The universal cover of a flat, connected, complete Riemannian manifold is Euclidean space, which allows for classification of all such manifolds as quotient spaces $\mathbb{R}^n/\Gamma$ for torsionfree crystallographic groups $\Gamma \leq O(n)\ltimes \mathbb{R}^n$. Are there any similar results pertaining to flat manifolds with boundary? A classification is probably out of reach, but even partial (in particular, positive) results would help. For instance, submanifolds of the above flat manifolds would be examples; are there others?
2026-03-28 10:40:37.1774694437
Flat Riemannian manifolds with boundary
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