Difference between revisions of "Cohn-Vossen transformation"
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| + | 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]]]. | ||
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====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> | ||
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| − | |||
====Comments==== | ====Comments==== | ||
| − | For this type of infinitesimal deformation one also uses the term infinitesimal bending. The mean surface | + | 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=16863
Cohn-Vossen transformation. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Cohn-Vossen_transformation&oldid=16863
This article was adapted from an original article by M.I. Voitsekhovskii (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article