hencethebookwillbeimmenselyusefultograduatestudentsandresearchersingeometry，inparticularRiemanniangeometry．

【目录】

Chapter1.PreliminaryInformation
1.Introduction
1.1.GeneralInformationabouttheSubjectofResearchandaSurveyofResults
1.2.SomeNotationandTerminology

2.TheConceptofaSpacewithIntrinsicMetric
2.1.TheConceptoftheLengthofaParametrizedCurve
2.2.ASpacewithIntrinsicMetric.TheInducedMetric
2.3.TheConceptofaShortestCurve
2.4.TheOperationofCuttingofaSpacewithIntrinsicMetric

3.Two-DimensionalManifoldswithIntrinsicMetric
3.1.Definition.TriangulationofaManifold
3.2.PastingofTwo-DimensionalManifoldswithIntrinsicMetric
3.3.CuttingofManifolds
3.4.ASideofaSimpleArcinaTwo-DimensionalManifold

4.Two-DimensionalRiemannianGeometry
4.1.DifferentiableTwo-DimensionalManifolds
4.2.TheConceptofaTwo-DimensionalRiemannianManifold
4.3.TheCurvatureofaCurveinaRiemannianManifold.IntegralCurvature.TheGauss-BonnetFormula
4.4.IsothermalCoordinatesinTwo-DimensionalRiemannianManifoldsofBoundedCurvature

5.ManifoldswithPolyhedralMetric
5.1.ConeandAngularDomain
5.2.DefinitionofaManifoldwithPolyhedralMetric
5.3.CurvatureofaSetonaPolyhedron.TurnoftheBoundary.TheGauss-BonnetTheorem
5.4.ATurnofaPolygonalLineonaPolyhedron
5.5.CharacterizationoftheIntrinsicGeometryofConvexPolyhedra
5.6.AnExtremalPropertyofaConvexCone.TheMethodofCuttingandPastingasaMeansofSolvingExtremalProblemsforPolyhedra
5.7.TheConceptofaK-Polyhedron

Chapter2.DifferentWaysofDefiningTwo-DimensionalManifoldsofBoundedCurvature
6.AxiomsofaTwo-DimensionalManifoldofBoundedCurvature.CharacterizationofsuchManifoldsbyMeansofApproximationbyPolyhedra
6.1.AxiomsofaTwo-DimensionalManifoldofBoundedCurvature
6.2.TheoremsontheApproximationofTwo-DimensionalManifoldsofBoundedCurvaturebyManifoldswithPolyhedralandRiemannianMetric
6.3.ProofoftheFirstTheoremonApproximation
6.4.ProofofLemma6.3.1
6.5.ProofoftheSecondTheoremonApproximation

7.AnalyticCharacterizationofTwo-DimensionalManifoldsof
BoundedCurvature
7.1.TheoremsonIsothermalCoordinatesinaTwo-DimensionalManifoldofBoundedCurvature
7.2.SomeInformationaboutCurvesonaPlaneandinaRiemannianmanifold
7.3.ProofsofTheorems7.1.1，7.1.2，7.1.3
7.4.OntheProofofTheorem7.3.1

Chapter3.BasicFactsoftheTheoryofManifoldsofBoundedCurvature
8.BasicResultsoftheTheoryofTwo-DimensionalManifoldsofBoundedCurvature
8.1.ATurnofaCurveandtheIntegralCurvatureofaSet
8.2.ATheoremontheContractionofaCone.AnglebetweenCurves.ComparisonTheorems
8.3.ATheoremonPastingTogetherTwo-DimensionalManifoldsofBoundedCurvature
8.4.TheoremsonPassagetotheLimitforTwo-DimensionalManifoldsofBoundedCurvature
8.5.SomeInequalitiesandEstimates.ExtremalProblemsforTwo-DimensionalManifoldsofBoundedCurvature
9.FurtherResults.SomeAdditionalRemarks
References