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.
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 (Burago, Burago & Ivanov 2001, Theorem 9.1.21).
- 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:
- 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.
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