Peterson correspondence

A correspondence between two surfaces under which their tangent planes at corresponding points are parallel. It was considered in general form by K.M. Peterson [1] in connection with the problem of deformation over a principal base. For example, there is a Peterson correspondence between a surface and it spherical image (cf. Spherical map), between a surface and its rotation indicatrix and between adjoint minimal surfaces (cf. Adjoint surface).

If two surfaces $ S $ and $ S ^ {*} $ in Peterson correspondence have a common parametrization, then their third fundamental forms are equal. The principal net for the asymptotic nets (cf. Asymptotic net) of $ S $ and $ S ^ {*} $ is conjugate (cf. Conjugate net) on each of them. This net is uniquely defined if the asymptotic nets do not have common families of lines, it degenerates if these nets are linked and it becomes undefined if these nets correspond to one another. Corresponding tangents to the lines of the principal net in $ S $ and $ S ^ {*} $ are parallel.

If the principal net is taken to be a $ ( u, v) $ coordinate net, then the position vectors $ x $ and $ x ^ {*} $ of $ S $ and $ S ^ {*} $ are related by

$$ x _ {u} ^ {*} = \rho x _ {u} ,\ \ x _ {v} ^ {*} = \sigma x _ {v} , $$

where the functions $ \rho $ and $ \sigma $ satisfy the system of equations

$$ \rho _ {v} E - \sigma _ {u} F = \frac{\sigma - \rho }{2} ,\ \ \rho _ {v} F - \sigma _ {u} G = \frac{\sigma - \rho }{2} G _ {u} , $$

i.e. $ \rho $ and $ \sigma $ depend only on the metric of $ S $( $ E $, $ F $ and $ G $ are the coefficients of the first fundamental form). One therefore naturally can apply a Peterson correspondence to a pair of isometric surfaces $ S $ and $ \widetilde{S} $: it gives another pair of isometric surfaces $ S ^ {*} $ and $ \widetilde{S} {} ^ {*} $ with the same normals $ n= n ^ {*} $ and $ \widetilde{n} = \widetilde{n} {} ^ {*} $, respectively. Also, the rotation diagrams for these surfaces are the same, and the base of the isometry for the new surfaces has the same spherical image as the original ones. For example, a sphere and a surface of constant positive curvature isometric with it correspond under a Peterson correspondence to isometric surfaces with corresponding lines of curvature, so-called Bonnet surfaces.

In particular, if the base of the isometry for $ S $ and $ S ^ {*} $ is a principal base, then it remains such. This is the so-called Peterson transformation of a surface deformed over a principal base into a surface of the same type. There is an extension of this transformation to the case of a family of nets [2].

A special case of a Peterson correspondence is that in which the principal net is a curvature net simultaneously in $ S $ and $ S ^ {*} $. This is called a Combescour correspondence.

If a Peterson correspondence is conformal, then either one of the surfaces is minimal and the other is a sphere (i.e. the Peterson correspondence is a spherical mapping), or the two surfaces are minimal and the conformal mapping vector satisfies the Laplace equation, or the surfaces are similar, or the surfaces are isothermal and are in Combescour correspondence.

There is a multi-dimensional generalization of the Peterson correspondence [4].

References

[1]	K.M. Peterson, "Об отношениях и средствах между кривыми поверхностями", Mat. Sb. , 1 (1866) pp. 391–438
[2]	V.I. Shulikovskii, "Classical differential geometry in a tensor setting" , Moscow (1963) (In Russian)
[3]	S.P. Finikov, "Bending and related geometrical problems" , Moscow-Leningrad (1937) (In Russian)
[4]	P.A. Shirokov, A.P. Shirokov, "Affine differentialgeometrie" , Teubner (1962) (Translated from Russian)