Reshetnyak gluing theorem

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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[edit]

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).

Notes[edit]

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

References[edit]