I am new here but tried to go through some of the posts on subject matter: I apologize if I am overlooking your input as I am sure you must have clarified already my naive doubts ! I just completed a first reading of Carlo Rovelli's Quantum Gravity book (hardcover edition, 2004). I find the part starting at p. 63 with “Given a manifold M, an active diffeomorphism…” and ending at p. 64 with “Beware the formal similarity between (2.135) and (2.136)” a bit hard to follow and the formalism not clear. E.g. why the same symbol for the active and passive maps while domains are different? Why P(x) in the last line of p. 63 seems to me not defined? (too obvious? Is it the composition of T with the inverse of the coordinate map x?) I found in the attachment an explication (from A. Zee, Einstein Gravity in a Nutshell) which, while very similar to Rovelli’s, looks to me much more clear (beside the statement “Suppose also that the diffeomorphism moves the number T(P) to the point Q” (what does that mean?)). I would like your comment if you also find that part of Rovelli’s book confusing (or it is just me!) and if you can help me mapping the symbols between Rovelli's and Zen's texts. I am also checking what Thomas Thiemann writes on the subject in his truly magnificent Modern Canonical Quantum General Relativity (can post that later) but I would like first to get Rovelli’s introduction. Thank you in advance. Any help is appreciated!