> We will take as a model of the domain of the fluid flow a compact Riemannian manifold ...
This might come as a bit of a shock to some, since the simplest 3d example is a 3-torus, and normal folks don't think about water flowing around in a 3-torus! We are saying goodbye to ordinary reality and embracing the joys of math here. Can you add a sentence or 2 to calm the reader?
It's technically a bit more tricky, but perfectly fine, to work with a compact Riemannian manifold _with boundary_. Then we can study water in a bottle.