My understanding of GR is very coordinate oriented which kind of drags me down when I try to answer more general questions. Can somebody recommend a book in Differential Geometry with application to GR? Here are my preferences. I don't like hand waving typical for some books 'written for physicists' - I need to see clear logical connection between the concepts defined not 'plausability arguments'. On the other hand, I don't like the hidden logic (the DaVinchy code lol) in many math books that just give you definitions and theorems without explaining the intuitive logic behind the scene. Such books, frankly speaking, do not anticipate the logical process of the reader and apparently do not care. I don't need monographies, 'phone books', 'bibles' or summaries of current research that leave me 'very informed' about stuff I don't actually understand. The book I am looking for must start with the most basic conceptual layer of Diff Geometry. It should anticipate that I am a beginner and not assume that I already know what it is supposed to teach me. It should contain lots of diagrams (cause I tend to think in terms of pictures) and lots of examples of application of Diff Geom to real life GR problems. I tend to learn most from examples of actual calculations. Also, the book needs to have exercises with answers or at least hints. The book has to be explicitly oriented towards applications to GR not a general monography in Diff Geom. The final goal is to intuitively and rigorously understand and work in practise with stuff like Lie derivatives, one forms, Killing vectors, foliation of spacetime into space and time etc. Does that perfect book exist?