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).