I am looking for an example of a non-complete isotropic manifold. This is because I believe isotropic $\Rightarrow$ homogeneous $\Rightarrow$ complete (with a proof similar to the one in the answer of this question). However, as both in Wikipedia (https://en.wikipedia.org/wiki/Isotropic_manifold) and on some online notes, I find the statement "isotropic and complete $\Rightarrow$ homogeneous" I deduce that this is false and that therefore there exists and example as the one of the title.
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All isotropic manifolds are complete (and homogeneous, if connected). See
Lemma 8.12.1 in
Wolf, Joseph A., Spaces of constant curvature. 3rd ed, Boston, Mass.: Publish or Perish, Inc. XV, 408 p. $ 10.75 (1974). ZBL0281.53034.
As Wolf observes, the proof of completeness is the same as for symmetric spaces. It is the same proof as the one sketched by Jason DeVito in a comment to his answer here.
Moishe Kohan
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@C.F.G: I meant the comment starting with "I guess." – Moishe Kohan May 27 '20 at 16:51
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This one? – C.F.G May 27 '20 at 17:35
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Ok thanks! I was pretty sure this was the case, but as I found this written both on my notes and on Wikipedia I had the doubt I was missing something! Clearly I suppose the manifold has to be connected though – Nicolò Cavalleri May 27 '20 at 20:04
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@NicolòCavalleri: Actually, connected is not needed for completeness (one needs this for homogeneity, of course). – Moishe Kohan May 27 '20 at 20:13
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Ok great I am gonna edit the Wikipedia article then – Nicolò Cavalleri May 27 '20 at 22:07