Learning differential/Riemannian geometry for PDEs

Solution 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.

At a more advanced level there is also:

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

Solution 2:

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.