# Regularity of metric of the double of a Riemannian manifold

Let $M$ be a Riemannian manifold with totally geodesic boundary $\partial M$. We let $\check{M}$ be its double, i.e. the disjoint union of $M$ with itself under identification of corresponding boundary points. It is then well known that $\check{M}$ is a smooth manifold. Can you tell me whether the metric is still smooth along the boundary or else what is its degree of regularity? Can you maybe also tell me what it means for the boundary to be an isoperimetric surface for $M$?

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Regarding isoperimetric hypersurfaces, these are hypersurfaces that minimize area among those that enclose a fixed volume. They are automatically stable constant mean curvature hypersurfaces (although the reverse implication generally fails). Also, in general, the totally geodesic boundary $\partial M$ is not isoperimetric in the double of $M$. For a nice introduction to the subject, take a look at A. Ros' survey.