Where can I find a good course on tensor/ricci calculus not focused on applications and physics?

I've been running into lots of tensor-theoretic stuff in differential geometry, so I don't know if proof-based differential geometry courses constitute a "Tensor Calculus for Mathematicians", or if they just happen to have the parts useful to differential geometry, like topology to analysis.

As of now, it seems like the vast majority of tensor calculus books are directed at theoretical physicists learning general relativity, and as such, they lack that mathematical rigor I've recently grown to know and love.

Also, textbooks are a preferred answer compared to video courses/lectures, as I find them easier to work through.

Note: I realize that a similar question has been asked here: Tensor Calculus. However, this seems to be looking for a less rigorous textbook that an engineer might use, rather than something you would write a lot of proofs for.

  • $\begingroup$ Check "A Short Course in General Relativity" by Foster & Nightingale, in particular the appendix on tensors. $\endgroup$
    – Ivo Terek
    Oct 12, 2020 at 21:59
  • $\begingroup$ @IvoTerek Really? I would think that it would be more oriented towards theoretical physics rather than mathematical rigor, considering the title. $\endgroup$ Oct 13, 2020 at 2:33
  • $\begingroup$ That being said, the back matter (which includes the said appendix) is free on SpringerLink, so I will take a look regardless. $\endgroup$ Oct 13, 2020 at 2:34
  • $\begingroup$ That's the book that made Einstein's notation click for me. But if this is not what you're looking for, then check "An Introduction to Riemannian Geometry with Applications to Mechanics and Relativity" by Godinho & Natario. Or "Differential Geometry: Connections, Curvature and Characteristic Classes" by Loring Tu. $\endgroup$
    – Ivo Terek
    Oct 13, 2020 at 3:16
  • $\begingroup$ Search for Saul A. Teukolsky works physics.cornell.edu/saul-teukolsky $\endgroup$
    – janmarqz
    Oct 13, 2020 at 4:20


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