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The original post is in StackExchange but no one has answered it yet. I personally think it is more related to the research area so I put it in MathOverflow. Below is the question in the original post:...

Let $(M,g)$ a smooth closed Riemannian manifold with non trivial first homology group $H^1(M,\mathbb{R})$. Any element of $H^1(M,\mathbb{Z})$ will define a riemannian $\mathbb{Z}$-cover of $M$ ...