Here’s an idea I had for a book. Maybe someone has already written it. If you know of such a book, please let me know.
Differential geometry has a huge ratio of definitions to theorems. It seems like you do nothing but study definitions for a semester or two in preparation for proving something later. It’s easy to lose sight of the geometry. I’d like to see a book that is a concrete complement more typical abstract books.
My suggestion is for someone to write a book that goes through a standard differential geometry book, like Spivak’s, and compute everything for a small number of example manifolds: at least a sphere and an ellipsoid, maybe a torus. The book would first go through everything on a sphere where things are simplest, then generalize to an ellipsoid. There would be a lot of applications to geodesy: to first approximation the earth is a sphere, to second approximation it is an ellipsoid.
Sometimes a calculation, such as arc length, is very simple on a sphere. It can be done just using trig. Then the analogous calculation on an ellipsoid is much harder. It is complicated enough to illustrate the machinery of differential geometry. However, we know the answers shouldn’t be much different from those for a sphere, so we have a way to see whether the results are reasonable. This book would not shy away from computational difficulties.
I imagine this book would have lots of illustrations. It might even come with physical models, such as a globe with an exaggerated equatorial bulge. The idea is to be as tangible as other books are abstract.
I don’t plan to write this book, at least not any time soon. Maybe if my consulting goes well I would have the time to work on it in the future, but now is not the time for me to write a book. In the mean time, if someone wants to scoop my idea, please do!