Introductory differential geometry text

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What is a very good differential geometry introductory text?

My only background is Calculus (spivak). however, I'm very interested in mastering differential geometry (at both the pure math and physics application level). Any recommendations?
 
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Well, you'll need a little bit more math to do differential geometry. Introductory differential geometry is about the same for pure math and physics, and they're both pretty rigorous. First, you'll need analysis. If you've MASTERED Spivak, you may have enough. When that happens, pick up John Lee's 'Introduction to Topological Manifolds'. Once you've finished that, then you'll be ready for the same author's 'Introduction to Smooth Manifolds'. I think these are the two best introductory texts on the matter. Even though the former book focuses heavily on topology, it also browses some material of what most would call differential geometry, so that you'll get a little taste of differential geometry just by going through the prereq texts.
 

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To study differential geometry in a rigorous style, you need to know at least rigorous analysis for functions on R. Mastering differential geometry at a rigorous level is not easy, and you need quite a lot of maturity and knowledge of analysis, algebra and topology.

To learn the ideas and methods of differential geometry for application to physics, you just need a background in multivariable calculus and some experience with abstract mathematics (maybe a course in abstract algebra). For this I recommend "Applied Differential Geometry" by William Burke.
 
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Barret O'Neill, Elementary Differential Geometry - a truly great book for beginning Diff Geo. Classical topics with both a modern and classical approach.

Singer and Thorpe, Lecture Notes on Elementary Topology and Geometry -
An undergraduate book designed to prepare for graduate work. Not only geometry but de Rham cohomology, covering spaces and fundamental group. The approach to connections is advanced. I love this book but would not read it only. I would read it along with Barret O'Neill. That will put you in great shape.

A good companion book is Struik's book on classical differential geometry. It does not hurt to know classical methods. This book is rich in examples and motivates intrinsic geometry from extrinsic. Great stuff on the differential geometry of curves also.

Hilbert and Cohn -Vossen's, Geometry and the Imagination is readable and an important classic.

For Physics sadly you need to learn tensor calculus in local coordinates and Special Relativity first. This sucks but there is no way around it. I would try the Taylor, Wheeler book Exploring Black Holes for starters and then move on to Thorne, Wheeler, and Misner's Gravitiation. They teach Diff Geo for GR with great care and always try to relate it to the Physics.

For Geometry in higher dimensions I don't know a good book. Thurston's Three Dimensional Geometry and Topology is amazing but not really on differential geometry per se. Much of it is hard but there are incredible examples in it and it is de rigeur if you want to learn about hyperbolic geometries. Get it and read it over time.

Finally, you need to learn some Complex Analysis so that you can at least understand the Uniformization Theorem. Every 2 dimensional surface is also a Riemann surface and the relation between its Riemannian geometry and its conformal structure is necessary and fascinating to know. This is a classical topic in the geometry of surfaces.
 
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Have you done Spivak's second calculus book? That may help.
 
Hopefully, you are still wanting to study differential geometry, as I just came upon this thread in a search of mine. If you have just finished calculus, then I highly recommend the following book for superb introduction to differential geometry:
https://www.amazon.com/dp/1852331526/?tag=pfamazon01-20 by Andrew Pressley
This book provides solutions to every exercise in the book, so it is ideal for self-study, and it requires the least amount of mathematics of the books I list (and probably of those above also).

The standard graduate level text on differential geometry is:
https://www.amazon.com/dp/0132125897/?tag=pfamazon01-20 by Manfredo do Carmo
But this is an advanced book, and probably an end goal.

phreak mentioned pick up John M. Lee's Introduction to Topological Manifolds and then his Introduction to Smooth Manifolds. I disagree. Although they are good texts, Lee gets very wordy and you will spend a lot of time getting through those two books. You will be much better off getting
https://www.amazon.com/dp/0387480986/?tag=pfamazon01-20 by Loring Tu
Tu even mentions that he tries to hold back on the topology so that he may teach you calculus on manifolds, without getting bogged down in topology. I learned from Lee in a course I took, but now I always pick up Tu. It is very clear and concise, while covering the same material as Lee. The exercises are more focused and have selected solutions in the back.

Also, these books are classified under calculus on manifolds, which is a little different than the standard differential geometry books such as Pressley's, do Carmo's, or O'Neill's, which phreak posted above.

If you liked Spivak, then look at
https://www.amazon.com/dp/0805390219/?tag=pfamazon01-20 by Michael Spivak
I've never read his Calculus book, but I think this book is written in a different style. It contains an introduction to calculus on manifolds (the material in Tu's and Lee's books). Although, this book is small (a little over a hundred pages), and he only develops the essential material so that he can get to integration on manifolds as quickly as possible. The exercises are essential to his book, and you will have to go to Tu's book or another to get a complete progression of the theory. It would be very good to go through this book. I want to myself.

My last recommendation is
https://www.amazon.com/dp/0817637079/?tag=pfamazon01-20 by Harold Edwards
This would be a precursor to Tu's or Lee's calculus on manifolds books. Differential forms are extremely important in physics, and this book will get you going.

The books most acceptable to you right now are Pressley's and Edwards', and then probably Tu's book. Check them out to see which material you like better.
 
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I wanted to mention that, for most of the books mentioned above, you need to know some linear algebra (in addition to the other prerequisites listed above). Also, Spivak's Calculus on Manifolds is a tough read for self-study. Not only is it terse, but it contains many typos and misprints. I learned this from working my way through it several years ago.
 
I wanted to mention that, for most of the books mentioned above, you need to know some linear algebra (in addition to the other prerequisites listed above). Also, Spivak's Calculus on Manifolds is a tough read for self-study. Not only is it terse, but it contains many typos and misprints. I learned this from working my way through it several years ago.
You're right, but I think that with Advanced Calculus by Edwards and Elementary Differential Geometry by Pressley you would be okay, especially if you've had multi-variable calculus (I don't know whether Spivak covers that or not in his Calculus text). In fact, Edwards develops the necessary linear algebra in chapter 4, after spending the first three chapters building intuition and motivation for the three main chapters 4-6, which provide the main theory. He also has many interesting applications of the theory to differential equations, complex analysis, physics, and more in chapter 8. He even derives Einstein's equation, E=mc2. Basic experience with matrices and determinants, plus multi-variable calculus should be enough for Pressley's book.
 
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I thought Spivak's "Calculus on Manifolds" was an excellent intro to integration on manifolds and basic structure of calculus on manifolds. There aren't but a few typos which are easily spotted by anyone who has completed the much more demanding "Calculus" by Spivak. Most of the work in "Calculus on Manifolds" is on generalizing the concepts that you have already proven in R to Rn and general manifolds.
 

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