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.
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.
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.
