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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
    Commented Nov 21, 2019 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
    Commented Nov 21, 2019 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
    Commented Nov 21, 2019 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
    Commented Nov 21, 2019 at 22:30
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    $\begingroup$ The book of do Carmo's, Differential Geometry of Curves and Surfaces, is nice and can be found online. Klingenberg's A Course in Differential Geometry is good too. For more advanced topic I've found "Geometry I: Basic Ideas and Concepts of Differential Geometry (Encyclopaedia of Mathematical Sciences)" to be an insightful and succinct introduction to many topics, and the books of Novikov and Fomenko are also good. Spivak's tomes are nice, if a little wordy for my taste. KN is a great reference book, when you have already understood everything, you can go there to refresh your memory. $\endgroup$
    – alvarezpaiva
    Commented May 9, 2021 at 16:24

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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
    Commented Nov 22, 2019 at 3:31
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Two (bright new) interesting books by Jean Gallier and Jocelyn Quaintance:

  • Differential Geometry and Lie Groups: A Computational Perspective.
  • Differential Geometry and Lie Groups: A second course.

A nice introduction:

  • First Steps in Differential Geometry: Riemannian, Contact, Symplectic, by Andrew McInerney
  • Differential Geometry: Curves - Surfaces - Manifolds, by Wolfgang Kühnel

Applied to physics:

  • Differential Geometry and Relativity Theory: An Introduction, by Richard L. Faber
  • General Relativity Without Calculus: A Concise Introduction to the Geometry of Relativity, by Jose Natario
  • Tensors, Differential Forms, and Variational Principles, by David Lovelock & Hanno Rund

and many, many more...

Update 1:

After the comment by @BenMcKay, I found a set of (not as well-known as they should) lectures by Nomizu himself, published by the Mathematical Society of Japan in 1956. These are titled: Lie Groups and Differential Geometry.

Just for your information, these notes are about 80 pages long, and has three chapters:

  • Differential manifolds.
  • Connections in fibre bundles.
  • Linear connections.

Another useful reference is:

  • Analysis, manifolds and physics Part I: Basics. by Y. Choquet-Buhat, C. DeWitt-Morett and M. Dillard-Bleick.

There is a Part II of this book, by the first two authors, focus in the applications.

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  • $\begingroup$ Similar to Kobayashi and Nomizu? $\endgroup$
    – Ben McKay
    Commented Mar 30, 2021 at 20:32
  • $\begingroup$ @BenMcKay You're right... but in some sense, there in no similar reference to the Kobayashi--Nomizu. However, I encounter a nice set of lectures fitting the conditions of the OP. I'm updating my answer to include it. $\endgroup$
    – Dox
    Commented May 9, 2021 at 9:44

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