In order for any system to evolve there has to be a reason. Objects do not "intrinsically expand" in the sense that I think you are trying to communicate. There could be thermal expansion/contraction, there could be an applied field acting on the object, etc. There has to be some cause for it to expand.Sort of. In physics we have to come up with a good explanation as why this sphere is expanding or why the space is expanding; I want to know if there's certain mathematical properties that can explain expansion as a geometrical phenomenon, not as a consequence of a force or energy. kind of like what Einstein did to gravity. with the difference that in relativity Einstein considers gravity as a result of presence of matter. I'm thinking expansion to be intrinsic. without any relation to matter. so there has to be some kind of intrinsic properties of geometry or topology for the space so it expands over time. That's why I want to know if there's any geometrical objects that are expanding solely because of intrinsic geometrical properties.

If you want an example for this it would be: curvature in space-time results in acceleration. that acceleration is a geometrical phenomenon. curvature is the intrinsic property of space-time that is has the key role for gravity here.

-Dan