Geodesics on manifold with corners

Question

Let $M$ be a smooth manifold, and $\Omega$ a bounded region of $M$ with smooth boundary. Suppose that $g_+$ and $g_-$ are smooth metrics on $M\setminus\Omega$ and $\overline\Omega$ respectively. And assume $g_-=g_+$ on $\partial\Omega$. Is there uniqueness for geodesics on $M$?

Answer 1

There is no unique continuation of geodesics in this setting (I assume this is what you mean in your question) unless you have more assumption about the boundary. Consider for instance the manifold $M$ obtained by doubling $\bar\Omega$ which is the Euclidean plane minus an open round disk. Doubling means: glue two copies of this domain along the common boundary circle.