Differential Geometry Book

• Geometry

Main Question or Discussion Point

Hello,
does anyone know an (more or less) easy differential geometry book for courses in generall relativity and quantum field theory? I'm looking for a book without proofs that focus on how to do calculations and also gives some geometrical intuition. I already looked at The Geometry of Physics: An Introduction, but it was too detailed for me.

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fresh_42
Mentor
Hello,
does anyone know an (more or less) easy differential geometry book ...
Yes.
... for courses in general relativity and quantum field theory?
No.
I'm looking for a book without proofs ...
No.
... that focus on how to do calculations and also gives some geometrical intuition.
Yes.
I already looked at The Geometry of Physics: An Introduction, but it was too detailed for me.
So for half of the requirements:
https://www.amazon.com/dp/0387903577/?tag=pfamazon01-20

George Jones
Staff Emeritus
Gold Member
quantum field theory?
By the differential geometry of "quantum field theory", do you mean fibre bundles, i.e., principal bundles and associated vector bundles?

I don't know anything about quantum field theory yet, I just heard that diffenetial geometry is used in it ;-). I'd like to read a seperate book on the mathematics beside an introduction to quantum field theory/general relativity.

@fresh_42 Thanks for your answer. It's not quite what I'm looking for right now, but I will save it for later if I want a deeper understanding of mathematics.

fresh_42
Mentor
@fresh_42 Thanks for your answer. It's not quite what I'm looking for right now, but I will save it for later if I want a deeper understanding of mathematics.
It's actually rather basic with a lot of drawings. It introduces all basic ideas and concepts, from smooth manifolds to curves and vector fields, and a lot of coordinate calculus. It ends where the more abstract concepts, which @George Jones mentioned, begin with. However, it contains all the fundamentals as geodesics, parallel transport, curvature etc. which the abstract concepts are developed from. As an introduction to differential geometry it is pretty good.

I didn't found tensors in the index, thats why I thought it might be a bit over the top.

fresh_42
Mentor
Yes, that's true. The excessive use of tensors, covariant and contravariant by physicists is mathematical nonsense. They are all vectors, transformations, multilinear forms, curvature, gradient or whatever, so mathematics doesn't just call them tensors unless they are part of a universal mapping problem. In so far, it is true. The book doesn't prepare you well for the notations used in physics. The objects are all there, but not their physical notation as "tensor"; coordinates are only used if necessary: vectors are written $\mathbf{v}$ and not $v^i$.

The objects are all there, but not their physical notation as "tensor"; coordinates are only used if necessary: vectors are written $\mathbf{v}$ and not $v^i$.
Thanks for making that clear. I didn't really like the index notation anyway, so I will go and buy it :-)

Demystifier
I'm looking for a book without proofs that focus on how to do calculations and also gives some geometrical intuition.
I didn't really like the index notation anyway,
The no-index notation is more suitable for doing proofs, while the index notation is more suitable for doing calculations.

A book on "elementary differential geometry" will cover the local and global differential geometry of curves and surfaces and is not going to get you very far towards the math required for GR, though it will help with intuition and mathematical maturity.

haushofer