# Textbook on Differential Geometry for General Relativity [duplicate]

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 that focuses on this aspect. Having said that, I'm ideally looking for a textbook that won't keep me densely involved with it; rather a textbook that I could hopefully juggle with the textbook on GR as well.

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.

Note: I am a student of physics with a mathematical leaning.

• Comments are not for extended discussion; this conversation has been moved to chat. – David Z Jul 6 '16 at 11:08
• This looks awfully close to this question, and if it's not a duplicate of that, maybe it's a math question? – David Z Jul 6 '16 at 11:09

I wish someone had recommended Paul Renteln's Manifolds, Tensors, and Forms. An Introduction for Mathematicians and Physicists. Unlike many mathematically inclined differential geometry textbooks, it works with an indefinite metric the whole way through.

Chapters one and two aren't very necessary and primarily form a review of linear algebra. It also defines the tensor product. It does contain quite a few gems however. One thing, which isn't quite a gem but in my opinion shows the usefulness of a mathematical point of view: relating raising/lowering the indices to the Riesz representation theorem.

Chapter three is a big chapter, covering the definition of a differentiable manifold, the definition of vectors, the definition of co-vectors, their relation, the definition of differential forms, and lots of applications. I like Renteln's approach because it uses the notion of a differentiable manifold at first, and only brings in geometric manifolds (where a metric is defined and there is a clear map between vectors and covectors) when they are needed. This makes the structure of the theory of differentiable manifolds MUCH clearer. In some physicist definitions of tensors it is never clear what is and what isn't dependent on the metric. It also uses, in my opinion, the best definition of a tensor: A vector is a tangent to a curve. A covector is an element of the dual space of vectors. A tensor is a tensor product of vectors and covectors. None of this "list of quantities that transform as such and such", the transformation laws pop out of Renteln's definitions.

Chapters four and five are on homology and cohomology. I found this stuff really beautiful but skimmed it for expedience - I wanted to get on with it to study general relativity quickly.

Chapter six covers integration on manifolds. It is necessary if you want to study, say, deriving general relativity from the action principles (action is an integral over spacetime), but is mostly optional.

Chapter seven covers vector bundles. This is necessary to understand how tensor fields on differentiable manifolds behave. Again, this section is very general and only brings in the metric when absolutely needed.

Chapter eight is, like chapter three, a very big chapter. It covers all the good stuff - the metric giving rise to the connection/covariant derivative, curvature, torsion, and all of those good covariant derivative identities you need to do any GR! This is the chapter you really need to get started with GR, but it is dependent on chapters three and seven.

Chapter nine is another short one, applying things in previous chapters, but is not so relevant to starting GR.

I spent a whole bunch of time on chapters three, seven, and eight, before jumping into a graduate level general relativity sequence of courses, and found that I was more than well enough prepared.

Doesn't Sean Carroll's book give recommendations? It has an extensive bibliography, and recommends Schutz among others. The Preface explains what pre-requisites are useful, and that "building a mathematical framework is the goal" of the early chapters (2 Curvature, 3 Manifolds). It contains 8 mathematical appendices. So you are unlikely to need any reference book.

I learnt from Schutz: Geometrical Methods of Mathematical Physics, combined Choquet-Bruhat, DeWitt-Morette (& I think one other): Analysis, Manifolds and Physics, and I found it a good combination.

GMoMP is mildly rigorous, and covers most of the material you need to get a pretty good handle on GR. AMaP is a much more serious approach to a larger subject area: I have not read it completely (or even mostly) but I found it invaluable to answer questions like 'OK, what does this really mean mathematically?' in other texts. GMoMP will help a physicist understand what continuity means in a topological space well enough to be dangerous, while AMaP will tell you about nets and ultrafilters.

Notes.

• I learned this stuff in the 80s (have now largely forgotten it), there may be more recent, better, texts;
• AMaP seems to have mutated into some multi-volume epic and I don't know how what I have relates to it (it might now be vol 1, but they might have split it, I'm not sure).
• Schutz and choquet-bruhat. You win. Phenomenal books! – user122066 Jul 6 '16 at 2:05