Reshetnyak gluing theorem

In metric geometry, the Reshetnyak gluing theorem gives information on the structure of a geometric object built by using as building blocks other geometric objects, belonging to a well defined class. Intuitively, it states that a manifold obtained by joining (i.e. "gluing") together, in a precisely defined way, other manifolds having a given property inherit that very same property.

The theorem was first stated and proved by Yurii Reshetnyak in 1968.

Statement
Theorem: Let $$X_i$$ be complete locally compact geodesic metric spaces of CAT curvature $$\leq \kappa$$, and $$C_i\subset X_i$$ convex subsets which are isometric. Then the manifold $$X$$, obtained by gluing all $$X_i$$ along all $$C_i$$, is also of CAT curvature $$\leq \kappa$$.

For an exposition and a proof of the Reshetnyak Gluing Theorem, see.