I am currently very confused about the "topological" prerequisites of Lee's Riemannian Geometry (RG) book, An Introduction to Riemannian Manifolds. I have heard that this is a "truly introductory" text for beginners in RG. But as for prerequisites, in his preface Lee states that his other two books on Topological and Smooth Manifolds are a sufficient preparation for his RG text. But from several online forums I've come to the realizations that this would be an "overkill" just for studying RG (e.g., one need not study all of his Topological Manifolds text to study most of his Smooth Manifolds text!). So, in general, his preface is confusing to me when discussing prerequisites.
I'm confident with my analysis background (inverse/implicit function theorem, etc.) As for the differential geometry stuff, I'm thinking about going over doCarmo's Curves & Surfaces text. I'm a beginner in DG and I prefer E. Kreyszig's Differential Geometry as it introduces tensors early on (something I'm highly interested in learning about). I'm also aware of another "standard" text on RG by doCarmo as well. So, here's what I need guidance on:
- Which of the books among Lee and doCarmo on RG is preferable if one wants to get a decent exposure to RG so that they can pursue research in related areas?
- How much topology does one need to know in order to tackle the texts by Lee and/or doCarmo on RG?
- For differential geometry, is Kreyszig's DG book sufficient to prepare for Lee and/or doCarmo's RG text(s)? Or is it preferable to study do Carmo's DG before tackling RG?
My ultimate goal is (broadly) mathematical physics, in particular, Relativity and Quantum Gravity. Moreover, I consider myself more of a "math"-person and so would not prefer texts that ruin mathematical rigor.