# Hopf-Rinow Theorem for Riemannian Manifolds with Boundary

I am a little rusty on my Riemannian geometry. In addressing a problem in PDE's I came across a situation that I cannot reconcile with the Hopf-Rinow Theorem. If $\Omega \subset \mathbb{R}^n$ is a bounded, open set with smooth boundary, then $\mathbb{R}^n - \Omega$ is a Riemannian manifold with smooth boundary. Since $\mathbb{R}^n - \Omega$ is closed in $\mathbb{R}^n$, it follows that $\mathbb{R}^n - \Omega$ is a complete metric space. However, the Hopf-Rinow Theorem seems to indicate that $\mathbb{R}^n - \Omega$ (endowed with the usual Euclidean metric) is not a complete metric space since not all geodesics $\gamma$ are defined for all time. Am I missing something here? Do the hypotheses of the Hopf-Rinow theorem have to be altered to accommodate manifolds with boundary?

• For manifolds with boundary, completeness as a metric space and completeness in the sense that all geodesic curves extend forever, are no longer equivalent. Clearly, the boundary can ensure that all Cauchy sequences converge (possibly to a boundary point), and just as clearly, there is going to be some geodesic curves that "run into" the boundary and cannot be continued beyond it. – Jeppe Stig Nielsen Jul 13 at 1:44

• @John I do not know precisely what you ask, but if I consider $M = \{ (x,y)\in\mathbb{R}^2 \mid y \le x^2 \}$ with the usual metric (inherited from $\mathbb{R}^2$), then starting from the point $p=(0,-1)$, or from any point of $M$, I guess, there is clearly going to be points in $M$ that I cannot hit. Each geodesic curve will reflect at most once on the boundary (parabola). But maybe you thought of compact manifolds with boundary? – Jeppe Stig Nielsen Jul 13 at 2:02