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.

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
