C3.3 Differentiable Manifolds (2023-24)
Topic outline
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These are Nigel Hitchin's 2014 Differentiable Manifolds lecture notes. I based my lectures on them, and they are the primary reference for the course.
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I think it would be better if you come to the lectures in person. It will make you a superior human being, and you can ask questions. However, I am aware that the lectures are being made available online and some idle lummockses will stay home and watch them in bed. It will probably be difficult to read the whiteboards on the videos.
Here are handwritten notes of what I intend to write on the board in the lectures (I will say more than this), in case of problems with reading the boards in the videos. I will add to this file to keep pace with the lectures.
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These are Jason Lotay's 2021 Differentiable Manifolds lecture notes. They are very useful, and you may prefer them to the Hitchin notes. I didn't base my lectures on them, though.
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This is an introductory sheet which will not be marked. It covers properties of smooth maps between Rᵐ and Rⁿ, and examples of manifolds defined by atlases.
The file 23C33Sheet0.pdf contains just the questions, whilst the file 23C33Soln0.pdf contains the questions and solutions. -
This sheet covers the material of lectures 1-4, with the topics manifolds, smooth maps, and partitions of unity. The relevant parts of the Hitchin lecture notes are Sections 2-3, and of the Lotay lecture notes Section 1.
Only Section B will be marked. The file 23C33Sheet1.pdf just contains the questions, whilst the file 23C33Soln1A+C.pdf contains the questions and the solutions to Sections A and C. -
This sheet covers the material of lectures 5-7, with the topics submersions, immersions and embeddings, the Whitney Embedding Theorem, vector fields and flows, and the Lie derivative. The relevant parts of the Hitchin lecture notes are Sections 3-4, and of the Lotay lecture notes Sections 2-3.
None of this sheet will be marked, I'm afraid. The class will concentrate on Section B unless people ask otherwise. The file 23C33Sheet2.pdf just contains the questions, whilst the file 23C33Soln2A+C.pdf contains the questions and the solutions to Sections A and C. -
This sheet covers the material of lectures 8-12, with the topics differential forms and the exterior derivative, de Rham cohomology, and orientations. The relevant parts of the Hitchin lecture notes are Sections 5-7, and of the Lotay lecture notes Sections 4, 5 and 7.
Only Section B will be marked. The file 23C33Sheet3.pdf just contains the questions, whilst the file 23C33Soln3A+C.pdf contains the questions and the solutions to Sections A and C. -
This sheet covers the material of lectures 13-16, with the topics manifolds with boundary, Stokes' Theorem, the degree of smooth maps, Riemannian metrics, isometries, and Killing fields. The relevant parts of the Hitchin lecture notes are Sections 7-9, and of the Lotay lecture notes Sections 6-8.
None of this sheet will be marked, I'm afraid. The class will concentrate on Section B unless people ask otherwise. The file 23C33Sheet4.pdf just contains the questions, whilst the file 23C33Soln4A+C.pdf contains the questions and the solutions to Sections A and C. -
Note: this is not relevant to undergraduates.
Some DPhil students doing this course may want to submit it as a 'broadening course', and so be assessed on it. Here is a list of miniprojects for this. You can do one over the Christmas vacation and submit it in January.
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