$(M,g)$ is a Riemannian manifold with non-empty boundary and $DM$ is the double of $M$, is there a Riemannian metric $G$ on $DM$ such that $g=i^*G$? ($i$ is the inclusion from $M$ to $DM$)
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Generally there isn't a natural one, although metrics do exist. There is no such Riemann metrics if you demand $i^* G = g$ for both inclusions of $M$ in its double. For example, consider the flat disc $D^2$. Having a metric on the double $S^2$ that restricts to two flat discs would violate the Gauss-Bonnet theorem. But in the world of metric spaces, there are metrics on the double, they're just not Riemann metrics.