I just finished watching Steve Brunton's video https://www.youtube.com/watch?v=Jt5R-Tm8cV8, and, when I see a new vector calculus topic, I immediately think, "What is the analogous concept on manifolds?" I did some Googling, and I was unable to get a straight answer to "What is a PDE on a manifold?" and "What is the solution to a PDE on a manifold?"
Of course, jet bundles came up, and it was said that "a PDE on a manifold is a function on a jet bundle $J^r(M)$", but what does this mean? A real-valued function on $J^r(M)$? Is the solution then a real-valued function on $M$ (a 0-form) which satisfies the "jet bundle function"? If so, what is a solution to the Navier-Stokes equation on an open, contractible subset of $\mathbb{R}^3$? Is Navier-Stokes a PDE?
Suppose we had some kind of "analog" to Navier-Stokes on $S^3$ instead of $\mathbb{R}^3$ (I'm not even sure what those words might mean). How would it be formulated? Would it be some kind of smooth map $\zeta: J^2(S^3) \to T(S^3)$? Would a solution be a smooth section of the tangent bundle $\xi: S^3 \to T(S^3)$? For a Riemannian manifold (which $S^3$ naturally is as a Lie group), could we reformulate the problem as a smooth map to the cotangent bundle $\zeta^{\flat}: J^2(S^3) \to T^*(S^3)$ and a solution as a 1-form $\omega = \xi^{\flat}: S^3 \to T^*(S^3)$ using the musical isomorphism? (I find it more natural to work with 1-forms as ODEs on manifolds that tangent vector fields, currently, sorry if the last couple of questions seem stilted.)
Sorry for all the super-basic questions, and thanks in advance.