# 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)

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).