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I know there have been threads on which books to learn DG/RG from but hopefully this is sufficiently different to avoid closure.

Can anyone recommend a book to learn DG/RG (whichever is appropriate) so that I can do PDEs on manifolds? At the moment I am reading through John M Lee's Introduction to Smooth Manifolds and I am wondering whether I really need to learn all the topics in it since it doesn't touch RG which I believe is more used in the theory of PDEs. Maybe there is a better text.

Also, any topics to particularly study or avoid would be useful.

Thanks

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Michael Taylor's three volumes of Partial Differential Equations does some PDEs on manifolds, and Jürgen Jost's Riemannian Geometry and Geometric Analysis may also be of interest to you. –  Henry T. Horton Jul 15 '12 at 21:58
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2 Answers

up vote 5 down vote accepted

Thierry Aubin, A Course in Differential Geometry.
An excerpt from its preface:

The aim of this book is to facilitate the teaching of differential geometry. This material is useful in other fields of mathematics, such as partial differential equations, to name one. We feel that workers in PDE would be more comfortable with the covariant derivative if they had studied it in a course such as the present one.


At a more advanced level there is also:

Thierry Aubin, Nonlinear Analysis on Manifolds: Monge-Ampère Equations

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@t.b. Thanks Theo –  Giuseppe Tortorella Jul 15 '12 at 22:24
    
+1. A good book by a late master,although terse and challenging. If you like that kind of presentation a la Warner or Walshap, then this is one of the best. –  Mathemagician1234 Jan 8 '13 at 18:53
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If you're assuming the Riemannian manifold has a fixed metric, then the most introductory source I've found is Folland, Introduction to Partial Differential Equations, which discusses aspects of PDEs on hypersurfaces and the Laplace-Beltrami operator, for example.

If you're looking for something more advanced, but which avoids getting into curvatures, then Grigor'yan's Heat kernel and analysis on Manifolds (AMS,2009) is excellent IMHO.

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