Manchak, JB
(2023)
Does the Curvature Structure of Spacetime Determine Its Topology?
[Preprint]
Abstract
We explore the title question. After some topological preliminaries, we define a "curvature isomorphism" between spacetimes. We introduce a hierarchy of curvature conditions and show that at a certain level, a curvature isomorphism must be a homeomorphism. The highest level of the hierarchy is satisfied by a spacetime if every smooth scalar function on its manifold is an invariant scalar curvature function. We show that such "maximally structured" spacetimes exist and that a curvature isomorphism between them must be a diffeomorphism. We highlight a number of connections between our project and the one in which the topology of spacetime is determined from a causal relation between spacetime points (Malament 1977). We emphasize that analogous results are obtained here by considering only invariant properties of spacetime points -- no relations needed.
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