# Relative geometry

The geometry of a configuration composed of two surfaces $S _ {0} : \mathbf n = \mathbf n ( u ^ {1} , u ^ {2} )$ and $S: \mathbf r = \mathbf r ( u ^ {1} , u ^ {2} )$ that are in Peterson correspondence. The analogy between this correspondence and the 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 ).
An examination of the derivation of the equations for the reference frame $\partial \mathbf r / \partial u ^ {1}$, $\partial \mathbf r / \partial u ^ {2}$, $\mathbf n$ leads to the concept of the interior relative geometry of a surface $S$( see ). 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 ).
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 $\mathbf n$ of the affine normal is characterized by the fact that the asymptotic net of the surface $S$ is a Chebyshev net (see ).
A further generalization of relative geometry is the theory of normalized surfaces (see ). With each point of a surface $S$ in a projective space two straight lines are connected: the first-order normal which passes through a point $A$ of the surface but having no other common points with the tangent plane $\alpha$, and the second-order normal belonging to $\alpha$ but not passing through $A$. Two interior geometries conjugated through an asymptotic net are defined on $S$. The construction of relative geometries allows many generalizations (see ).