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.^[1]

Statement

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

For an exposition and a proof of the Reshetnyak Gluing Theorem, see (Burago, Burago & Ivanov 2001, Theorem 9.1.21).