Let $(M^2,g)$ be a Riemannian manifold, with manifold boundary $\partial M$. We assume that the metric degenerates at the boundary, in the sense that the (Gauss) curvature diverges like $K \to +\infty$ as one approaches it.

Question. Is it true that 'nearly all' geodesics avoid the boundary? Perhaps those going to the boundary are meagre in some sense?

• In general, it seems intuitively reasonable that 'most' geodesics would avoid regions where the curvature is 'large and positive'. (This is of course a bit vague—I was unable to come up with a precise formulation.)
• The question is motivated by a comment made in passing by Robert Bryant in his answer to this question, where he discussed the geodesics of a very specific metric.