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''in a domain of a mapping''
 
''in a domain of a mapping''
  
An [[Orthogonal net|orthogonal net]] in a domain <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062310/m0623101.png" /> of an <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062310/m0623102.png" />-dimensional [[Manifold|manifold]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062310/m0623103.png" /> (which can be, in particular, a Euclidean space) that is mapped onto a net, also orthogonal, by a [[Diffeomorphism|diffeomorphism]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062310/m0623104.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062310/m0623105.png" /> onto a domain <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062310/m0623106.png" /> in the same or another Riemannian manifold <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062310/m0623107.png" />. The directions tangential to the lines of the principal net of the mapping at the point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062310/m0623108.png" /> are the principal directions of the ellipsoid of deformation of the induced mapping <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062310/m0623109.png" /> of the tangent space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062310/m06231010.png" /> onto the tangent space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062310/m06231011.png" />. When <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062310/m06231012.png" />, generally speaking, the principal net of a mapping is not holonomic. If the mapping <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062310/m06231013.png" /> is conformal, then any orthogonal net in the domain <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062310/m06231014.png" /> serves as a principal net.
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An [[Orthogonal net|orthogonal net]] in a domain $  G $
 +
of an $  n $-
 +
dimensional [[Manifold|manifold]] $  M $(
 +
which can be, in particular, a Euclidean space) that is mapped onto a net, also orthogonal, by a [[Diffeomorphism|diffeomorphism]] $  f: G \rightarrow G  ^  \prime  $
 +
of $  G $
 +
onto a domain $  G  ^  \prime  $
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in the same or another Riemannian manifold $  M  ^  \prime  $.  
 +
The directions tangential to the lines of the principal net of the mapping at the point $  x \in G $
 +
are the principal directions of the ellipsoid of deformation of the induced mapping $  f _  \star  : T _ {x} \rightarrow T _ {f(} x)  ^  \prime  $
 +
of the tangent space $  T _ {x} $
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onto the tangent space $  T _ {f(} x)  ^  \prime  $.  
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When $  n > 2 $,  
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generally speaking, the principal net of a mapping is not holonomic. If the mapping $  f $
 +
is conformal, then any orthogonal net in the domain $  G $
 +
serves as a principal net.
  
 
====References====
 
====References====
 
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  V.V. Ryzhkov,  "Differential geometric point correspondences between spaces"  ''Itogi Nauk. Ser. Mat. Geom. 1963'' , '''3'''  (1965)  pp. 65–107  (In Russian)</TD></TR></table>
 
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  V.V. Ryzhkov,  "Differential geometric point correspondences between spaces"  ''Itogi Nauk. Ser. Mat. Geom. 1963'' , '''3'''  (1965)  pp. 65–107  (In Russian)</TD></TR></table>

Latest revision as of 07:59, 6 June 2020


in a domain of a mapping

An orthogonal net in a domain $ G $ of an $ n $- dimensional manifold $ M $( which can be, in particular, a Euclidean space) that is mapped onto a net, also orthogonal, by a diffeomorphism $ f: G \rightarrow G ^ \prime $ of $ G $ onto a domain $ G ^ \prime $ in the same or another Riemannian manifold $ M ^ \prime $. The directions tangential to the lines of the principal net of the mapping at the point $ x \in G $ are the principal directions of the ellipsoid of deformation of the induced mapping $ f _ \star : T _ {x} \rightarrow T _ {f(} x) ^ \prime $ of the tangent space $ T _ {x} $ onto the tangent space $ T _ {f(} x) ^ \prime $. When $ n > 2 $, generally speaking, the principal net of a mapping is not holonomic. If the mapping $ f $ is conformal, then any orthogonal net in the domain $ G $ serves as a principal net.

References

[1] V.V. Ryzhkov, "Differential geometric point correspondences between spaces" Itogi Nauk. Ser. Mat. Geom. 1963 , 3 (1965) pp. 65–107 (In Russian)
How to Cite This Entry:
Mapping, principal net of a. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Mapping,_principal_net_of_a&oldid=17526
This article was adapted from an original article by V.T. Bazylev (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article