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.
