Cohn-Vossen transformation

A correspondence between a pair of isometric surfaces and and an infinitesimal deformation of the so-called mean surface : If and are the radius (position) vectors of the surfaces and , then the radius vector of is given by , and the field of velocities of the infinitesimal deformation is . It was introduced by S.E. Cohn-Vossen [1]. If and are smooth surfaces and if the angle between the semi-tangents and to the curves on and corresponding under the isometry is less than , then turns out to be smooth. This fact has enabled one to reduce in a number of cases the study of the isometry of and to the study of infinitesimal deformations (cf. Infinitesimal deformation) of . For fixed points on and on the Cohn-Vossen transformation defines a Cayley transformation of the orthogonal matrix , representing the isometry of the tangent space to to that of , into a skew-symmetric matrix describing the infinitesimal deformation of .

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 is the special case of the mixture of isometric surfaces and defined by the points in space dividing the segments joining corresponding points (under the isometry) in the ratio . 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).