Introductory Book on Differential Geometry

[fys iks!]

Hey,

I was wondering if anyone could recommend an introductory book for differential geometry. I am studying general relativity and need some help with this topic.

Thanks.

 

[Daverz]

This one may be helpful:

https://www.amazon.com/dp/082471749X/?tag=pfamazon01-20

 

[Landau]

Do you want a pure mathematical treatment, or a text for physicists?
Modern Differential Geometry for Physicists by Isham is a good one.

 

[dx]

"Applied Differential Geometry" by William Burke

 

[Fredrik]

"Introduction to smooth manifolds", and "Riemannian manifolds: an introduction to curvature" by John M. Lee. These books are great. The only problem is that you need both of them. Everything about connections, geodesics, covariant derivative and curvature is in the second book. It don't think there can be a better place to read about those things than this book.

I like Isham's book too. I think he covers some concepts, in particular the tangent space, even better than Lee. But it's not really an introduction to the differential geometry you need to understand GR. It's an introduction to the differential geometry you need to understand Yang-Mills theory. So he talks a lot about Lie groups and fiber bundles, which you don't need right now, and doesn't say much about geodesics and curvature.

 

[fys iks!]

Thanks for the suggestions!

Isham's book was a little to advanced. I was able to find one though to suit my needs. It doesn't look like it is too popular but its called "Introduction To Differential Geometry and Reimannian Geometry By: Erwin Kreyszig"

 

[Fredrik]

Isham's book was a little to advanced.

I very much doubt that, but as I said, it's going in the wrong direction for you. It's preparing the reader for Yang-Mills theory, not GR.

I was able to find one though to suit my needs. It doesn't look like it is too popular but its called "Introduction To Differential Geometry and Reimannian Geometry By: Erwin Kreyszig"

His functional analysis book is very popular, so he seems to know how to write good books. I'm sure it's fine, but it's hard to beat Lee. Hm, there's also a book by a guy named Manfredo Perdigão do Carmo that's getting good reviews at Amazon.

 

[Landau]

I very much doubt that

It does require more mathematical prequisites than most physics students have.

His functional analysis book is very popular, so he seems to know how to write good books. I'm sure it's fine,

His FA book is ok. But if we're talking about https://www.amazon.com/dp/0486667219/?tag=pfamazon01-20 (perhaps an expanded version including Riemannian Geometry?), I have to warn OP: this is pretty old-fashioned! Everything is done in local coordinates, and mostly in three dimensions. Compare this book with something like Lee, and you'll see they are almost disjoint.

but it's hard to beat Lee. Hm, there's also a book by a guy named Manfredo Perdigão do Carmo that's getting good reviews at Amazon.

Lee is pretty good in my opinion, but of course it's not the only one. These all have their merits:

Tu - An Introduction to Manifolds
Spivak - A comprehensive introduction to differential geometry Vol I
Darling - Differential Forms and Connections
Lang - Introduction to Differentiable Manifolds
Conlon - Differentiable Manifolds
Barden, Thomas - An Introduction to Differential Manifolds

 

[Fredrik]

It does require more mathematical prequisites than most physics students have.

Don't they all?

I have to warn OP: this is pretty old-fashioned! Everything is done in local coordinates, and mostly in three dimensions.

Ughh...that sucks. I want everything to be as coordinate-independent as possible. I retract my "I'm sure it's fine" comment.