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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]]]).
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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]]]).
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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]]]).
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The geometry of a configuration composed of two surfaces $  S _ {0} :  \mathbf n = \mathbf n ( u  ^ {1} , u  ^ {2} ) $
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and  $  S: \mathbf r = \mathbf r ( u  ^ {1} , u  ^ {2} ) $
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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]]]).
  
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]]]).
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An examination of the derivation of the equations for the reference frame  $  \partial  \mathbf r / \partial  u  ^ {1} $,
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$  \partial  \mathbf r / \partial  u  ^ {2} $,
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$  \mathbf n $
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leads to the concept of the interior relative geometry of a surface  $  S $(
 +
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  $  \mathbf n $
 +
of the affine normal is characterized by the fact that the asymptotic net of the surface  $  S $
 +
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 $  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 [[#References|[4]]]).
  
 
====References====
 
====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>
 
<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>

Revision as of 08:10, 6 June 2020


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)
How to Cite This Entry:
Relative geometry. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Relative_geometry&oldid=14857
This article was adapted from an original article by A.P. Norden (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article