| 《微分几何基础(英文版)》是由世界图书出版公司出版的。 |
| foreword acknowledgments part ⅰ general differential theory chapter ⅱ differential calculus 1.categories 2.topological vector spaces 3.derivatives and composition of maps 4.integration and taylor's formula 5.the inverse mapping theorem chapter ⅱ manifolds 1.atlases, charts, morphisms 2.submanifolds, immersions, submersions 3.partitions of unity 4.manifolds with boundary chapter ⅲ vector bundles 1.definition, pull backs 2.the tangent bundle 3.exact sequences of bundles 4.operations on vector bundles 5.splitting of vector bundles chapter ⅳ vector fields and differential equations 1.existence theorem for differential equations 2.vector fields, curves, and flows 3.sprays 4.the flow of a spray and the exponential map 5.existence of tubular neighborhoods 6.uniqueness of tubular neighborhoods chapter ⅴ operations on vector fields and differential forms 1.vector fields, differential operators, brackets 2.lie derivative 3.exterior derivative 4.the poincare lemma. 5.contractions and lie derivative 6.vector fields and l-forms under self duality 7.the canonical 2-form 8.darboux's theorem chapter ⅵ the theorem ol frobenius 1.statement of the theorem 2.differential equations depending on a parameter 3.proof of the theorem 4.the global formulation 5.lie groups and subgroups part ⅱ metrics, covariant derivatives, and riemannian geometry chapter ⅶ metrics 1.definition and functoriality 2.the hilbert group 3.reduction to the hiibert group 4.hilbertian tubular neighborhoods 5.the morse-palais lemma 6.the riemannian distance 7.the canonical spray chapter ⅷ covarlent derivatives and geodesics 1.basic properties 2.sprays and covariant derivatives 3.derivative along a curve and parallelism 4.the metric derivative 5.more local results on the exponential map 6.riemannian geodesic length and completeness chapter ⅸ curvature 1.the riemann tensor 2.jacobi lifts. 3.application of jacobi lifts to texp 4.convexity theorems. 5.taylor expansions part ⅲ volume forms and integration index |
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