# Cohn-Vossen transformation

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).

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Cohn-Vossen transformation.

*Encyclopedia of Mathematics.*URL: http://encyclopediaofmath.org/index.php?title=Cohn-Vossen_transformation&oldid=46385