View source for 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 [[#References|[1]]] in connection with the problem of [[Deformation over a principal base|deformation over a principal base]]. For example, there is a Peterson correspondence between a surface and it spherical image (cf. [[Spherical map|Spherical map]]), between a surface and its [[Rotation indicatrix|rotation indicatrix]] and between adjoint minimal surfaces (cf. [[Adjoint surface|Adjoint surface]]).

If two surfaces <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072460/p0724601.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072460/p0724602.png" /> in Peterson correspondence have a common parametrization, then their third fundamental forms are equal. The principal net for the asymptotic nets (cf. [[Asymptotic net|Asymptotic net]]) of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072460/p0724603.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072460/p0724604.png" /> is conjugate (cf. [[Conjugate net|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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072460/p0724605.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072460/p0724606.png" /> are parallel.

If the principal net is taken to be a <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072460/p0724607.png" /> coordinate net, then the position vectors <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072460/p0724608.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072460/p0724609.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072460/p07246010.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072460/p07246011.png" /> are related by

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072460/p07246012.png" /></td> </tr></table>

where the functions <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072460/p07246013.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072460/p07246014.png" /> satisfy the system of equations

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072460/p07246015.png" /></td> </tr></table>

i.e. <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072460/p07246016.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072460/p07246017.png" /> depend only on the metric of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072460/p07246018.png" /> (<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072460/p07246019.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072460/p07246020.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072460/p07246021.png" /> are the coefficients of the [[First fundamental form|first fundamental form]]). One therefore naturally can apply a Peterson correspondence to a pair of isometric surfaces <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072460/p07246022.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072460/p07246023.png" />: it gives another pair of isometric surfaces <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072460/p07246024.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072460/p07246025.png" /> with the same normals <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072460/p07246026.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072460/p07246027.png" />, 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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072460/p07246028.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072460/p07246029.png" /> 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 [[#References|[2]]].

A special case of a Peterson correspondence is that in which the principal net is a curvature net simultaneously in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072460/p07246030.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072460/p07246031.png" />. 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 [[#References|[4]]].

====References====
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  K.M. Peterson,   ''Mat. Sb.'' , '''1'''  (1866)  pp. 391–438</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  V.I. Shulikovskii,   "Classical differential geometry in a tensor setting" , Moscow  (1963)  (In Russian)</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top">  S.P. Finikov,   "Bending and related geometrical problems" , Moscow-Leningrad  (1937)  (In Russian)</TD></TR><TR><TD valign="top">[4]</TD> <TD valign="top">  P.A. Shirokov,   A.P. Shirokov,   "Affine differentialgeometrie" , Teubner  (1962)  (Translated from Russian)</TD></TR></table>