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I am looking for a reference for this result.

Let $S$ be a metric space such that

  • it is homeomorphic to a two-dimensional manifold,
  • it is 2-homogeneous: given two pairs of points $(x,y)$ and $(x',y')$ such that $d(x,y) = d(x',y')$ there exists an isometry of $S$ that sends $x$ to $x'$ and $y$ to $y'$.
  • it is a geodesic space: given two points $x$ and $y$, for $l=d(x,y)$ there exists a curve $\gamma:[0,l]\rightarrow S$ such that $d(\gamma(s), \gamma(t)) =|s-t|$, $\gamma(0) = x$, $\gamma(l) = y$.

Then $S$ is isometric to either the Euclidean plane, some hyperbolic plane, a sphere or its quotient by $\{\mathrm{id}, -\mathrm{id}\}$.

The result appears in a survey in French of Étienne Ghys in a volume entitled "L'héritage scientifique de Poincaré". There is no attribution, only a vague sketch of proof. I browsed Ghys' papers in search of this result without success.

coudy
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    I think it is in Busemann's work, either Geometry of Geodesics, or Metric methods in Finsler spaces and in the foundations of geometry – Ben McKay Sep 11 '20 at 08:22
  • It's not true as stated: you should replace "isometric" with "homothetic". – YCor Sep 11 '20 at 09:02
  • @YCor Indeed there are several hyperbolic and spherical spaces, classified up to isometry by the area of their unit disc. And only one euclidean space. So isometric to one of these. – coudy Sep 11 '20 at 10:29
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    See theorem 7 in the survey https://arxiv.org/abs/1412.7893, and also https://arxiv.org/abs/2007.11917. – Igor Belegradek Sep 11 '20 at 12:10
  • @McKay thanks, indeed the two books of buseman contain results pretty close to Ghys'statement, e.g. 55.3 in "geometry of geodesics". I am not sure how Ghys' hypotheses imply axiom IV of Busemann's G-space but the reference to Berestovskii by Igor seems to sort it out. Busemann is citing Tits (1952) for the two-dimensional case but I can't find the article at the moment. – coudy Sep 11 '20 at 17:09
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    I mentioned this same theorem of Busemann’s at the end of an answer to another question, https://mathoverflow.net/a/346560. But the taxicab metric shows that Ghys’s first and third hypotheses on their own will not imply Busemann’s axiom for uniqueness of prolongation. –  Sep 12 '20 at 06:22

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