Ellis Kolchin developed differential Galois theory in the 1950s. It seems to be a powerful tool that can decide the solvability and the form of the solutions to a given differential equation.
Why isn't differential Galois theory widely used in differential geometry? It is plausible that we can solve some problems of differential/integral geometry using this theory.
So, what is the major pullback in this theory that prevents its wide application to other fields rather than discrete geometry (e.g., Diophantine geometry)?