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Let $(M, g)$ be a simply-connected $n$-dimensional Riemannian manifold with constant sectional curvature equal to $k$ (note that $M$ need not be complete). I would like to show that there is an isometric immersion from $M$ into the $n$-dimensional model space of constant sectional curvature.

I do not how to start proving this. I know that if $M$ is complete, then $M$ is actually isometric to the corresponding model space, and I also know that manifolds with equal sectional curvature are locally isometric, but I do not how to use these to construct an immersion of $M$ into the corresponding model space.

S.T.
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  • Yes, such an immersion is called a developing map; see references given here – Moishe Kohan Dec 14 '21 at 06:35
  • This is quite a late comment, but you say you're already aware that if $M$ is also complete then it is in fact globally isometric to a model space. Review the proof of this fact (e.g. the proof of theorem 12.4 in Lee's Intro to Riemannian Manifolds) and consider how completeness is used. What happens if the condition is removed? – Itserpol Jun 21 '25 at 15:56

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