Reshetnyak gluing theorem - Alchetron, the free social encyclopedia

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.

Statement

Theorem: Let X i be complete locally compact geodesic metric spaces of CAT curvature ≤ κ , and C i ⊂ 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 ≤ κ .

For an exposition and a proof of the Reshetnyak Gluing Theorem, see (Burago, Burago & Ivanov 2001, Theorem 9.1.21).

References

Reshetnyak gluing theorem Wikipedia

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