# Every connected complete (?) manifold isotropic at every point is homogeneous

I am trying to prove the statement in the title, which was proved already in this question: Isotropic Manifolds, but hypothesis are not clearly stated (the manifold is not assumed to be either connected nor complete and the statement is certainly not true for manifolds with multiple connected components). In my proof, I basically consider the equivalence relation defined by "$$p \sim q$$ iif there is an isometry which brings $$p$$ to $$q$$". Then I prove equivalence classes to be open by considering a totally normal neighborhood for a point and using the argument in the linked question (which to me only works in a totally normal neighborhood), and then I conclude using connectedness.

However, I never used the completeness hypothesis in this proof (which is suggested by Wikipedia: https://en.wikipedia.org/wiki/Isotropic_manifold) and I wonder whether it is necessary. Is there a counterexample of a connected isotropic manifold which is not complete and not homogeneous? If not, is it all the same necessary to include completeness in the hypothesis? To me, it should at most be part of the thesis, as every homogeneous manifold is complete. In case it is necessary, where is the flaw in my proof?