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
Class or The Class may refer to:
Common uses not otherwise categorized
* Class (biology), a taxonomic rank
* Class (knowledge representation), a collection of individuals or objects
* Class (philosophy), an analytical concept used differentl ...
. Intuitively, it states that a
manifold
In mathematics, a manifold is a topological space that locally resembles Euclidean space near each point. More precisely, an n-dimensional manifold, or ''n-manifold'' for short, is a topological space with the property that each point has a n ...
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
Yurii Grigorievich Reshetnyak (russian: Ю́рий Григо́рьевич Решетня́к, 26 September 1929 – 17 December 2021) was a Soviet and Russian mathematician and academician.
He worked in geometry and the theory of functions of ...
in 1968.
[See the original paper by or the book by .]
Statement
Theorem: Let
be
complete
Complete may refer to:
Logic
* Completeness (logic)
* Completeness of a theory, the property of a theory that every formula in the theory's language or its negation is provable
Mathematics
* The completeness of the real numbers, which implies t ...
locally compact In topology and related branches of mathematics, a topological space is called locally compact if, roughly speaking, each small portion of the space looks like a small portion of a compact space. More precisely, it is a topological space in which ev ...
geodesic
In geometry, a geodesic () is a curve representing in some sense the shortest path ( arc) between two points in a surface, or more generally in a Riemannian manifold. The term also has meaning in any differentiable manifold with a connection. ...
metric space
In mathematics, a metric space is a set together with a notion of ''distance'' between its elements, usually called points. The distance is measured by a function called a metric or distance function. Metric spaces are the most general settin ...
s of
CAT curvature , and
convex subsets which are
isometric
The term ''isometric'' comes from the Greek for "having equal measurement".
isometric may mean:
* Cubic crystal system, also called isometric crystal system
* Isometre, a rhythmic technique in music.
* "Isometric (Intro)", a song by Madeon from ...
. Then the
manifold
In mathematics, a manifold is a topological space that locally resembles Euclidean space near each point. More precisely, an n-dimensional manifold, or ''n-manifold'' for short, is a topological space with the property that each point has a n ...
, obtained by gluing all
along all
, is also of CAT curvature
.
For an exposition and a proof of the Reshetnyak Gluing Theorem, see .
Notes
References
*, translated in English as:
**.
*.
Theorems in geometry
Metric geometry
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