In physics, homogenous is used to mean that some quantity does not change with position.
This is less general than the use of homogeneous in mathematics, I think.
Question: So how does one define a metric space which is "homogeneous" in the sense of physics, i.e. its (metric) geometry is the same at every point?
(E.g. hyperbolic, Euclidean, and spherical geometries.)
There are definitions of a homogeneous metric space (see here or here), but both seem to be more general than what I am trying to describe above.
Attempt: Given a metric space $(X,d)$, denote the isometry group by $Iso(X)$, and for each $x \in X$, denote the subgroup of $Iso(X)$ consisting of elements fixing $x$ ($f(x) = x, f \in Iso(X)$), by $Iso(X,x)$.
Then a metric space "has the same geometry at each point" if and only if, for any two points $x_1, x_2 \in X$, one has that $Iso(X,x_1) \cong Iso(X, x_2)$.
Euclidean space satisfies this condition, since $Iso(X,x) \cong O(n)$ for any point $x$ in $n$-dimensional Euclidean space (I think). I don't know if hyperbolic and spherical geometries satisfy this too.