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I recently read that (at least in $2+1$ dimensions but maybe it's true in general) the fact that all the component of Riemann tensor are identically 0: \begin{equation} R_{\alpha\beta\mu\nu} = 0, \end{equation} does not necessarily implies that spacetime is Minkowskian. It said the following:

If the topology is trivial, the local flatness ensured by $R_{\alpha\beta\mu\nu} = 0$ can be extended for the hole space, so the space is Minkowski's. But if the topology is not trivial, flatness is just local and interesting things can happen!

For me, the Riemann tensor was precisely the tool to asserts flatness or not, since Christoffel's symbols can be non-zero even for Minkowski's spacetime, they depend "too much" of the choice of coordinates.

  1. Could you give some examples of spacetimes that are locally flat (in the sense described above, not the ones concerned by the equivalence principle of course)?

  2. Given a metric, how do we know if the topology is trivial or not?

  3. This seems important, what else should I know about those spacetimes? What interesting things can happen?

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    $\begingroup$ Where did you read this? It's hard to comment without knowing the context, though I'd guess it's referring to the conical defects caused by point masses in 2+1D GR. $\endgroup$ Dec 1, 2020 at 8:47
  • $\begingroup$ I read that in a thesis that I don't feel confortable to share. After that, they talked about homotopy and conical defect, indeed. But this passage is very unclear for me. I know the basics of homotopy theory. $\endgroup$
    – xpsf
    Dec 1, 2020 at 8:52
  • $\begingroup$ Do you know about the Aharonov-Bohm effect? $\endgroup$ Dec 1, 2020 at 16:16
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    $\begingroup$ @xpsf Although not Riemannian geometry, you can view the vector potential in the AB affect as a $\mathrm{U}(1)$ connection, and the magnetic field as its curvature. In the setup of the AB effect the magnetic field (thus the curvature) vanishes, but still, parallel transport on a loop around the cylinder is not trivial. This is a case of null holonomy allowed by the fact that the 3-space with the cylinder cut out has a topological defect. Such a thing can happen in GR/Riemannian geometry too - you can have path-dependent parallel transport even with a flat connection... $\endgroup$ Dec 1, 2020 at 18:39
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    $\begingroup$ ... if the loop encircles some topological defect, but the holonomy of the loop in this case depends only on the homotopy class of the loop. $\endgroup$ Dec 1, 2020 at 18:39

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  1. Think of a cone without the tip, e.g. , $$ds^2~=~-dt^2+ dr^2 +r^2d\theta^2, $$ $$ t~\in~\mathbb{R}, \qquad r~\in~\mathbb{R}_+, \qquad \theta~\in~[0,\theta_0],$$ where the angle $\theta=\theta_0$ is identified with the angle $\theta=0$. The Riemann curvature tensor vanishes everywhere, and this spacetime is locally isometric to the Minkowski spacetime in 2+1D. However, a geodesic may intersect itself if $\theta_0\neq 2\pi$, unlike in Minkowski spacetime.

  2. Generalization of the cone example to other dimensions is straightforward.

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  • $\begingroup$ So the main point here is the fact that there's a missing point (the origin in this case). This makes the topology different from the trivial topology of Minkowski's spacetime (and in particular the fundamental homotopy group is different too). Is that right? But is it possible to have such a conical without default without the singularity mass at the top ? Does this still satisfies Einstein's equations ? I'm referring to inspirehep.net/literature/192056 . $\endgroup$
    – xpsf
    Dec 1, 2020 at 10:28
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The metric and the Riemann tensor are local constructs, which means that they completely specify the geometry of your manifold. These quantities don't say anything about the topology of the manifold, which is a global (in contrast to geometric/local) construct. Thus, you can in principle change the topology of a space while keeping $R_{\alpha\beta\mu\nu}=0$.

  1. Think for example of the flat Euclidean space with a toroidal topology. That is, imagine an infinite cubic lattice where each cell of the lattice has sides $\{L_x,L_y,L_z\}$, and where the opposing faces are identified (in physical terms, you have periodic boundary conditions on your space). This space is geometrically trivial, but topologically nontrivial.

  2. You don't in general, since the metric has information about the local geometry of the space, and not about its topological properties. A cone and a flat 2D surface are two manifolds with the same metric, but different topologies.

  3. This depends on what you are really trying to learn. As an example familiar to me, I can mention that the observational search for the topology of the universe was a hot topic of research in the beginning of this century. To this day, we still don't know that the topology of our universe is.

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