I've been studying differential geometry by myself for some time now. I studied a fair amount of the basic general theory and gone through a lot of the exercises from several textbooks.
Lately I started to realize how huge (and daunting) differential geometry really is. I think I have a pretty solid grasp of the basic objects. For example I can switch from global to local description with comfort, and do most symbolic calculations using the basic objects of the theory and not get confused about what i'm doing since i understand the operations and the context (e.g. proving Fundamental theorem of Riemannian geometry, proving general properties of connections and their curvature tensors, proving various identities about different derivations, proving frobenius theorem etc.).
My problem lies with concrete examples and computations. I've had little to no experience with those and frankly they quite scare me. The only examples I know and tampered with before are spheres and projective spaces (and a pinch of some matrix groups and grassmanians). And even with them i feel my experience is quite brief. Most of the textbook problems I solved were general theory, which is great and rewarding, but I do feel unbalanced at the moment.
Why is the pool of examples I found so far in textbooks so small?
Is it the case you need a lot of machinary before you can really tackle more examples?
What would you recommend me to do?
Edit: Here are the main books I've studied from (I admit I wasn't completely thorough - however I'd rarely skip an exercise problem from a chapter i've been reading):
- guilliam and pollack
- Liviu - geometry of manifolds (roughly the first third of the book).
- Jefferey Lee - manifolds and differential geometry (up to chapter 8).