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.
Statement Theorem: Let X i {\displaystyle X_{i}} be complete locally compact geodesic metric spaces of CAT curvature ≤ κ {\displaystyle \leq \kappa } , and C i ⊂ X i {\displaystyle C_{i}\subset X_{i}} convex subsets which are isometric. Then the manifold X {\displaystyle X} , obtained by gluing all X i {\displaystyle X_{i}} along all C i {\displaystyle C_{i}} , is also of CAT curvature ≤ κ {\displaystyle \leq \kappa } . For an exposition and a proof of the Reshetnyak Gluing Theorem, see (Burago, Burago & Ivanov 2001, Theorem 9.1.21).
Notes
References Reshetnyak, Yu. G. (1968), "Nonexpanding maps in spaces of curvature not greater than K", Sibirskii Matematicheskii Zhurnal (in Russian), 9 (4): 918–927, MR 0244922, Zbl 0167.50803, translated in English as: Reshetnyak, Yu. G. (1968), "Inextensible mappings in a space of curvature no greater than K", Siberian Mathematical Journal, 9 (4): 683–689, Bibcode:1968SibMJ...9..683R, doi:10.1007/BF02199105, Zbl 0176.19503. Burago, Dmitri; Burago, Yuri; Ivanov, Sergei (2001), A course in metric geometry, Graduate Studies in Mathematics, vol. 33, Providence, RI: American Mathematical Society, pp. xiv+415, ISBN 978-0-8218-2129-9, MR 1835418, Zbl 0981.51016.