# Where do I start learning differential geometry?

1. Feb 14, 2015

### PWiz

I've recently finished tackling differential equations. I want to start learning general relativity, but from what I've read, I need to have a firm footing in differential geometry first. So where do I start learning DG? I really don't want to do a half-hearted job in an attempt to quickly jump onto GR, so does anyone know some good websites where I can properly practice it? It is out of portion stuff for me so I don't have any book to refer to, but I want to do it anyways. I would also love to know the difference between topological analysis and differential geometry.

2. Feb 14, 2015

### micromass

The mathematical language of differential geometry consists of manifolds. But the least you can say is that manifolds are pretty abstract and confusing. In order to really grasp manifolds, you'll need to know more about curves and surfaces, because you can easily grasp and visualize that. I recommend the book by Pressley to start off with differential geometry.

https://www.amazon.com/Elementary-Differential-Geometry-Undergraduate-Mathematics/dp/184882890X

Then you have enough motivations to start manifolds, but you won't know enough math. Manifolds rely on topology (which basically studies the shape of a space without taking distances, angles, etc. into account). But in order to start topology, you'll need to know a bit about metric spaces. So I suggest

https://www.amazon.com/Metric-Spaces-Springer-Undergraduate-Mathematics/dp/1846283698 (you can read this together with Pressley)

Then you can start the awesome 3-volume text of Lee:

https://www.amazon.com/Introduction-Topological-Manifolds-Graduate-Mathematics/dp/1441979395/
https://www.amazon.com/Introduction-Smooth-Manifolds-Graduate-Mathematics/dp/1441999817
https://www.amazon.com/Riemannian-Manifolds-Introduction-Curvature-Mathematics/dp/0387983228

For questions/guidance/mentoring, feel free to PM me anytime!

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