when a space (or spacetime) is said to be maximally symmetric, does this mean that it is homogeneous?
Does this mean a flat Euclidean space is more symmetric than spaces with closed and open curvature (hyperspheric and hyperbolic)?
VKint said:Thus, in addition to a full complement of ordinary Killing vectors, Euclidean space is also blessed with a maximal collection of conformal Killing vectors, making it "uber-maximally symmetric," unlike hyperspheres and hyperboloids. So, indeed, there is a "third kind of invariance," as you say, having to do with changes of scale, that Euclidean space possesses but other so-called maximally symmetric spaces do not.
Noether would say for every symmetry there is an inertia or conservation law, so what is law that results from scale symmetry?
A maximally symmetric space is a mathematical concept that describes a space where every point is equivalent in terms of its properties and the relationships between points. This means that the space has the same properties, such as curvature, at every point and that any transformation of the space, such as rotation or translation, does not change its overall structure.
Some examples of maximally symmetric spaces include spheres, Euclidean spaces, and hyperbolic spaces. These spaces have constant curvature, meaning that the curvature is the same at every point, and they exhibit symmetry under various transformations.
Maximally symmetric spaces are relevant to science because they provide a framework for understanding the properties of physical space and the relationships between objects in that space. They are also used in various branches of science, such as physics and cosmology, to model and study the behavior of the universe.
In general relativity, a maximally symmetric space is significant because it is a solution to Einstein's field equations, which describe the curvature of space-time in the presence of matter and energy. This means that a maximally symmetric space can be used to model the universe and make predictions about its behavior.
Maximally symmetric spaces are closely related to symmetry in physics because they exhibit symmetry under various transformations, such as rotations and translations. This symmetry is important in understanding the laws of nature and how they apply uniformly throughout space and time.