Selecting Differential Geometry Exercises

Question

I'm self-studying differential geometry with Do Carmo's books "Differential Geometry of Curves and Surfaces" and "Riemannian Geometry" and I find those books very good, however I feel a little confused when selecting which exercises to do.

What's the best way to select exercises when studying that kind of math? I know this question seems silly, it's like : "how can someone don't know which exercises to do?", but it's just the case that there's no time to work on all of them, so I feel a little confused in which to work more.

Thanks in advance, and sorry again if the question is not fitted to this website.

Answer 1

Usually exercises are ordered according to level of difficulty: In terms of the order in which they are listed, the lower numbers are usually more a matter of understanding more concretely, and often more computational then the later problems, where the emphasis is more on conceptual understanding.

So I'd recommend "warming up" with some of the earlier exercises; select scattered problems in towards the middle.

But I'd emphasize working as many higher numbered exercises as you have the time to do. You'll have the deepest understanding of the material if you emphasize and work through the latter problems. They usually require a solid mastery of the section they cover, and often require that you synthesized the material, and are able to use what you've learned more "flexibly" than simple rote repetition/computation requires.

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One additional suggestion: try googling the title of the text you're using (or perhaps author name(s), colon, .edu). You will likely stumble upon class syllabi for courses using that text(s), and in many, you'll find a list of exercises assigned. You'll likely also encounter a course website for a course using the text(s), where assignments are often posted.