Differential Geometry for General Relativity 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.
 A: 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


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*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


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*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


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*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.


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*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.


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*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.
A: 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.
A: 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
