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I'm currently studying some basic theory about manifolds from the book 'An introduction to manifolds' by Loring W. Tu.

The problem I have with this book is that there are very little exercises, and the exercises that are included are quite easy and do not really serve to provide you with any real insight by forcing you to think deeply about the material.

I do like the pace and assumed prerequisites of the book.

So I'm looking for something similar to this book but with more and harder/more interesting exercises.

Thanks.

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  • $\begingroup$ It's a little hard to say what a hard intro differential geometry exercise should be. In some sense you're setting up notation (a formidable amount, granted) for a few hundred pages. Have you looked at Guillemin-Pollack? That's a fun book. They do favor proving geometric theorems over building up the formalism (everything is in $\mathbb{R}^n$, for example) but you might like it. Combining it with Tu's book could be great. $\endgroup$ – Hoot Jul 18 '15 at 21:03
  • $\begingroup$ @Hoot That might be a fair point. All theorems so far are basically just pretty straightforward substitutions of one definition into the next, so I can imagine that it's hard to come up with nice exercises. Still hope they exist though $\endgroup$ – user2520938 Jul 18 '15 at 21:05
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I would recommend John Lee's trilogy:

  • Introduction to Topological Manifolds
  • Introduction to Smooth Manifolds
  • Riemannian Manifolds: An Introduction to Curvature
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    $\begingroup$ My one big problem with this series is that the problems tend to be quite easy. Given that that's one of the OP's complaints, I'm not sure if it's a good recommendation here. $\endgroup$ – user98602 Jul 18 '15 at 20:57
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    $\begingroup$ @MikeMiller So far I've found most books from the GTM-series to be more-or-less at the right level for myself (I've similarly found that most books in the UTX-series vary a lot more in both their level and in overall quality). So thanks Spenser, I'll definitely take a look at those suggestions. $\endgroup$ – user2520938 Jul 18 '15 at 21:02
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    $\begingroup$ I don't think the books in GTM have all that much in common. You've been lucky, maybe! $\endgroup$ – Hoot Jul 18 '15 at 22:47

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