This are my notes from Math 203a (Fall 1982) and Math203b (Spring 1983), taught by Raoul Bott, Harvard University.
I didn't appreciate it at the time, but Bott taught a sort of "coordinate free approach" to differential geometry, one that avoids explicit computations as much as possible. This is a powerful method of developing Riemannian geometry to give as much intuition as possible and avoid computation clutter. Every once in a while one has to write down a curvature tensor, Christoffel symbols, etc.
I find these notes helpful to review, even though these days AI tools do a (pretty) good job at producing coordinate free formulas, etc. However, I'm not sure how much such tools realize that one can start the coordinate free approach by defining the cotangent space at a point, P, as m_P/m_P^2, for the maximum ideal m_P associated to a point.
[This approach via maximum ideals is used in modern algebraic geometry. It is also used in C. Zeeman's notes on "Castrophe Theory" (a somewhat sensationalistic name for a part of singularity theory, inspired by a monograph of Rene Thom). This is a fascinating, but rather long story...]
These notes may contains errors, misconceptions, etc. They are offered as is.
Copyright 2025, Joel Friedman. all rights reserved. Free to download and print for your personal use and education needs.
I often had to get up around 5:30am those days, and I was often sleepy during class, despite Bott's inspiring lectures. Usually I sat next to a budding artisit, who would occasionally make sketches when I was sleeping.
[The budding artist would have made a first-rate mathematician, although I'm less certain about their prospects as a professional artist. Sadly, our artist decided to professionally pursue neither art nor math, but I believe they landed in a happy and rewarding career nonetheless.]
Our artist will remain anonymous unless I hear from them and they express to be acknowledged for the sketches in these notes...