Learning differential/Riemannian geometry for PDEs

Question

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

Answer 1

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.

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At a more advanced level there is also:

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