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 space obtained by joining (i.e. "gluing") together, in a precisely defined way, other spaces having a given property inherit that very same property.
The theorem was first stated and proved by Yurii Reshetnyak in 1968.
Theorem: Let be complete locally compact geodesic metric spaces of CAT curvature , and convex subsets which are isometric. Then the manifold , obtained by gluing all along all , is also of CAT curvature .
For an exposition and a proof of the Reshetnyak Gluing Theorem, see .