Namespaces
Variants
Actions

Difference between revisions of "Cohn-Vossen transformation"

From Encyclopedia of Mathematics
Jump to: navigation, search
(Importing text file)
 
m (tex encoded by computer)
 
Line 1: Line 1:
A correspondence between a pair of isometric surfaces <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023040/c0230401.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023040/c0230402.png" /> and an infinitesimal deformation of the so-called mean surface <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023040/c0230403.png" />: If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023040/c0230404.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023040/c0230405.png" /> are the radius (position) vectors of the surfaces <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023040/c0230406.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023040/c0230407.png" />, then the radius vector <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023040/c0230408.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023040/c0230409.png" /> is given by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023040/c02304010.png" />, and the field of velocities <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023040/c02304011.png" /> of the infinitesimal deformation <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023040/c02304012.png" /> is <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023040/c02304013.png" />. It was introduced by S.E. Cohn-Vossen [[#References|[1]]]. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023040/c02304014.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023040/c02304015.png" /> are smooth surfaces and if the angle between the semi-tangents <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023040/c02304016.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023040/c02304017.png" /> to the curves on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023040/c02304018.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023040/c02304019.png" /> corresponding under the isometry is less than <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023040/c02304020.png" />, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023040/c02304021.png" /> turns out to be smooth. This fact has enabled one to reduce in a number of cases the study of the isometry of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023040/c02304022.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023040/c02304023.png" /> to the study of infinitesimal deformations (cf. [[Infinitesimal deformation|Infinitesimal deformation]]) of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023040/c02304024.png" />. For fixed points <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023040/c02304025.png" /> on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023040/c02304026.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023040/c02304027.png" /> on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023040/c02304028.png" /> the Cohn-Vossen transformation defines a Cayley transformation of the orthogonal matrix <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023040/c02304029.png" />, representing the isometry of the tangent space to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023040/c02304030.png" /> to that of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023040/c02304031.png" />, into a skew-symmetric matrix describing the infinitesimal deformation of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023040/c02304032.png" />.
+
<!--
 +
c0230401.png
 +
$#A+1 = 37 n = 0
 +
$#C+1 = 37 : ~/encyclopedia/old_files/data/C023/C.0203040 Cohn\AAnVossen transformation
 +
Automatically converted into TeX, above some diagnostics.
 +
Please remove this comment and the {{TEX|auto}} line below,
 +
if TeX found to be correct.
 +
-->
 +
 
 +
{{TEX|auto}}
 +
{{TEX|done}}
 +
 
 +
A correspondence between a pair of isometric surfaces  $  F _ {1} $
 +
and  $  F _ {2} $
 +
and an infinitesimal deformation of the so-called mean surface $  F _ {\textrm{ m } }  $:  
 +
If $  x _ {1} $
 +
and $  x _ {2} $
 +
are the radius (position) vectors of the surfaces $  F _ {1} $
 +
and $  F _ {2} $,  
 +
then the radius vector $  x _ {\textrm{ m } }  $
 +
of $  F _ {\textrm{ m } }  $
 +
is given by $  ( x _ {1} + x _ {2} )/2 $,  
 +
and the field of velocities $  z $
 +
of the infinitesimal deformation $  Z $
 +
is $  ( x _ {1} - x _ {2} )/2 $.  
 +
It was introduced by S.E. Cohn-Vossen [[#References|[1]]]. If $  F _ {1} $
 +
and $  F _ {2} $
 +
are smooth surfaces and if the angle between the semi-tangents $  \tau _ {1} $
 +
and $  \tau _ {2} $
 +
to the curves on $  F _ {1} $
 +
and $  F _ {2} $
 +
corresponding under the isometry is less than $  \pi $,  
 +
then $  F _ {\textrm{ m } }  $
 +
turns out to be smooth. This fact has enabled one to reduce in a number of cases the study of the isometry of $  F _ {1} $
 +
and $  F _ {2} $
 +
to the study of infinitesimal deformations (cf. [[Infinitesimal deformation|Infinitesimal deformation]]) of $  F _ {\textrm{ m } }  $.  
 +
For fixed points $  M _ {1} $
 +
on $  F _ {1} $
 +
and $  M _ {2} $
 +
on $  F _ {2} $
 +
the Cohn-Vossen transformation defines a Cayley transformation of the orthogonal matrix $  O $,  
 +
representing the isometry of the tangent space to $  F _ {1} $
 +
to that of $  F _ {2} $,  
 +
into a skew-symmetric matrix describing the infinitesimal deformation of $  F _ {\textrm{ m } }  $.
  
 
The Cohn-Vossen transformation can be generalized to the case of spaces of constant curvature [[#References|[2]]].
 
The Cohn-Vossen transformation can be generalized to the case of spaces of constant curvature [[#References|[2]]].
Line 5: Line 48:
 
====References====
 
====References====
 
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  S.E. Cohn-Vossen,  "Some problems of differential geometry in the large" , Moscow  (1959)  (In Russian)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  A.V. Pogorelov,  "Extrinsic geometry of convex surfaces" , Amer. Math. Soc.  (1972)  (Translated from Russian)</TD></TR></table>
 
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  S.E. Cohn-Vossen,  "Some problems of differential geometry in the large" , Moscow  (1959)  (In Russian)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  A.V. Pogorelov,  "Extrinsic geometry of convex surfaces" , Amer. Math. Soc.  (1972)  (Translated from Russian)</TD></TR></table>
 
 
  
 
====Comments====
 
====Comments====
For this type of infinitesimal deformation one also uses the term infinitesimal bending. The mean surface <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023040/c02304033.png" /> is the special case <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023040/c02304034.png" /> of the mixture of isometric surfaces <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023040/c02304035.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023040/c02304036.png" /> defined by the points in space dividing the segments joining corresponding points (under the isometry) in the ratio <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023040/c02304037.png" />. The study of these mixtures is an important tool in the isotopy problem of convex surfaces (cf. [[Convex surface|Convex surface]] and [[#References|[2]]], Chapt. 3, Para. 3).
+
For this type of infinitesimal deformation one also uses the term infinitesimal bending. The mean surface $  F _ {\textrm{ m } }  $
 +
is the special case $  \lambda = 1/2 $
 +
of the mixture of isometric surfaces $  F _ {0} $
 +
and $  F _ {1} $
 +
defined by the points in space dividing the segments joining corresponding points (under the isometry) in the ratio $  \lambda : ( \lambda - 1) $.  
 +
The study of these mixtures is an important tool in the isotopy problem of convex surfaces (cf. [[Convex surface|Convex surface]] and [[#References|[2]]], Chapt. 3, Para. 3).

Latest revision as of 17:45, 4 June 2020


A correspondence between a pair of isometric surfaces $ F _ {1} $ and $ F _ {2} $ and an infinitesimal deformation of the so-called mean surface $ F _ {\textrm{ m } } $: If $ x _ {1} $ and $ x _ {2} $ are the radius (position) vectors of the surfaces $ F _ {1} $ and $ F _ {2} $, then the radius vector $ x _ {\textrm{ m } } $ of $ F _ {\textrm{ m } } $ is given by $ ( x _ {1} + x _ {2} )/2 $, and the field of velocities $ z $ of the infinitesimal deformation $ Z $ is $ ( x _ {1} - x _ {2} )/2 $. It was introduced by S.E. Cohn-Vossen [1]. If $ F _ {1} $ and $ F _ {2} $ are smooth surfaces and if the angle between the semi-tangents $ \tau _ {1} $ and $ \tau _ {2} $ to the curves on $ F _ {1} $ and $ F _ {2} $ corresponding under the isometry is less than $ \pi $, then $ F _ {\textrm{ m } } $ turns out to be smooth. This fact has enabled one to reduce in a number of cases the study of the isometry of $ F _ {1} $ and $ F _ {2} $ to the study of infinitesimal deformations (cf. Infinitesimal deformation) of $ F _ {\textrm{ m } } $. For fixed points $ M _ {1} $ on $ F _ {1} $ and $ M _ {2} $ on $ F _ {2} $ the Cohn-Vossen transformation defines a Cayley transformation of the orthogonal matrix $ O $, representing the isometry of the tangent space to $ F _ {1} $ to that of $ F _ {2} $, into a skew-symmetric matrix describing the infinitesimal deformation of $ F _ {\textrm{ m } } $.

The Cohn-Vossen transformation can be generalized to the case of spaces of constant curvature [2].

References

[1] S.E. Cohn-Vossen, "Some problems of differential geometry in the large" , Moscow (1959) (In Russian)
[2] A.V. Pogorelov, "Extrinsic geometry of convex surfaces" , Amer. Math. Soc. (1972) (Translated from Russian)

Comments

For this type of infinitesimal deformation one also uses the term infinitesimal bending. The mean surface $ F _ {\textrm{ m } } $ is the special case $ \lambda = 1/2 $ of the mixture of isometric surfaces $ F _ {0} $ and $ F _ {1} $ defined by the points in space dividing the segments joining corresponding points (under the isometry) in the ratio $ \lambda : ( \lambda - 1) $. The study of these mixtures is an important tool in the isotopy problem of convex surfaces (cf. Convex surface and [2], Chapt. 3, Para. 3).

How to Cite This Entry:
Cohn-Vossen transformation. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Cohn-Vossen_transformation&oldid=46385
This article was adapted from an original article by M.I. Voitsekhovskii (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article