现代芬斯勒几何初步
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作者沈一兵,沈忠民 著
出版社高等教育出版社
ISBN9787040444247
出版时间2016-01
版次1
装帧精装
开本16开
纸张胶版纸
页数393页
字数99999千字
定价98元
上书时间2024-03-25
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基本信息
书名:现代芬斯勒几何初步
定价：98元
作者:沈一兵,沈忠民 著
出版社：高等教育出版社
出版日期：2016-01-01
ISBN：9787040444247
字数：530000
页码：393
版次：1
装帧：精装
开本：16开
商品重量：
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内容提要
This comprehensive book is aintroductioto the basics of Finsler geometry with recent developments iits area. It includes local geometry as well as global geometry of Finsler manifolds. IPart Ⅰ， the authors discuss differential manifolds， Finsler metrics， the Cherconnection， Riemanniaand non- Riemanniaquantities. Part Ⅱ is writtefor readers who would like to further their studies iFinsler geometry. It covers projective transformations，comparisotheorems， fundamental group， minimal immersions，harmonic maps， Einsteimetrics， conformal transformations，amongst other related topics.The authors made great efforts to ensure that the contents are accessible to senior undergraduate students， graduate students， mathematicians and scientists.
目录
PrefaceFoundations1． Differentiable Manifolds1．1 Differentiable manifolds1．1．1 Differentiable manifolds1．1．2 Examples of differentiable manifolds1．2 Vector fields and tensor fields1．2．1 Vector bundles1．2．2 Tensor fields1．3 Exterior forms and exterior differentials 1．3．1 Exterior differential operators1．3．2 de Rham theorem1．4 Vector bundles and connections1．4．1 Connectioof the vector bundle 1．4．2 Curvature of a connectionExercises2． Finsler Metrics2．1 Finsler metrics2．1．1 Finsler metrics2．1．2 Examples of Finsler metrics2．2 Cartatorsion2．2．1 Cartatorsion2．2．2 Deicke theorem2．3 Hilbert form and sprays2．3．1 Hilbert form2．3．2 Sprays2．4 Geodesics2．4．1 Geodesics2．4．2 Geodesic coefficients2．4．3 Geodesic completenessExercises3． Connections and Curvatures3．1 Connections3．1．1 Cherconnection3．1．2 Berwald metrics and Landerg metrics3．2 Curvatures3．2．1 Curvature form of the Cherconnection3．2．2 Flag curvature and Ricci curvature3．3 Bianchi identities3．3．1 Covariant differentiation3．3．2 Bianchi identities3．3．3 Other formulas3．4 Legendre transformation3．4．1 The dual norm ithe dual space3．4．2 Legendre transformation3．4．3 ExampleExercises4． S-Curvature4．1 Volume measures4．1．1 Busemann-Hausdorff volume element4．1．2 The volume element induced from SM4．2 S-curvature4．2．1 Distortion4．2．2 S-curvature and E-curvature4．3 Isotropic S-curvature4．3．1 Isotropic S-curvature and isotropic E-curvature4．3．2 Randers metrics of isotropic S-curvature4．3．3 Geodesic flowExercises5． RiemanCurvature5．1 The second variatioof arc length5．1．1 The second variatioof length5．1．2 Elements of curvature and topology5．2 Scalar flag curvature5．2．1 Schur theorem5．2．2 Constant flag curvature5．3 Global rigidity results5．3．1 Flag curvature with special conditions5．3．2 Manifolds with non-positive flag curvature5．4 Navigation5．4．1 Navigatioproblem5．4．2 Randers metrics and navigation5．4．3 Ricci curvature and EinsteimetricsExercisesFurther Studies6． Projective Changes6．1 The projective equivalence6．1．1 Projective equivalence6．1．2 Projective invariants6．2 Projectively flat metrics6．2．1 Projectively flat metrics6．2．2 Projectively fiat metrics with constant flag curvature6．3 Projectively fiat metrics with almost isotropic S-curvature6．3．1 Randers metrics with almost isotropic S-curvature6．3．2 Projectively flat metrics with almost isotropicS-curvature6．4 Some special projectively equivalent Finsler metrics6．4．1 Projectively equivalent Randers metrics6．4．2 The projective equivalence of （α， β）-metrics6．4．3 The projective equivalence of quadratic （α， β）metricsExercises7． ComparisoTheorems7．1 Volume comparisotheorems for Finsler manifolds7．1．1 The Jacobiaof the exponential map7．1．2 Distance functioand comparisotheorems7．1．3 Volume comparisotheorems7．2 Berger-Kazdacomparisotheorems7．2．1 The Kazdainequality7．2．2 The rigidity of reversible Finsler manifolds7．2．3 The Berger-KazdacomparisotheoremExercises8． Fundamental Groups of Finsler Manifolds8．1 Fundamental groups of Finsler manifolds8．1．1 Fundamental groups and covering spaces8．1．2 Algebraic norms and geometric norms8．1．3 Growth of fundamental groups8．2 Entropy and finiteness of fundamental group8．2．1 Entropy of fundamental group8．2．2 The first Betti number8．2．3 Finiteness of fundamental group8．3 Gromov pre-compactness theorems8．3．1 General metric spaces8．3．2 δ-Gromov-Hausdorff convergence8．3．3 Pre-compactness of Finsler manifolds8．3．4 Othe Gauss-Bonnet-ChertheoremExercises9． Minimal Immersions and Harmonic Maps9．1 Isometric immersions9．1．1 Finsler submanifolds9．1．2 The variatioof the volume9．1．3 Non-existence of compact minimal submanifolds9．2 Rigidity of minimal submanifolds9．2．1 Minimal surfaces iMinkowski spaces9．2．2 Minimal surfaces i（α， β）-spaces9．2．3 Minimal surfaces ispecial Minkowskia（α， β）spaces9．3 Harmonic maps9．3．1 A divergence formula9．3．2 Harmonic maps9．3．3 Compositiomaps9．4 Second variatioof harmonic maps9．4．1 The second variation9．4．2 Stress-energy tensor9．5 Harmonic mapetweecomplex Finsler manifolds9．5．1 Complex Finsler manifolds9．5．2 Harmonic mapetweecomplex Finsler manifolds9．5．3 Holomorphic mapsExercises10． EinsteiMetrics10．1 Projective rigidity and m-th root metrics10．1．1 Projective rigidity of Einsteimetrics10．1．2 m-th root Einsteimetrics10．2 The Ricci rigidity and Douglas-Einsteimetrics10．2．1 The Ricci rigidity10．2．2 Douglas （α， β）-metrics10．3 Einstei（α， β）-metrics10．3．1 Polynomial （α， β）-metrics10．3．2 Kropina metricsExercises11． Miscellaneous Topics11．1 Conformal changes11．1．1 Conformal changes11．1．2 Conformally flat metrics11．1．3 Conformally flat （α， β）-metrics11．2 Conformal vector fields11．2．1 Conformal vector fields11．2．2 Conformal vector fields oa Randers manifold11．3 A class of critical Finsler metrics11．3．1 The Einstein-Hilbert functional11．3．2 Some special g-critical metrics11．4 The first eigenvalue of Finsler Laplaciaand the generalized maximal principle11．4．1 Finsler Laplaciaand weighted Ricci curvature11．4．2 Lichnerowicz-Obata estimates11．4．3 Li-Yau-Zhong-Yang type estimates11．4．4 Mckeatype estimatesExercisesAppendix A Maple ProgramA．1 Spray coefficients of two-dimensional Finsler metricsA．2 Gauss curvatureA．3 Spray coefficients of （α， β）-metricsBibliographyIndex
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