We know that any $2$-dimensional manifold $M$ with constant negative curvature is locally isometric to hyperbolic plane $\mathbb{H}_2$. Can this be extended to a local isometry to its closure as well, i.e $\overline{\mathbb{H}_2} = \mathbb{H}_2\bigcup \partial\mathbb{H}_2$? If not, when is this possible?
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Extension of isometry to the closure
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