Timeline for When can a connection Induce a Riemannian metric for which it is the Levi-Civita connection?
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| Oct 18, 2021 at 18:08 | comment | added | Mike Cocos | Hi Ben. There are a multitude of examples of connections which are locally metric but not globally metric. Just look at the affinely flat manifolds. The Euler class of a locally metric connection DOES NOT depend on the choice of the local metric: researchgate.net/publication/… | |
| Oct 17, 2021 at 15:53 | comment | added | Ben McKay | Bill Thurston's example of a connection on the 2-sphere which is locally but not globally Levi-Civita shows that there is no global topological invariant which ensures that a locally Levi-Civita connection is globally Levi-Civita. Your Euler form is only defined for each local choice of metric compatible with the connection, so you get an Euler form on each open set where you have your local metric, but how do they patch to a global Euler form if you have to change local metric in different open sets? | |
| Dec 2, 2019 at 18:04 | history | answered | Mike Cocos | CC BY-SA 4.0 |