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](https://terrytao.wordpress.com/2011/06/17/hilberts-fifth-problem-and-gleason-metrics/):



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> Any two-point homogeneous locally compact length-metric space is a Riemannian manifold.
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Is there a reference for this statement?



Locally compact + two-point homogeneous => Riemannian
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.


