Euler463

I had this idea and I think it belongs to Algebraic Topology but here it is: Imagine some object C in some non-Euclidean space K such that object C is constantly being transformed in some important way every time this object moves in K but only because it moves through K. If it moved through some space M, then it would transform but differently or not at all. Is there a theory about objects like these and spaces that manipulate objects.