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 P: 11 Thank you, your post really helped a lot. It seems to me that Wald isn't completely consistent with his own definitions of a metric and a metric tensor since he use the 2 words as synonyms. I am not quite sure what you mean when you say that the tensor field F takes a point to the pair $$(p,g_p)$$ because we want the tensor to be a section of a fiber bundle. I am not familiar with the term fiber bundle.