Suppose $(M_1,g_1)$ and $(M_2,g_2)$ are (semi)Riemannian manifolds with boundaries (resp.) $\Sigma_1$ and $\Sigma_2$. The metrics $g_{1,2}$ yield induced metrics $\gamma_{1,2}$ on those boundaries $\Sigma_{1,2}$ so that the pairs $(\Sigma_{1,2},\gamma_{1,2})$ are also (semi)Riemannian manifolds. Suppose these manifolds are isometric, the isometry being $i:(\Sigma_{1},\gamma_{1}) \to (\Sigma_{2},\gamma_{2})$, then it is natural to sew together $M_1$ and $M_2$ by identifying every point $x \in \Sigma_1$ with $i(x)\in \Sigma_2$. The sewn-together manifold is a manifold without boundary but the metrics $g_1$ and $g_2$ do not necessarily match nicely at the junction in the directions normal to the junction surface, so the sewn manifold $\overline{M}$ fails to inherit a full (semi)Riemannian inner product on its tangent space, although that failure is confined to a co-dimension-1-submanifold. In particular the notions of (Lorentzian) distance along a curve and distance-extremizing curves survive. All this remains the same if we connect more than two manifolds with boundary along boundary segments.
Is there a name (or perhaps an alternative characterisation) for these geometric objects?