Lectures on Differential Geometry I
Description
This textbook is designed for a one-semester introductory course in Differential Geometry. It covers the fundamentals of differentiable manifolds, explores Lie groups and homogeneous spaces, and concludes with rigorous proofs of Stokes' Theorem and the de Rham Theorem. The material closely follows the author's lectures at ETH Zürich.
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Will J. Merry obtained his doctorate from the University of Cambridge. He carried out research at Cambridge and ETH in Zürich and published a number of papers in symplectic and contact topology and Hamiltonian dynamics. He made several key contributions to the theory of Rabinowitz Floer homology and its applications. He was recognized as a lecturer who helped his students understand not only the course material but also the way it fitted into the whole landscape of mathematics.
Content
Chapter 1. Smooth manifolds.- Chapter 2. Tangent spaces.- Chapter 3. Partition of unity.- Chapter 4. The derivative.- Chapter 5. The tangent bundle.- Chapter 6. Submanifolds.- Chapter 7. The Whitney theorems.- Chapter 8. Vector fields.- Chapter 9. Flows.- Chapter 10. Lie groups.- Chapter 11. The Lie algebra of a Lie group.- Chapter 12. Smooth actions of Lie groups.- Chapter 13. Homogeneous spaces.- Chapter 14. Distributions and integrability.- Chapter 15. Foliations and the Frobenius theorem.- Chapter 16. Bundles.- Chapter 17. The fibre bundle construction theorem.- Chapter 18. Associated bundles.- Chapter 19. Tensor and exterior algebras.- Chapter 20. Sections of vector bundles.- Chapter 21. Tensor fields.- Chapter 22. The Lie derivative revisited.- Chapter 23. The exterior differential.- Chapter 24. Orientations and manifolds with boundary.- Chapter 25. Smooth singular cubes.- Chapter 26. Stokes' theorem.- Chapter 27. The Poincaré lemma and the de Rham theorem.
