View source for 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]]].

====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>

====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|Convex surface]] and [[#References|[2]]], Chapt. 3, Para. 3).