# 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 . 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 $F _ {\textrm{ m } }$.