The discussion centers on the concepts of expansion and dilation in Riemannian geometry, particularly in relation to closed balls. Expansion is defined as a set of points at time t2 containing those at time t1, independent of geometry. Dilation, often associated with metric expansion in cosmology, requires a clearer context to be properly understood. The ambiguity in the original question highlights the necessity for precise terminology in mathematical discussions.
PREREQUISITESMathematicians, physicists, and students of geometry seeking clarity on the distinctions between expansion and dilation in Riemannian contexts.
andrewkirk said:Expansion is a concept that does not even require topology, let alone geometry, to define it. All we need is set theory. An object is expanding if and only if for any two times t1 and t2 such that t1 < t2, the set of points covered by the object at time t2 properly contains the set of points covered by the object at time t1. So the geometry does not affect whether the object is judged to be expanding.
andrewkirk said:Expansion is a concept that does not even require topology, let alone geometry, to define it. All we need is set theory. An object is expanding if and only if for any two times t1 and t2 such that t1 < t2, the set of points covered by the object at time t2 properly contains the set of points covered by the object at time t1. So the geometry does not affect whether the object is judged to be expanding.
Does it? Metric expansion is typically used in cosmology, but it then refers to a particular family of spatial submanifolds of a Lorentzian manifold. The OP seems to imply talking about Riemannian geometry. To me it just seems as if the OP has not written a well defined question and may think he is asking something but in reality that something is going to be interpreted differently by different people. There is a need of context from OP (ie, I agree with #4).PeroK said:Okay, but this was posted under "Differential Geometry", which assumes we are talking about some sort of metric expansion.
Cosmology doesn't have a monopoly on the term "metric"! In any case, "ball" or "sphere" are only defined in sets with a metric.Orodruin said:Does it? Metric expansion is typically used in cosmology, but it then refers to a particular family of spatial submanifolds of a Lorentzian manifold. The OP seems to imply talking about Riemannian geometry. To me it just seems as if the OP has not written a well defined question and may think he is asking something but in reality that something is going to be interpreted differently by different people. There is a need of context from OP (ie, I agree with #4).
This is not the point. The point is that the question is ambiguous.PeroK said:Cosmology doesn't have a monopoly on the term "metric"! In any case, "ball" or "sphere" are only defined in sets with a metric.