View source for Relative geometry

The geometry of a configuration composed of two surfaces <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080990/r0809901.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080990/r0809902.png" /> that are in [[Peterson correspondence|Peterson correspondence]]. The analogy between this correspondence and the [[Spherical map|spherical map]] makes it possible to introduce the concepts of a relative area, Gaussian and mean curvature, etc., and in particular of a relatively-minimal surface (see [[#References|[1]]]).

An examination of the derivation of the equations for the reference frame <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080990/r0809903.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080990/r0809904.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080990/r0809905.png" /> leads to the concept of the interior relative geometry of a surface <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080990/r0809906.png" /> (see [[#References|[2]]]). This is the geometry of an affine connection (or more precisely, an equi-affine connection) without torsion. The concept of a second-order geometry similar to the geometry of the spherical map has been introduced (see [[#References|[3]]]).

Within relative geometry it is possible to include in an overall scheme not only the geometry of Euclidean surfaces and pseudo-Euclidean spaces, but also the geometry of affine differential geometry. The vector <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080990/r0809907.png" /> of the affine normal is characterized by the fact that the asymptotic net of the surface <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080990/r0809908.png" /> is a Chebyshev net (see [[#References|[3]]]).

A further generalization of relative geometry is the theory of normalized surfaces (see [[#References|[4]]]). With each point of a surface <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080990/r0809909.png" /> in a projective space two straight lines are connected: the first-order normal which passes through a point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080990/r08099010.png" /> of the surface but having no other common points with the tangent plane <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080990/r08099011.png" />, and the second-order normal belonging to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080990/r08099012.png" /> but not passing through <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080990/r08099013.png" />. Two interior geometries conjugated through an asymptotic net are defined on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080990/r08099014.png" />. The construction of relative geometries allows many generalizations (see [[#References|[4]]]).

====References====
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  E. Müller,   ''Monatsh. Math. und Physik'' , '''31'''  (1921)  pp. 3–19</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  A.P. Norden,   "Sur l'inclusion des théories métriques et affines des surfaces dans la géométrie des systèmes spécifiques"  ''C.R. Acad. Sci. Paris'' , '''192'''  (1931)  pp. 135–137</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top">  A.P. Norden,   "On the intrinsic geometry of second kind hypersurfaces in affine space"  ''Izv. Vyzov. Mat.'' , '''4'''  (1958)  pp. 172–183  (In Russian)</TD></TR><TR><TD valign="top">[4]</TD> <TD valign="top">  A.P. Norden,   "Spaces with an affine connection" , Nauka , Moscow-Leningrad  (1976)  (In Russian)</TD></TR></table>