Literature on differential geometry, suggestions?

[saminator910]

I am reading Spivak, Calculus on manifolds, and I have a basic working knowledge of topology through Mendelson, "Introduction to Topology", I want to learn more about differential geometry, especially co variant derivatives, levi-civita connections, Ricci and Rieman curvature tensors. I know about the fundamental forms, and Rieman metrics. I am interested in general relativity but It's impossible for me to learn anything substantial about it without learning more about differential geometry. By the way, I am very familiar with differential forms, differentiable manifolds, and the classic multivariable stuff. Any suggestions?

 

[yenchin]

Try "Riemannian Manifolds: An Introduction to Curvature" by John M. Lee.

 

[saminator910]

thanks, any other suggestions?

 

[robphy]

ONeill's Semi-Riemannian Geometry

 

[blue_raver22]

As a fellow scientist, I would suggest exploring other textbooks on differential geometry such as "Differential Geometry of Curves and Surfaces" by Manfredo Do Carmo or "Riemannian Geometry" by Peter Petersen. These texts provide a more in-depth and comprehensive understanding of the topics you mentioned, including covariant derivatives, Levi-Civita connections, and curvature tensors.

Additionally, I would recommend supplementing your reading with online resources and lectures from experts in the field. This will not only provide a different perspective on the material but also allow for a deeper understanding through visual aids and demonstrations.

In terms of general relativity, I would suggest starting with "Gravitation" by Charles Misner, Kip Thorne, and John Wheeler. This textbook provides a thorough introduction to the mathematical concepts and principles of general relativity.

Lastly, I would encourage you to actively engage with the material by attempting practice problems and discussing your understanding with others, whether it be through study groups or online forums. This will solidify your understanding and allow for a more comprehensive grasp of the subject matter.