Please recommend some good books of differential geometry for a physics student.
Physics books typically jump right into Riemannian geometry without discussing e.g. local surface theory. A couple books for background that will help give you a more intuitive feel for the math:
I still like Schutz, even though his emphasis is not Riemannian geometry:
Frankel is pretty readable and covers an interesting selection of topics:
An older text, Bishop & Goldberg, is pretty concise, but I like it for that and the price is right:
If the goal is to understand relativity, I would first seek out treatments of differential geometry by a mathematically-oriented relativist... then to others when needed.
Some names (in no particular order... some found in the URL I pasted above):
Schutz, Faber, and Frankel (as named above)
Burke, Isham, Sachs&Wu, O'Neill, Crampin, Marsden, Choquet-Bruhat, Hawking&Ellis, ....
see also https://www.physicsforums.com/showthread.php?t=168568
I am reading Frankel's book. But it is the first edition. Is the change between the first edition and the second edition very big and significant?
I don't have the editions on hand to compare....
however, the Amazon review (from the URL above) says
"Key highlights of his new edition are the inclusion of three new appendices that cover symmetries, quarks, and meson masses; representations and hyperelastic bodies; and orbits and Morse-Bott Theory in compact Lie groups."
Based on that, it seems that:
for applications to GR, I think the second edition covers as much as the first.
Separate names with a comma.