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.
A closed ball in Riemannian geometry is a set of points that are all within a certain distance (called the radius) from a given center point. In this context, the distance is defined using the Riemannian metric, which is a way of measuring distances on curved surfaces.
Dilating or expanding a closed ball allows us to study the behavior of geometric objects on a larger or smaller scale. It can also help us understand how the geometry of a space changes when we change the scale of measurement.
The dilation or expansion of a closed ball is calculated using the Riemannian metric, which assigns a distance to every pair of points in a space. To dilate a closed ball, we multiply the radius of the ball by a constant factor, while to expand a closed ball, we divide the radius by a constant factor.
The dilation or expansion of a closed ball is directly related to the curvature of the space it is embedded in. In general, the larger the curvature of a space, the more the closed ball will expand when dilated and the more it will contract when expanded.
The volume and surface area of a closed ball will change when it is dilated or expanded. In general, the volume of the ball will increase when dilated and decrease when expanded, while the surface area will decrease when dilated and increase when expanded.