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Alexandrov geometry studies non smooth analogues of Riemannian manifolds with curvature bounded from below or above. It includes spaces with curvature bounded below (briefly $\mathrm{CBB}[\kappa]$) and spaces with curvature bounded above (briefly $\mathrm{CAT}[\kappa]$).

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Gluing Alexandrov spaces along parts of boundary

I'm familiar with the Petrunin gluing theorem that states that gluing two Alexandrov spaces $M_1,M_2\in Alex(k)$ along their boundaries via an isometry $:\partial M_1\rightarrow \partial M_2$ results …
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