Writing a review for something that everybody knows its high quality would be a waste of time, but perhaps not anymore - younger people should know the 'standard candles'.

Unless you are in a place where all this material you can attend from lectures, this is the book that if you are (or want to be) a mathematical physicist must try to read 'a little every day', hoping that eventually things will start focusing and you will catch up.

It should be considered in a sense as THE modern analogue of Synge & Schild's Tensor Calculus - it has the same selection of topics but now all on manifolds: Analysis on Manifolds, Riemannian geometry, Integration, Connections, plus distributions and aplications to PDEs and selected topics of infinite-dim geometry.

So you have here a source-book that will not only allow you to formulate, in a modern way, physical laws (differential geometry) but also help you to study them (PDEs).

It is a profitable reading for someone who is somewhat versatile with elementary abstract mathematics, say at the level of Geroch's Mathematical Physics (algebra, topology, measure theory, functional analysis), but once you get going, you never stop!

Start off from chapter 3 and get back (or look up Geroch) if you need an explanation of a word you don't understand or have forgotten.

After you have a basic understanding a Riemannian geometry from this book you'll hopefully be able to reach Mme Choquet's new book on GR and the Einstein Equations, it is a continuation and uses the same notation, or volume 2 of the book under review on various topics in mathematical physics.

Most importantly, keep your cool and don't get intimidated!

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