I found this thread What is a covector and what is it used for?, in which the top answer states "Many of us got very confused with the notions of tensors in differential geometry not because of its algebraic structure or definition, but because of confusing old notation".

I am currently working through Barret Oniell: Elementary Differential Geometry 2nd Edition and going through all of the problems, it feels rather like fumbling in a dark room for a light switch (at the end of chapter 4). I find myself having to review all sorts of different material using different notations and try to piece everything together in a sensible way on my own. I also have purchased Wolfgang Kuhnel's book on the matter, but before I spend another few months headed down that road, I think it best to ask this advice...

What is a thorough, articulate and reliable (i.e. not riddled with mistakes) book on the subject matter suitable for self study that uses modern (by which I understand to be less confusing) notation? Preferably it would be available for purchase in an electronic format if possible.

  • $\begingroup$ Well, let me tell you that the modern notation used in differential geometry is still the "confusing old notation" which you seem to refer to. It so happens that "in practice", a lot of the notation is useful for fast computations or even insight. It is, however, mostly terrible for learning and filled up with abuses. I don't think there is much to be done about it. $\endgroup$ – Aloizio Macedo Sep 27 '17 at 3:02
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    $\begingroup$ You can look at An Introduction to Manifolds by Loring Tu, Riemannian Manifolds; An introduction to curvature by John Lee or Manifolds and differential geometry by Jeffrey Lee. $\endgroup$ – 512122 Sep 27 '17 at 3:09
  • $\begingroup$ I don't know what can say to you, since I don't know in what differential geometry course are you interested and since I've forgotten this subject. Thus take this recommendation as a belief. I know the nature of books written by Jürgen Jost, because in home I've Postmodern Analysis and I've read in the past Partial Differential Equations and I like those. Thus look at the Table of contents from this Springer link for Jürgen Jost, Riemannian Geometry and Geometric Analysis, Springer (2011). Good luck. $\endgroup$ – user243301 Sep 27 '17 at 3:23
  • $\begingroup$ Even though modern notation might be a huge improvement over old notation, it's still a challenge to absorb all the modern notation. I think Barrett O'Neil's notation is fairly modern and clear. $\endgroup$ – littleO Sep 27 '17 at 4:07
  • $\begingroup$ People are recommending more advanced texts to you (with the exception of Pressley). O'Neill makes you do differential forms, which happens to be my personal favorite but is hard going for someone who's not learned them already. You might check out my (free) undergraduate text, available in .pdf form from the link in my profile. $\endgroup$ – Ted Shifrin Sep 27 '17 at 5:45

Introduction to Smooth Manifolds by John Lee is the best book I know for a modern introduction to Differential Geometry, another good book is Introduction to Manifolds by Tu.

Lee's book is really good good for self study in that it contains a very thorough exposition, a plethora of examples, and many good exercises.

  • $\begingroup$ Tu is a very nice book to begin with. $\endgroup$ – g.s Sep 27 '17 at 8:37
  • $\begingroup$ Thanks I'll give those a look :) $\endgroup$ – Mark Sep 27 '17 at 12:49

Pressley would be the best bet. It also has all the solutions worked out. However, it doesn't discuss any material on connections, bundles, and tensors. It truly is an undergraduate text, but a very good one.

  • $\begingroup$ Having solutions worked out is really something that would help. Sounds like a great place to start. Thanks $\endgroup$ – Mark Sep 27 '17 at 12:50

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