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## Main Question or Discussion Point

I'm reading 'A Most Incomprehensible Thing: Notes towards a very gentle introduction to the mathematics of relativity' by Collier and came across the following paragraph in the chapter on tensors:

Personally I definitely intend to learn differential geometry, but I'm still curious on the above questions.

I'm a bit skeptical about the "useful but not essential" part, but I'd still like to get the opinion of the members here. Let's say a student of GR (or other branches like QFT or quantum gravity candidate theories, etc.) simply memorizes the rules of tensor manipulation without paying much attention to the underlying differential geometry - would they struggle at any point? Would this handicap them in any way (or maybe I should ask how seriously would this handicap them)?Later on, we'll meet the rules of tensor algebra, including operations such as scaling [...] and contraction [...]. Differential geometry is the theoretical foundation to these rules. However, just as you don't need to be an automotive engineer in order to drive a car, you don't have to know all the underlying mathematics if you want to manipulate tensors - just a working knowledge of the rules of tensor manipulation, which you can more or less learn by rote. So, much of this section is 'under the bonnet' detail -useful but not essential.

Personally I definitely intend to learn differential geometry, but I'm still curious on the above questions.