Reshetnyak gluing theorem explained

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

X _{i} be complete locally compact geodesic metric spaces of CAT curvature \leq \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 \leq \kappa .

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

References

• , translated in English as:
  • .
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Notes and References

1. See the original paper by or the book by .