Reshetnyak gluing theorem

In metric geometry, the Reshetnyak gluing theorem gives information on the structure of a geometric object build 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 $X_{i}$ 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 $X$, obtained by gluing all $X_{i}$ along all $C_{i}$, 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).

Notes

1. See the original paper by Reshetnyak (1968) or the book by Burago, Burago & Ivanov (2001, Theorem 9.1.21).

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:
  • "Inextensible mappings in a space of curvature no greater than K", Siberian Mathematical Journal, 9 (4): 683–689, 1968, doi:10.1007/BF02199105, Zbl 0176.19503.
• Burago, Dmitri; Burago, Yuri; Ivanov, Sergei (2001), A course in metric geometry, Graduate Studies in Mathematics, 33, Providence, RI: American Mathematical Society, pp. xiv+415, ISBN 0-8218-2129-6, MR 1835418, Zbl 0981.51016.