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I'm going to start self-studying General Relativity from Sean Caroll's Spacetime and Geometry: An Introduction to General Relativity. I'd like to have a textbook on Differential Geometry/Calculus on Manifolds for me on the side.

I do like mathematical rigor, and I'd like a textbook whose focus caters to my need. Having said that, I don't want a exchaustive mathematics textbook (although I'd appreciate one) that'll hinder me from going back to the physics in a timely manner.

I looked for example at Lee's textbook but it seemed too advanced. I have done courses on Single and Multivariable Calculus, Linear Algebra, Analysis I and II and Topology but I'm not sure what book would be the most useful for me given that I have a knack of seeing all results formally.

P.S: I'm a student of physics with a mathematical leaning.

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I wanted to recommend Lee, but since you said it's too advanced... Well, to be fair, while his book is quite extensive, it is a very pedagogically written one too, so if you wish to study manifolds, at one point at least, you should read it.

I am not sure that's what you are looking for, but there are some GR books that discuss differential geometry in bit more detail and rigour than Carroll's book, these would be for example

  • Wald: General Relativity
  • Straumann: General Relativity With Applications To Astrophysics
  • Hawking & Ellis: The Large-Scale Structure Of Spacetime

The last third of Straumann's book is essentially differential geometry, and he is quite rigorous.

For pure math books you could try

  • Spivak: A Comprehensive Introduction To Differential Geometry

This is essentially a 5-volume grimoire, however it builds everything up quite slowly and pedagogically, and makes an attempt to build a bridge between the old formalism (indices, coordinates, etc.) and the modern one

  • Isham: Modern Differential Geometry For Physicists

This one does not actually treat Riemannian geometry as far as I recall, but was written specifically for physics people, and also it has a nice account of principal bundles.

  • Boothby: An Introduction To Differentiable Manifolds And Riemannian Geometry

About as advanced as Lee, I believe. Also this book does treat Riemannian geometry, as you can infer from the title.

  • Warner: Foundations Of Differentible Manifolds and Lie Groups

  • Kobayashi & Nomizu: Foundations Of Differential Geometry

This is a very advanced book that is quite hard to read, so I'd suggest visiting this later. However, it is also quite essential. Despite the fact that this (two-volume) book is quite old, it is still the standard reference in the field. The contents of volume 1 is what would interest you more, probably, as the most of Riemannian geometry is being treated there.

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  • $\begingroup$ Lee's book looks great, but it seemed really daunting after I had read the pre-requisites. It'd be great if you could elaborate on the pre-requisites required to tackle Lee's textbook? $\endgroup$ – Junaid Aftab Jul 6 '16 at 0:59
  • $\begingroup$ @JunaidAftab You should have a good grasp of linear algebra and multivariable calculus. He also assumes some knowledge of topology, however I could work through myself most of his book by having a very basic knowledge of topology. You might wanna skip de Rham cohomology and other algebraic topology-related stuffs on first reading, the rest should be quite palatable. $\endgroup$ – Bence Racskó Jul 6 '16 at 6:52
  • $\begingroup$ I have had a course on topology, but I'd definitely want to go back and review stuff. I, for example, don't really know much about regular, normal spaces and the deluge of results that go with classifying topological spaces since I didn't focus much on that unit of the course. I definitely need to revise topology since it's quite different and hard. What about algebra? Do I need that? Also, how much knowledge of linear algebra is required? I have done the first course, not a second course that presents everything (in real and complex vector spaces) with proofs. $\endgroup$ – Junaid Aftab Jul 6 '16 at 6:59
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Check out Barrett O'Neill's book on semi-Riemannian geometry. This book is written exactly for your purposes: it discusses manifolds with symmetric nonsingular metrics, and in particular spacetime metrics. There are even chapters on cosmology and the Schwarzchild metric.

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Semi-Riemannian Geometry with Applications to General Relativity by Barrett O'Neill is my recommendation. He's very thorough, and doesn't skip the details, which is great for someone new to the subject. He introduces general relativity later on once he's covered all the necessary semi-Riemannian geometry

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