Lectures on Differential Geometry
Lectures on Differential Geometry
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DOI：
10.2307/2315376
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发表时间：
1964
期刊：
影响因子：
--
通讯作者：
S. Sternberg
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文献类型：
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作者：
S. Sternberg
1.第一章向量空间的张量积2.向量空间的张量代数3.反变与对称代数4.第五章外代数外部方程微分流形：1.定义2. 3.第三章Sard定理4单位分割，逼近定理5。切空间6.主束7。张量束8.向量场和李导数流形上的积分：1。运算符$d $2。链与集成3.密度的整合4. $0 $和$n $维上同调，5次。Frobenius定理6.达布定理7.哈密顿结构变分法：1。第二章勒让德变换必要条件3。保护法4.充分条件5.共轭点和焦点，Jacobi条件6。第七章.完整性8. Lie groups：1.定义2.不变形式和李代数3。法向坐标，指数图4。封闭的亚组5.不变量度量6.形式与价值观在一个向量空间微分几何的欧几里德空间：1。欧氏空间的结构方程2.子流形的结构方程3.黎曼流形的结构方程4. 5.第五章.第二种基本形式6.表面的几何$G $-结构：1。主要和相关的捆绑包，连接2. $G $-结构3.延长4.有限类型的结构5. $G $-结构上的连接6.喷射的线性连接附录I：两个存在定理附录II：大纲理论的整合$E ^n $附录III：代数模型的传递微分几何附录IV：可积性问题的几何结构参考索引。
Algebraic Preliminaries: 1. Tensor products of vector spaces 2. The tensor algebra of a vector space 3. The contravariant and symmetric algebras 4. Exterior algebra 5. Exterior equations Differentiable Manifolds: 1. Definitions 2. Differential maps 3. Sard's theorem 4. Partitions of unity, approximation theorems 5. The tangent space 6. The principal bundle 7. The tensor bundles 8. Vector fields and Lie derivatives Integral Calculus on Manifolds: 1. The operator $d$ 2. Chains and integration 3. Integration of densities 4. $0$ and $n$-dimensional cohomology, degree 5. Frobenius' theorem 6. Darboux's theorem 7. Hamiltonian structures The Calculus of Variations: 1. Legendre transformations 2. Necessary conditions 3. Conservation laws 4. Sufficient conditions 5. Conjugate and focal points, Jacobi's condition 6. The Riemannian case 7. Completeness 8. Isometries Lie Groups: 1. Definitions 2. The invariant forms and the Lie algebra 3. Normal coordinates, exponential map 4. Closed subgroups 5. Invariant metrics 6. Forms with values in a vector space Differential Geometry of Euclidean Space: 1. The equations of structure of Euclidean space 2. The equations of structure of a submanifold 3. The equations of structure of a Riemann manifold 4. Curves in Euclidean space 5. The second fundamental form 6. Surfaces The Geometry of $G$-Structures: 1. Principal and associated bundles, connections 2. $G$-structures 3. Prolongations 4. Structures of finite type 5. Connections on $G$-structures 6. The spray of a linear connection Appendix I: Two existence theorems Appendix II: Outline of theory of integration on $E^n$ Appendix III: An algebraic model of transitive differential geometry Appendix IV: The integrability problem for geometrical structures References Index.