TITLE: Learning differential/Riemannian geometry for PDEs
QUESTION [11 upvotes]: 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

REPLY [3 votes]: 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.