How much Differential Geometry do you need to do PDE? Ever since I took functional analysis and I read in Brezis's famous book how to apply this to PDE I have been extremely interested in PDE and I have kept reading more and more of the standard material on PDE. This is all, of course, in $\mathbb{R}^n$, so you don't really need to know any differential geometry, but I saw that there are people doing PDE on Riemannian manifolds for instance. I for one am not a big differential geometry fan, I do know the basics, but I don't enjoy it much and I am not really eager to study it further. However, I can't help but wonder if I am going to enocunter differential geometry if I am going to further study PDE and whether it would be helpful to improve my background in differential geometry if PDE is what I am interested in.
Disclaimer: I am well aware of the fact that PDE has many subfields and one of them is even called "Geometric PDE" (or Geometric analysis if you prefer) and is basically a field that deals precisely with what I am trying to "run away" from, but my question is more of the sort "do all PDE involve differential geometry if you immerse yourself deep enough into their study?".
 A: Well, as you are correct to point out, the answer depends a lot of which subfield of PDE you study.  That being said, there is a huge portion of PDE for which differential geometry plays no essential role.  In fact, even in the areas where one could argue that it does appear, PDE practitioners often reformulate the problems in such a way that the geometric aspects don't resemble the corresponding problems in differential geometry.
Here's a concrete example.  In the study of elliptic boundary value problems we spend a lot of time studying PDEs in open, bounded subsets $\Omega \subseteq \mathbb{R}^n$.  In this context we specify that some elliptic PDE is satisfied in $\Omega$ and that some boundary conditions are satisfied on $\partial \Omega$.  In order to develop the theory we often need to know that the boundary $\partial \Omega$ is "nice" enough.  In differential geometric terms, this boils down to having $\partial \Omega$ be a sufficiently regular submanifold, which in principle seems to suggest we should think of $\bar{\Omega}$ as a manifold with boundary, which has all sorts of technical overhead about local charts, transition maps, induced orientations, etc.  However, in practice, we can usually get away with a very simplistic description of the boundary as "locally the graph of a sufficiently regular function," and this is good enough for most PDE purposes such as a priori estimates.  Of course, this formulation is one of the equivalent ways of describing a manifold, but the point is that it is arguably the least sophisticated.  Moreover, the more interesting geometric quantities (curvature, etc) don't play any / much of a role in most of this theory and so can be effectively ignored.
