Your question seems related to the “problem of space”, also known as the Riemann-Helmholtz (or Helmholtz-Lie) problem which approximately speaking consists of identifying (hopefully “visually obvious”) properties of the group of motions of space that characterize Euclidean, spherical and/or hyperbolic geometry. Various solutions have been proposed, notably by Lie and Engel, by Kolmogorov and by Tits (for the latter: see “Sur certaines classes d'espaces homogènes de groupes de Lie” (1955; =item [27] in vol. 1 of his collected works), section IV.E).
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