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We began an introductory course on Differential Geometry this semester but the text we are using is Kobayashi/Nomizu, which I'm finding to be a little too advanced for an undergraduate introductory course in DG. There are also no graded homeworks, quizzes, or exams so a text with solved problems would be preferred.

Textbook recommendations for introductory DG books is not a new question here, but I was specifically looking for books that follow a similar formalism as Kobayashi/Nomizu.

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    $\begingroup$ Specific topics: tensor fields, fibre bundles, connections, and Riemannian geometry - but I feel that I am missing some lower level motivation / intuition for these topics. Would love to see these in 2 - 3 dimensions. Not particularly interested in the details of defining differentiable manifolds. In general, would you say that KN is not a good book to first learn the material from? $\endgroup$ – still.ill.at.ease Nov 21 at 19:28
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    $\begingroup$ I would strongly advise against using a book without any pictures or diagrams to teach geometry at an undergraduate level. KN is written at a much more advanced level, so I wouldn't recommend it as a starting point. Personally, I really like John Lee's Introduction to Riemannian Geometry, but that doesn't have some of the topics you are looking for. $\endgroup$ – Gabe K Nov 21 at 21:04
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    $\begingroup$ If this is an intro diff geometry I would do only curves and surfaces. Unfortunately most intro books in diffgeometry actually do multivariable calculus instead. One exception is Toponogov's textbook, but likely it is too hard for your students. I was teaching such course couple of times and we wrote some notes --- they are not really ready, so use it on your own risk: anton-petrunin.github.io/comparison-geometry/… $\endgroup$ – Anton Petrunin Nov 21 at 21:20
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    $\begingroup$ A book I really enjoyed reading was Mathematical Gauge Theory by Hamilton. Though the book has a long arc towards explaining particle physics, the introductory chapters are great for understanding Lie groups, principle fiber bundles, connections, and curvature. $\endgroup$ – Mnifldz Nov 21 at 22:30
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    $\begingroup$ Did you check other posts tagged dg.differential-geometry+textbook-recommendation and the posts tagged differential-geometry+book-recommendation on Mathematics? $\endgroup$ – Martin Sleziak Nov 22 at 7:13
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Yikes, that's brutal - Kobayashi-Nomizu is an excellent reference text, but using it in a first course on the subject is a bit like learning English from the Oxford English Dictionary. For instance: chapter 2 is about connections on principal bundles, chapter 3 is about linear / affine connections, and chapter 4 is about the special case of Riemannian connections; this is conceptually an elegant way to build the theory, but pedagogically it is exactly backwards.

I second the suggestions in the comments to at least start with curves and surfaces. If the course has to go beyond that then it gets tough - there are good books about curves and surfaces, and there are good books about connections on vector bundles, but there aren't many that do both subjects in a unified way. In fact the only example that I know is Loring Tu's Differential Geometry: Connections, Curvature, and Characteristic Classes, which covers both branches of the subject and bridges the gap with explicit calculations involving Riemannian connections on surfaces in $\mathbb{R}^3$. It has a modest number of problems at the end of each chapter, and they're generally pretty good if not numerous.

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I second some of the other recommendations (Tu and Hamilton's books both seem very good from my quick look at them). Another option is the two books by Gregory Naber: Topology, Geometry and Gauge Fields: Foundations and Topology, Geometry and Gauge Fields: Interactions. They're both very clear, extremely explicit in their proofs and calculations, and at least make an attempt to have some exercises for the reader.

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    $\begingroup$ I really like these books as well, and in fact I once taught a reading course for an undergraduate based just on chapter 0 of the "Foundations" volume. By the end of it you come to see a principal bundle equipped with a connection 1-form - normally a pretty abstract construction - as a concrete geometric object which emerges naturally from considerations in physics. The books aren't quite as good for Riemannian geometry, in my opinion, but they could still make for a great advanced undergraduate course. $\endgroup$ – Paul Siegel Nov 22 at 3:31

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