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 [1]).

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 [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 [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 $ \mathbf n $ of the affine normal is characterized by the fact that the asymptotic net of the surface $ S $ is a Chebyshev net (see [3]).

A further generalization of relative geometry is the theory of normalized surfaces (see [4]). 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 [4]).

References

[1]	E. Müller, Monatsh. Math. und Physik , 31 (1921) pp. 3–19
[2]	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
[3]	A.P. Norden, "On the intrinsic geometry of second kind hypersurfaces in affine space" Izv. Vyzov. Mat. , 4 (1958) pp. 172–183 (In Russian)
[4]	A.P. Norden, "Spaces with an affine connection" , Nauka , Moscow-Leningrad (1976) (In Russian)