I want to describe motion along geodesics in a foam-like structure where the bubbles wouldn't be spherical as their locally discontinuous expansion—every part of the foam expands at different rates—follows a quadratic rather than linear function, and the bubbles' surfaces aren't planes but 3d space, so the entire thing requires 4 spatial dimensions.
If you dimension-reduce or flatlander-ize it then you might visualize it as a rubber surface that gets stretched continuously with local extrema that expand faster than the main surface so you end up with blisters, bubbles or protuberances everywhere.
Note that expansion as well as motion are physical concepts involving a temporal rather than spatial dimension. Not exactly sure what pure mathematicians do—from what I've seen you don't really talk about 'time'?
Anyway the question is what this might remind people of or which branch of geometry/topology would deal with such ideas?