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Let $(X,d)$ be a metric space, and let $x \in X$. Consider the following definitions:

  • $Iso(x) = \{ f: X \to X \mid f(x)=x,\, f\text{ is an isometry} \}$.
  • $SurSim(x) = \{ f: X \to X \mid f(x)=x,\, f\text{ is a surjective similitude} \}$.
  • $Sim(x) = \{ f: X \to X \mid f(x)=x,\, f\text{ is a similitude} \}$.

Obviously, for every $x \in X$, $Iso(x) \subseteq SurSim(x) \subseteq Sim(x)$, but the inclusions may not necessarily be strict. Also note that while $Iso(x)$ and $SurSim(x)$ are necessarily groups (see here), $Sim(x) $ is not.

Consider the following statements (which in absolute geometry are related to the parallel postulate, see here for details):

$A$. For every ($\forall$) $x \in X$, one has that $Iso(x) \subsetneq Sim(x)$.
$B$.There exists ($\exists$) an $x \in X$ such that $Iso(x) \subsetneq Sim(x)$.
$\neg B.$ For every ($\forall$) $x\in X$, one has that $Iso(x) = Sim(x)$.

Define $\tilde{A}$ and $\tilde{B}$ to be the same as the above, but replacing $Sim(x)$ with $SurSim(x)$.

In the context of absolute geometry, $A \iff B \iff \tilde{A} \iff \tilde{B}$, and thus likewise $\neg B \iff \neg \tilde{B}$. However, absolute geometry is a highly restrictive assumption (e.g. see here), and there is no reason to assume that all of these equivalences hold in arbitrary metric spaces.

Question: Do any of the statements/properties $A, B, \tilde{A}, \tilde{B}, \neg B,$ or $\neg \tilde{B}$ have names within the context of general metric spaces, i.e. metric geometry?

They make sense for arbitrary metric spaces, since the concepts of isometry and similitude make sense for arbitrary metric spaces.

Also, if you know of any examples, besides Euclidean geometry and hyperbolic geometry, of metric spaces satisfying any of these statements, please feel free to mention them. (Assume, if necessary for the sake of simplicity, that $SurSim(x) = Sim(x)$ always, so that $A \iff \tilde{A}$, $B \iff \tilde{B}$.)

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