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I am trying to understand whether the Einstein field equation uniquely determines the topology of spacetime. As far as I know, given a metric, we can always find the induced topology. However, I was wondering if this topology is unique. To put it another way, metric completely encodes the local data about the manifold but does it also completely encode the global data about the manifold?

For example, $\mathbb{R}^2$ and $T^2$ both admit a flat metric. Is it possible to distinguish these two topologies using the field equation?


marked as duplicate by Ben Crowell general-relativity May 13 at 2:12

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    $\begingroup$ There might be some clever way of doing this, but surely one needs to specify the manifold before you can go about specifying the metric on it? $\endgroup$ – jacob1729 May 12 at 20:11
  • $\begingroup$ The field equations all reduce to $0=0$ for your two empty spacetimes, so the answer is No. The field equations do not determine the topology. However, I think some features of the topology are encoded in the spectrum of differential operators on the spacetimes. $\endgroup$ – G. Smith May 12 at 21:05
  • $\begingroup$ possible duplicate: How does GR determine the topology of spacetime?. $\endgroup$ – AccidentalFourierTransform May 12 at 21:19
  • $\begingroup$ Field equations are not the only thing you need to specify a manifold, you also need some sort of initial data, so the answer to the title question is trivially no. $\endgroup$ – A.V.S. May 12 at 21:19