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 [math]\displaystyle{ X_i }[/math] be complete locally compact geodesic metric spaces of CAT curvature [math]\displaystyle{ \leq \kappa }[/math], and [math]\displaystyle{ C_i\subset X_i }[/math] convex subsets which are isometric. Then the manifold [math]\displaystyle{ X }[/math], obtained by gluing all [math]\displaystyle{ X_i }[/math] along all [math]\displaystyle{ C_i }[/math], is also of CAT curvature [math]\displaystyle{ \leq \kappa }[/math].
For an exposition and a proof of the Reshetnyak Gluing Theorem, see (Burago Burago).
Notes
- ↑ See the original paper by (Reshetnyak 1968) or the book by (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.
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