Differential geometry textbook to supplement a General Relativity self-study/refresher course

Question

I have taken an undergraduate course in GR via Thomas Moore's A General Relativity Workbook. To prepare for a research experience, I have been advised to take a GR course, so I will be refreshing myself on the work I did through this course. I would like to supplement my work with differential geometry by studying from one of the following books:

Riemannian Geometry by Manfredo Perdigão do Carmo (unfortunately don't have his earlier Curves and Surfaces book),

Differential Geometry and Its Applications by John Opera,

Elements of Differential Geometry by Millman and Parker,

notes on Differential Geometry by Hicks,

Elementary Differential Geometry by Barrett O'Neil,

An Introduction to Differentiable Manifolds and Riemannian Geomtry by Boothew,

as well as the graduate-level general relativity texts,

General Relativity by Wald, and

Gravitation by Misner, Thorne, and Wheeler.

Any recommendations of a favorite among these?

similar question

Answer 1

Gravitation by Misner, Thorne and Wheeler.

If you want to check out a book not on your list, but which is by far the most thorough and rigorous treatment of the differential geometric and other mathematical aspects of General Relativity, I suggest you obtain a copy of The Large Scale Structure of Space-Time by Hawking and Ellis.

Answer 2

For a differential geometric viewpoint of General Relativity I would suggest General Relativity by Wald which is a basic introduction to the subject in the most elegant way. Along with that you could refer to selected topics from Gravitation by Misner, Thorne and Wheeler.

Apart from the above two books, as mentioned in another answer, you could refer The Large Scale Structure of Space-Time by Hawking and Ellis. This one is compact, rigorous and discusses the mathematical aspects of General Relativity with much care.

Answer 3

https://www.academia.edu/38949108/Geometria_diferencial_de_curvas_y_superficies_Manfredo_P._do_Carmo_Comprimido

I have a spanish version. Here in Brazil, Do Carmo is considered the founder of Brazilian differential geometry.

Answer 4

Maybe you could also take a look at A Mathematical Introduction to General Relativity.

The aim is to present general relativity to undergraduate mathematics students in a mathematically rigorous fashion, and so it first develops all the needed differential geometry (assuming familiarity with linear algebra and multivariable calculus). A pdf file of some chapters can be found on the publisher's website above. Also a google preview is linked here .