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A metric space $M$ is called two-point homogeneous if for any pair of points $(p,q)$ in $M$ any distance preserving map $f\colon\{p,q\}\to M$ can be extended to an isometry $\bar f\colon M\to M$.

The following statement seems to be an easy corollary of Gleason--Yamabe theorem:

Any two-point homogeneous locally compact length-metric space is a Riemannian manifold.

Is there a reference for this statement?

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Tits proves in [Tits, J. Sur certaines classes d'espaces homogènes de groupes de Lie, Acad. Roy. Belg. Cl. Sci. Mém. Coll. in 8$^\circ$ 29 (1955), no. 3, 268 pp. MR0076286], page 220, the following. If $M$ is a locally compact and connected metric space which is 2-point homogeneous (in your sense) then $M$ is isometric to euclidean space or to a rank 1 Riemannian symmetric space. I don't know any newer references. Helgason mentions in his book some related results.

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  • $\begingroup$ (Thank you, I did not think that two-point homogeneous is a standard term --- it was stupid of me) + I just found slightly older paper which does almost the same thing Wang, Hsien-Chung Two-point homogeneous spaces. Ann. of Math. (2) 55 (1952), 177–191. $\endgroup$
    – Anton Petrunin
    Commented Aug 9, 2022 at 13:22
  • $\begingroup$ Did you mean Helgason's "Differential geometry, Lie groups, and symmetric spaces"? $\endgroup$
    – Anton Petrunin
    Commented Aug 9, 2022 at 13:32
  • $\begingroup$ Yes, that's the book I meant. $\endgroup$
    – Linus
    Commented Aug 24, 2022 at 7:26

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