Namespaces
Variants
Actions

Difference between revisions of "Glueing method"

From Encyclopedia of Mathematics
Jump to: navigation, search
(Importing text file)
 
(TeX)
 
Line 1: Line 1:
 +
{{TEX|done}}
 
''in the theory of surfaces''
 
''in the theory of surfaces''
  
 
A method for constructing surfaces isometric to a given one. The glueing method has applications in proving the realizability of abstractly-defined convex metrics, in questions of bending convex surfaces, and in quantitative estimates of bendability. Its strength lies in the fact that it works in situations where differential equations are powerless.
 
A method for constructing surfaces isometric to a given one. The glueing method has applications in proving the realizability of abstractly-defined convex metrics, in questions of bending convex surfaces, and in quantitative estimates of bendability. Its strength lies in the fact that it works in situations where differential equations are powerless.
  
Aleksandrov's theorem: Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g044/g044490/g0444901.png" /> be closed domains in a manifold with an intrinsic metric of positive curvature, bounded by a finite number of curves with bounded rotational variation. Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g044/g044490/g0444902.png" /> be the manifold consisting of the domains <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g044/g044490/g0444903.png" /> with their boundaries identified in such a way that:
+
Aleksandrov's theorem: Let $G_1,\ldots,G_n$ be closed domains in a manifold with an intrinsic metric of positive curvature, bounded by a finite number of curves with bounded rotational variation. Let $G$ be the manifold consisting of the domains $G_i$ with their boundaries identified in such a way that:
  
1) the identified segments of the boundaries of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g044/g044490/g0444904.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g044/g044490/g0444905.png" /> have equal length;
+
1) the identified segments of the boundaries of $G_i$ and $G_j$ have equal length;
  
2) the sum of the rotations of the identified segments of the boundaries of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g044/g044490/g0444906.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g044/g044490/g0444907.png" /> from the sides of these domains is non-negative;
+
2) the sum of the rotations of the identified segments of the boundaries of $G_i$ and $G_j$ from the sides of these domains is non-negative;
  
3) the sum of the angles made by the sectors at the identified points of the domains <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g044/g044490/g0444908.png" /> with the sides of these domains does not exceed <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g044/g044490/g0444909.png" />.
+
3) the sum of the angles made by the sectors at the identified points of the domains $G_k$ with the sides of these domains does not exceed $2\pi$.
  
Then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g044/g044490/g04449010.png" /> has an intrinsic metric of positive curvature that coincides with the metrics of the domains in neighbourhoods of corresponding points.
+
Then $G$ has an intrinsic metric of positive curvature that coincides with the metrics of the domains in neighbourhoods of corresponding points.
  
 
====References====
 
====References====
 
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  A.D. Aleksandrov,  V.A. Zalgaller,  "Two-dimensional manifolds of bounded curvature"  ''Proc. Steklov Inst. Math.'' , '''76'''  (1962)  ''Trudy Mat. Inst. Steklov.'' , '''76'''  (1962)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  A.V. Pogorelov,  "Die Verbiegung konvexer Flächen" , Akademie Verlag  (1957)  (Translated from Russian)</TD></TR></table>
 
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  A.D. Aleksandrov,  V.A. Zalgaller,  "Two-dimensional manifolds of bounded curvature"  ''Proc. Steklov Inst. Math.'' , '''76'''  (1962)  ''Trudy Mat. Inst. Steklov.'' , '''76'''  (1962)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  A.V. Pogorelov,  "Die Verbiegung konvexer Flächen" , Akademie Verlag  (1957)  (Translated from Russian)</TD></TR></table>

Latest revision as of 21:03, 30 April 2014

in the theory of surfaces

A method for constructing surfaces isometric to a given one. The glueing method has applications in proving the realizability of abstractly-defined convex metrics, in questions of bending convex surfaces, and in quantitative estimates of bendability. Its strength lies in the fact that it works in situations where differential equations are powerless.

Aleksandrov's theorem: Let $G_1,\ldots,G_n$ be closed domains in a manifold with an intrinsic metric of positive curvature, bounded by a finite number of curves with bounded rotational variation. Let $G$ be the manifold consisting of the domains $G_i$ with their boundaries identified in such a way that:

1) the identified segments of the boundaries of $G_i$ and $G_j$ have equal length;

2) the sum of the rotations of the identified segments of the boundaries of $G_i$ and $G_j$ from the sides of these domains is non-negative;

3) the sum of the angles made by the sectors at the identified points of the domains $G_k$ with the sides of these domains does not exceed $2\pi$.

Then $G$ has an intrinsic metric of positive curvature that coincides with the metrics of the domains in neighbourhoods of corresponding points.

References

[1] A.D. Aleksandrov, V.A. Zalgaller, "Two-dimensional manifolds of bounded curvature" Proc. Steklov Inst. Math. , 76 (1962) Trudy Mat. Inst. Steklov. , 76 (1962)
[2] A.V. Pogorelov, "Die Verbiegung konvexer Flächen" , Akademie Verlag (1957) (Translated from Russian)
How to Cite This Entry:
Glueing method. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Glueing_method&oldid=18059
This article was adapted from an original article by M.I. Voitsekhovskii (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article