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Are there examples of complete Riemannian manifolds which are not connected ?

This question follows my previous question. The more I think about it and the less I'm convinced it exists.

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    $\begingroup$ $[0,1]\cup[2,3]\subset\mathbb R$ will work. More generally, the disjoint union of two copies of any compact manifold will work. $\endgroup$ – Jonas Dahlbæk Aug 1 '14 at 10:55
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    $\begingroup$ @user161825 The closed interval $[0,1]$ is a manifold with boundary. Sometimes it is implicitly understood that a manifold has no boundary unless stated otherwise. Two copies of $\mathbb R$ is a good example in the same spirit if boundaries are to be avoided. $\endgroup$ – Joonas Ilmavirta Aug 1 '14 at 12:40
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The disjoint union of two copies of $S^1$, for example, is a Riemannian manifold which is complete but not connected.

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