Comments on Differential Geometry Class
In the semester of 2024 Spring, I took Differential Geometry by Professor Hisham Sati.
The course is undoubtedly intense and covers a broad range of advanced topics. Professor Sati has designed it with the rigor of a graduate-level course, as he himself mentioned.
Professor Sati is both personable and generous in his grading approach. However, I sometimes feel that the course could be better prepared. His method of teaching primarily involves transcribing notes directly from his materials to the board. Interestingly, my peers have noted that his approach in topology was more engaging.
All course notes are handwritten, which was unexpected. It's apparent that Professor Sati invests considerable effort into preparing these notes, though they occasionally contain typos and may not be as polished as published textbooks. Nevertheless, his handwriting is clear, and his examples are easy to understand. The direct transcription of notes to the board can be challenging to follow at times, which sometimes affects my ability to stay engaged. Additionally, while the notes are concise, they are not always conducive to independent study. This prompted me to create several blogs specifically to discuss and clarify concepts in differential geometry. Resources, Quotien Space, etc.
Professor Sati tailored the course content to be accessible by integrating elements of linear algebra and multivariable calculus. However, despite not requiring algebra, category theory, or topology as prerequisites, these topics are frequently referenced, which can be confusing. For instance, terms like 'homeomorphism' and 'isomorphism' still puzzle me.
The course structure includes:
- Review of linear algebra and calculus 3
- Implicit function theorem
- Extension of multivariable calculus to matrix functions, covering general linear, orthogonal, and special orthogonal groups, among others
- Multilinear algebra and tensors
- Manifolds: from extrinsic to intrinsic concepts
- Differential maps between manifolds
- Tangent spaces
- Immersions, submersions, and submanifolds
- Vector bundles, including fiber bundles and vector fields on manifolds
- Differential forms
- Riemannian geometry
- Riemannian functionals and general relativity
This structure reflects the complexity and depth of the topics covered in the course.