微分几何基础 英文版PDF电子书下载
- 电子书积分:16 积分如何计算积分?
- 作 者:SergeLang编著
- 出 版 社:世界图书北京出版公司
- 出版年份:2010
- ISBN:9787510005404
- 页数:535 页
PART Ⅰ General Differential Theory 1
CHAPTER Ⅰ Differential Calculus 3
1.Categories 4
2.Topological Vector Spaces 5
3.Derivatives and Composition of Maps 8
4.Integration and Taylor's Formula 12
5.The Inverse Mapping Theorem 15
CHAPTER Ⅱ Manifolds 22
1.Atlases,Charts,Morphisms 22
2.Submanifolds,Immersions,Submersions 25
3.Partitions of Unity 33
4.Manifolds with Boundary 39
CHAPTER Ⅲ Vector Bundles 43
1.Definition,Pull Backs 43
2.The Tangent Bundle 51
3.Exact Sequences of Bundles 52
4.Operations on Vector Bundles 58
5.Splitting of Vector Bundles 63
CHAPTER Ⅳ Vector Fields and Differential Equations 66
1.Existence Theorem for Differential Equations 67
2.Vector Fields,Curves,and Flows 88
3.Sprays 96
4.The Flow of a Spray and the Exponential Map 105
5.Existence of Tubular Neighborhoods 110
6.Uniqueness of Tubular Neighborhoods 112
CHAPTER Ⅴ Operations on Vector Fields and Differential Forms 116
1.Vector Fields,Differential Operators,Brackets 116
2.Lie Derivative 122
3.Exterior Derivative 124
4.The Poincaré Lemma 137
5.Contractions and Lie Derivative 139
6.Vector Fields and 1-Forms Under Self Duality 143
7.TheCanonical 2-Form 149
8.Darboux's Theorem 151
CHAPTER Ⅵ The Theorem of Frobenius 155
1.Statement of the Theorem 155
2.Differential Equations Depending on a Parameter 160
3.Proof of the Theorem 161
4.The Global Formulation 162
5.Lie Groups and Subgroups 165
PART Ⅱ Metrics,Covariant Derivatives,and Riemannian Geometry 171
CHAPTER Ⅶ Metrics 173
1.Definition and Functoriality 173
2.The Hilbert Group 177
3.Reduction to the Hilbert Group 180
4.Hilbertian Tubular Neighborhoods 184
5.The Morse-Palais Lemma 186
6.The Riemannian Distance 189
7.The Canonical Spray 192
CHAPTER Ⅷ Covariant Derivatives and Geodesics 196
1.Basic Properties 196
2.Sprays and Covariant Derivatives 199
3.Derivative Along a Curve and Parallelism 204
4.The Metric Derivative 209
5.More Local Results on the Exponential Map 215
6.Riemannian Geodesic Length and Completeness 221
CHAPTER Ⅸ Curvature 231
1.The Riemann Tensor 231
2.Jacobi Lifts 239
3.Application of Jacobi Lifts to Texpx 246
4.Convexity Theorems 255
5.Taylor Expansions 263
CHAPTER Ⅹ Jacobi Lifts and Tensorial Splitting of the Double Tangent Bundle 267
1.Convexity of Jacobi Lifts 267
2.Global Tubular Neighborhood of a Totally Geodesic Submanifold 271
3.More Convexity and Comparison Results 276
4.Splitting of the Double Tangent Bundle 279
5.Tensorial Derivative of a Curve in TX and of the Exponential Map 286
6.The Flow and the Tensorial Derivative 291
CHAPTER Ⅺ Curvature and the Variation Formula 294
1.The Index Form,Variations,and the Second Variation Formula 294
2.Growth of a Jacobi Lift 304
3.The Semi Parallelogram Law and Negative Curvature 309
4.Totally Geodesic Submanifolds 315
5.Rauch Comparison Theorem 318
CHAPTER Ⅻ An Example of Seminegative Curvature 322
1.POSn(R) as a Riemannian Manifold 322
2.The Metric Increasing Property of the Exponential Map 327
3.Totally Geodesic and Symmetric Submanifolds 332
CHAPTER ⅩⅢ Automorphisms and Symmetries 339
1.The Tensorial Second Derivative 342
2.Alternative Definitions of Killing Fields 347
3.Metric Killing Fields 351
4.Lie Algebra Properties of Killing Fields 354
5.Symmetric Spaces 358
6.Parallelism and the Riemann Tensor 365
CHAPTER ⅩⅣ Immersions and Submersions 369
1.The Covariant Derivative on a Submanifold 369
2.The Hessian and Laplacian on a Submanifold 376
3.The Covariant Derivative on a Riemannian Submersion 383
4.The Hessian and Laplacian on a Riemannian Submersion 387
5.The Riemann Tensor on Submanifolds 390
6.The Riemann Tensor on a Riemannian Submersion 393
PART Ⅲ Volume Forms and Integration 395
CHAPTER ⅩⅤ Volume Forms 397
1.Volume Forms and the Divergence 397
2.Covariant Derivatives 407
3.The Jacobian Determinant of the Exponential Map 412
4.The Hodge Star on Forms 418
5.Hodge Decomposition of Differential Forms 424
6.Volume Forms in a Submersion 428
7.Volume Forms on Lie Groups and Homogeneous Spaces 435
8.Homogeneously Fibered Submersions 440
CHAPTER ⅩⅥ Integration of Differential Forms 448
1.Sets of Measure 0 448
2.Change of Variables Formula 453
3.Orientation 461
4.The Measure Associated with a Differential Form 463
5.Homogeneous Spaces 471
CHAPTER ⅩⅦ Stokes'Theorem 475
1.Stokes'Theorem for a Rectangular Simplex 475
2.Stokes'Theorem on a Manifold 478
3.Stokes'Theorem with Singularities 482
CHAPTER ⅩⅧ Applications of Stokes'Theorem 489
1.The Maximal de Rham Cohomology 489
2.Moser's Theorem 496
3.The Divergence Theorem 497
4.The Adjoint of d for Higher Degree Forms 501
5.Cauchy's Theorem 503
6.The Residue Theorem 507
APPENDIX The Spectral Theorem 511
1.Hilbert Space 511
2.Functionals and Operators 512
3.Hermitian Operators 515
Bibliography 523
Index 531
- 《市政工程基础》杨岚编著 2009
- 《零基础学会素描》王金著 2019
- 《计算机网络与通信基础》谢雨飞,田启川编著 2019
- 《生物质甘油共气化制氢基础研究》赵丽霞 2019
- 《花时间 我的第一堂花艺课 插花基础技法篇》(日)花时间编辑部编;陈洁责编;冯莹莹译 2020
- 《Photoshop CC 2018基础教程》温培利,付华编著 2019
- 《看视频零基础学英语口语》宋德伟 2019
- 《胃癌基础病理》(日)塚本彻哉编者;宫健,刘石译者 2019
- 《卓有成效的管理者 中英文双语版》(美)彼得·德鲁克许是祥译;那国毅审校 2019
- 《高校转型发展系列教材 素描基础与设计》施猛责任编辑;(中国)魏伏一,徐红 2019
- 《市政工程基础》杨岚编著 2009
- 《家畜百宝 猪、牛、羊、鸡的综合利用》山西省商业厅组织技术处编著 1959
- 《《道德经》200句》崇贤书院编著 2018
- 《高级英语阅读与听说教程》刘秀梅编著 2019
- 《计算机网络与通信基础》谢雨飞,田启川编著 2019
- 《看图自学吉他弹唱教程》陈飞编著 2019
- 《法语词汇认知联想记忆法》刘莲编著 2020
- 《培智学校义务教育实验教科书教师教学用书 生活适应 二年级 上》人民教育出版社,课程教材研究所,特殊教育课程教材研究中心编著 2019
- 《国家社科基金项目申报规范 技巧与案例 第3版 2020》文传浩,夏宇编著 2019
- 《流体力学》张扬军,彭杰,诸葛伟林编著 2019
- 《TED说话的力量 世界优秀演讲者的口才秘诀》(坦桑)阿卡什·P.卡里亚著 2019
- 《小手画出大世界 恐龙世界》登亚编绘 2008
- 《近代世界史文献丛编 19》王强主编 2017
- 《课堂上听不到的历史传奇 世界政治军事名人 初中版》顾跃忠等编著 2015
- 《指向核心素养 北京十一学校名师教学设计 英语 七年级 上 配人教版》周志英总主编 2019
- 《365奇趣英语乐园 世界民间故事》爱思得图书国际企业 2018
- 《近代世界史文献丛编 36》王强主编 2017
- 《北京生态环境保护》《北京环境保护丛书》编委会编著 2018
- 《近代世界史文献丛编 11》王强主编 2017
- 《近代世界史文献丛编 18》王强主编 2017