Reshetnyak gluing theorem

Short description: On the structure of a geometric object built by joining other geometric objects

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 $\le \kappa$, 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 $\le \kappa$.

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

References

• Reshetnyak, Yu. G. (1968), "Nonexpanding maps in spaces of curvature not greater than K" (in Russian), Sibirskii Matematicheskii Zhurnal 9 (4): 918–927 , 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, doi:10.1007/BF02199105 .
• 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 978-0-8218-2129-9, https://books.google.com/books?id=afnlx8sHmQIC .