I was wondering whether it is possible to get a fairly good understanding of the subject of differential geometry through learning General Relativity (eg from Woodhouse's book)?
Ie the sort of understanding where you know and can use the main results but dont have to go through all the depths of rigor (eg calculus vs analysis).
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Books on general relativity tend to be choc-a-bloc with index notation and it can get a bit confusing.
My advice is first read some notes on the geometry of curves and surfaces to get the general ideas about geometry and then go for a differential geometry book.
Then once you go to general relativity, you'll not only have a firm grip of geometry, you can understand the physics better.
It is definitely possible. However, in the long term, it is beneficial to study differential and Riemannian geometry. That being said these notes (http://www.damtp.cam.ac.uk/user/hsr1000/lecturenotes_2012.pdf) are an excellent start down the right path. They are written by Prof. Harvey Reall at DAMTP, Cambridge for the graduate mathematics course in GR. They explain all the necessary bits of differential geometry to gain a good grasp on GR from a mathematical standpoint.
As a previous commenter noted, Barrett O'Neill's Semi-Riemannian Geometry is an excellent text. I would also recommend Tu's Introduction to Manifolds.
You could also consult one of the following: Wald's General Relativity (a classic and has a fair amount of geometry in the appendices), Hawking and Ellis Large Scale Structure of Spacetime (the introductory chapter is differential geometry for GR).
Hope this helps.