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If the metric space is locally compact and intrinsic, then you get only spheres, Euclidean spaces and hyperbolic spaces. [See Metric methods in Finsler... by Busemann and Sur certaines classes d'espaces... by Tits (1955); thanks to Linus for the reference]

Without assuming local compactness, the same conclusion holds assuming local uniqueness of geodesics [See Metric foundations of geometry. I by Birkhoff]. Without this extra assumption you also get the so-called universal $\mathbb{R}$-trees of finite valence; they are complete, but not separable.

If the metric is not intrinsic you get discrete spaces and yet Cantor-like spaces build on them (who knows what else).

Comments

• Urysohn universal space the property holds for compact subsets.

• The real projective space is not three-point-homogeneous --- a closed geodesic contains three points on equal distance from each other, and there is an isometric three-point set that does not lie on a closed geodesic.

• See also a related question.

If the metric space is locally compact and intrinsic, then you get only spheres, Euclidean spaces and hyperbolic spaces. [See Metric methods in Finsler... by Busemann and Sur certaines classes d'espaces... by Tits (1955); thanks to Linus for the reference]

Without assuming local compactness, the same conclusion holds assuming local uniqueness of geodesics [See Metric foundations of geometry. I by Birkhoff]. Without this extra assumption you also get the so-called universal $\mathbb{R}$-trees of finite valence; they are complete, but not separable.

If the metric is not intrinsic you get discrete spaces and yet Cantor-like spaces build on them (who knows what else).

Comments

• In the Urysohn universal space $\mathbb{U}$ the property holds for compact subsets; that is, any distance-preserving map $K\to\mathbb{U}$ defined on a compact subset $K\subset \mathbb{U}$ can be extended to an isometry $\mathbb{U}\leftrightarrow\mathbb{U}$.

• The real projective space is not three-point-homogeneous --- a closed geodesic contains three points on equal distance from each other, and there is an isometric three-point set that does not lie on a closed geodesic.

• See also a related question.

In other words, you are interested in the spaces where isometric subsets are congruent.

If the metric space is locally compact and intrinsic and simply-connected then you get only spheres, Euclidean spaces and hyperbolic spaces. If not simply-conected, then in addition you get real projective spaces. [See Sur certaines classes d'espaces homogènes de groupes de Lie by Tits (1955); thanks to Linus for the reference]

Without assuming local compactness, the same conclusion holds assuming local uniqueness of geodesics [See Metric foundations of geometry. I by Birkhoff]. Without this extra assumption you also get the so-called universal $\mathbb{R}$-trees of finite valence. Here is a related question.

If the metric is not intrinsic you get discrete spaces and yet Cantor-like spaces build on them (who knows what else).

Yet in Urysohn universal space the property holds for compact subsets.

If the metric space is locally compact and intrinsic, then you get only spheres, Euclidean spaces and hyperbolic spaces. [See Metric methods in Finsler... by Busemann and Sur certaines classes d'espaces... by Tits (1955); thanks to Linus for the reference]

Without assuming local compactness, the same conclusion holds assuming local uniqueness of geodesics [See Metric foundations of geometry. I by Birkhoff]. Without this extra assumption you also get the so-called universal $\mathbb{R}$-trees of finite valence; they are complete, but not separable.

If the metric is not intrinsic you get discrete spaces and yet Cantor-like spaces build on them (who knows what else).

Comments

• Urysohn universal space the property holds for compact subsets.

• The real projective space is not three-point-homogeneous --- a closed geodesic contains three points on equal distance from each other, and there is an isometric three-point set that does not lie on a closed geodesic.

• See also a related question.

In other words, you are interested in the spaces where isometric subsets are congruent.

If the metric space is locally compact and intrinsic and simply-connected then you get only spheres, Euclidean spaces and hyperbolic spaces. If not simply-conected, then in addition you get real projective spaces. [See Sur certaines classes d'espaces homogènes de groupes de Lie by Tits (1955); thanks to Linus for the reference]

Without assuming local compactness, you also get the so-called universal $\mathbb{R}$-trees of finite valence. Here is a related question.

If the metric is not intrinsic you get discrete spaces and yet Cantor-like spaces build on them (who knows what else).

Yet in Urysohn universal space the property holds for compact subsets.

In other words, you are interested in the spaces where isometric subsets are congruent.

If the metric space is locally compact and intrinsic and simply-connected then you get only spheres, Euclidean spaces and hyperbolic spaces. If not simply-conected, then in addition you get real projective spaces. [See Sur certaines classes d'espaces homogènes de groupes de Lie by Tits (1955); thanks to Linus for the reference]

Without assuming local compactness, the same conclusion holds assuming local uniqueness of geodesics [See Metric foundations of geometry. I by Birkhoff]. Without this extra assumption you also get the so-called universal $\mathbb{R}$-trees of finite valence. Here is a related question.

If the metric is not intrinsic you get discrete spaces and yet Cantor-like spaces build on them (who knows what else).

Yet in Urysohn universal space the property holds for compact subsets.