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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.

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You probably want more of a Riemannian geometry book. I'd probably go with Lee's "Riemannian Manifolds", but he also has "Introduction to Smooth Manifolds".

There is also Bott and Tu's "Differential forms in Algebraic Topology".

Munkres has a book "Analysis on Manifolds" which is not as dry as the class I took, where it was the text. (But that's not saying much.)

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  • $\begingroup$ These are good recommendations, I'd add Jeffrey Lee's text on Differential Geometry is also useful. The gory details about the exterior algebra in his text have been useful to me at times. $\endgroup$
    – James S. Cook
    Commented Oct 13, 2020 at 6:26
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I would like to add a classic:

Bishop, R. and Goldberg, S. (1980). Tensor Analysis on Manifolds. Dover Publications.

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