Equivalence and Determinism in Light of Topologically-Induced Structure

Abstract Contemporary philosophy of spacetime has thus far focused predominantly on the metrical properties of spacetime, with substantially less attention paid to its topological aspects. In this article, we go some way to rectifying this situation, by introducing philosophers to a class of ‘(almost) asymmetric’ manifolds which possess flat geodesically complete Riemannian metrics yet admit only finite (or trivial) isometry groups. The resulting spacetimes exhibit a unique dual nature: they are locally homogeneous (indistinguishable from standard flat space) but globally inhomogeneous, such that every spatial point is individuated from its neighbours. We argue that this topology-induced structure has at least two significant philosophical upshots. First, we show that there are surprising upshots for existing popular notions of theoretical equivalence when one considers theories set on these manifolds: many theories which might appear prima facie inequivalent in fact become categorically equivalent on such topological settings. Second, we demonstrate that theories set on these manifolds satisfy what Manchak et al. (2026) dub “de re* determinism” and “rigidity”, thereby avoiding (e.g.) the indeterminism which some associate with Leibnizian spacetimes.