Looking to start studying real mathematics.

[Terilien]

hi physics forum. As many of you know, I've been studying general relatvity(its going quite well), but I'd like to delve into real rigorous mathematics. so essentially I'm looking for introductiosn to various topics, especially topology abstract algebra and differential geometry. Could someone tell me where to start and more importantly, how to go about doing it all?

thanks in advance.

Oh and my problems with certain concepts have vanished. It was due to notation.

 

[yenchin]

There are some good books on differential geometry, for example Do Carmo's two books: "Differential Geometry of Curves and Surfaces" and "Riemannian Geometry".

If you want to see more rigorous maths applied to GR, you should (if you have not already) try Barrett O'Neill's texts on "Semi-Riemannian Geometry with Applications to Relativity", and "Geometry of Kerr Black Holes".

Another book you may want to try is "Symmetries and Curvature Structure in General Relativity" by G. S. Hall. It's mainly a math book (not physics).

I think it might be useful to start with GR-related setting (since you are already studying GR), but now focusing more on the mathematical structures, hence the recommendations.

I am sure other more experienced people can give better advise though.

 

[Crosson]

Have you already done a rigorous version of (real) analysis?

 

[Terilien]

no but it would be nice!

 

[mathwonk]

i do not know of any generally accepted intro to topology, which is a very large subject, but Michael Artin's Algebra is excellent for beginning abstract algebra.

topology has many sides, and one often begins with the most boring aspect, namely general, or point set topology. I myself read Kelley for that many decades ago. and many students have started with a book by munkres.

the more interesting aspects are differential and algebraic topology, or perhaps also geometric topology.

a nice little very elementary introduction, but substantial, is by chinn and steenrod. some other lovely and elementary but excellent books are by andrew wallace: intro to alg top, and diff top, first steps. thurston has a nice intro to geometry: three dimensional geometry and topology.

my compliments to you for seeking actual recommendations of good books, unlike some posters here who waste their time and ours simply railing against what they claim are all the bad books out there.

actually given that it is largely a labor of love, and requires a long apprenticeship and unrewarded effort to be able to produce one, i am impressed at the number of wonderful books available.