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Manifold structure of Busemann G-spaces

Determine whether every Busemann G-space—i.e., a metric space satisfying Busemann’s axioms (finite compactness, existence of geodesic segments with betweenness, local extendability of geodesics, and uniqueness of geodesic extension)—is a topological manifold (locally homeomorphic to Euclidean space).

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Background

Busemann introduced G-spaces as metric spaces equipped with geodesic and extendability properties, without assuming differentiability or manifold structure. The axioms formalize geodesic behavior in a synthetic setting and underpin many of Busemann’s characterizations of classical geometries.

The article emphasizes that it is unknown in general whether such spaces possess the local Euclidean structure required of manifolds, noting that the question has been resolved only under special hypotheses, with significant contributions surveyed by Andreev and Berestovskii.

References

We note that it is not assumed here that the metric space is a manifold. In fact, it is an important open problem, which is presently solved only under some special extra hypotheses, to know whether a G-space is a manifold; see the reviews by Andreev [8] and Berestovskii [11] who made important contributions to this problem.

— Geometry in the twentieth century: A return to Euclid -- The work of Herbert Busemann (2406.01109 - Papadopoulos, 3 Jun 2024) in Section 2 (Leibniz, Busemann and a question of Euclidean geometry)