Posted: Tue Sep 29, 2015 8:08 pm UTC
I'm working with an infinite-dimensional affine space, and I want to do some geometry with it. It seems like the standard approach is to choose a finite-dimensional "submanifold" where everything works nicely, but I can't find any ways of choosing these submanifolds that don't feel extremely ad hoc. What I'd like to do is define a vector field on the entire affine space, then integrate these vector fields and work with the resulting integral curves. Is this possible? I know how to do this in finite-dimensional spaces, but I have no idea which theorems still apply once I switch to the infinite case.