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Difference between revisions of "Relative geometry"

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$  \partial  \mathbf r / \partial  u  ^ {2} $,  
 
$  \partial  \mathbf r / \partial  u  ^ {2} $,  
 
$  \mathbf n $
 
$  \mathbf n $
leads to the concept of the interior relative geometry of a surface $ S $(
+
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]]]).
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 $
 
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 $
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is a Chebyshev net (see [[#References|[3]]]).
 
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 $
+
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 $
+
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 $,  
+
of the surface but having no other common points with the tangent plane $\alpha$,  
and the second-order normal belonging to $ \alpha $
+
and the second-order normal belonging to $\alpha$
but not passing through $ A $.  
+
but not passing through $A$.  
Two interior geometries conjugated through an asymptotic net are defined on $ S $.  
+
Two interior geometries conjugated through an asymptotic net are defined on $S$.  
 
The construction of relative geometries allows many generalizations (see [[#References|[4]]]).
 
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>

Latest revision as of 10:54, 17 March 2023


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=52762
This article was adapted from an original article by A.P. Norden (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article