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I was going through the lectures of F. Schuller in the International Winter School on Gravity and Light 2015, and I finally understand thing due to the differential geometry chartless formalism. But I can't find a good textbook that goes in the same direction; there are always indexes and charts from start.
$\begingroup$This resource recommendation post asks for essentially the same thing. Also, since you're following (the excellent) Schuller lectures, I know of these very nice lectures notes based on them.$\endgroup$
$\begingroup$I don't see a whole lot of difference between Schuller's approach and Wald's "General relativity", but if you're looking for something that's more typical of pure differential geometry texts, I would say Frankel's "Geometry of Physics" is about as close as I've seen without losing the physics side of it.$\endgroup$
$\begingroup$I am not a theoretical physicist, but I doubt that you are doing yourself much of a favor if you are trying to formulate general relativity (or any of physics) without coordinates and indices. Why? Because defining inner and outer products is not enough to fix the dimensionality of the representation/manifold, but physics relies heavily on the fact that space is three dimensional. In other words... every time you need to calculate something that depends on dimension you have to bring a dimension fixing relation into play. I doubt that that is trivial in general. I might be wrong.$\endgroup$
