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Difference between revisions of "Liouville net"

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A net of parametrized curves on a surface such that the line element of the surface has the form
 
A net of parametrized curves on a surface such that the line element of the surface has the form
 
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$$
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059630/l0596301.png" /></td> </tr></table>
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ds^2 = (U+V)(du^2 + dv^2)
 
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$$
where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059630/l0596302.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059630/l0596303.png" />. In every rectangle formed by two pairs of curves of the different families, the two geodesic diagonals have the same length. Surfaces that carry a Liouville net are Liouville surfaces (cf. [[Liouville surface|Liouville surface]]). For example, central surfaces of the second order are Liouville surfaces. The Liouville net was introduced by J. Liouville in 1846 (see [[#References|[1]]], Prop. 3).
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where $U = U(u)$, $V = V(v)$. In every rectangle formed by two pairs of curves of the different families, the two geodesic diagonals have the same length. Surfaces that carry a Liouville net are [[Liouville surface]]s. For example, central surfaces of the second order are Liouville surfaces. The Liouville net was introduced by [[Joseph Liouville|J. Liouville]] in 1846 (see [[#References|[1]]], Prop. 3).
 
 
====References====
 
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  G. Monge,  "Application de l'analyse à la géométrie" , Bachelier  (1850)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  V.I. Shulikovskii,  "Classical differential geometry in a tensor setting" , Moscow  (1963)  (In Russian)</TD></TR></table>
 
 
 
 
 
 
 
====Comments====
 
 
 
  
 
====References====
 
====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  W. Blaschke,  K. Leichtweiss,  "Elementare Differentialgeometrie" , Springer  (1973)</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  M. Berger,  B. Gostiaux,  "Differential geometry: manifolds, curves, and surfaces" , Springer  (1988)  (Translated from French)</TD></TR></table>
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<table>
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<TR><TD valign="top">[1]</TD> <TD valign="top">  G. Monge,  "Application de l'analyse à la géométrie" , Bachelier  (1850)</TD></TR>
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<TR><TD valign="top">[2]</TD> <TD valign="top">  V.I. Shulikovskii,  "Classical differential geometry in a tensor setting" , Moscow  (1963)  (In Russian)</TD></TR>
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<TR><TD valign="top">[a1]</TD> <TD valign="top">  W. Blaschke,  K. Leichtweiss,  "Elementare Differentialgeometrie" , Springer  (1973)</TD></TR>
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<TR><TD valign="top">[a2]</TD> <TD valign="top">  M. Berger,  B. Gostiaux,  "Differential geometry: manifolds, curves, and surfaces" , Springer  (1988)  (Translated from French) {{ZBL|0629.53001}}</TD></TR>
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</table>

Latest revision as of 19:10, 24 March 2023

2020 Mathematics Subject Classification: Primary: 53A05 [MSN][ZBL]

A net of parametrized curves on a surface such that the line element of the surface has the form $$ ds^2 = (U+V)(du^2 + dv^2) $$ where $U = U(u)$, $V = V(v)$. In every rectangle formed by two pairs of curves of the different families, the two geodesic diagonals have the same length. Surfaces that carry a Liouville net are Liouville surfaces. For example, central surfaces of the second order are Liouville surfaces. The Liouville net was introduced by J. Liouville in 1846 (see [1], Prop. 3).

References

[1] G. Monge, "Application de l'analyse à la géométrie" , Bachelier (1850)
[2] V.I. Shulikovskii, "Classical differential geometry in a tensor setting" , Moscow (1963) (In Russian)
[a1] W. Blaschke, K. Leichtweiss, "Elementare Differentialgeometrie" , Springer (1973)
[a2] M. Berger, B. Gostiaux, "Differential geometry: manifolds, curves, and surfaces" , Springer (1988) (Translated from French) Zbl 0629.53001
How to Cite This Entry:
Liouville net. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Liouville_net&oldid=17151
This article was adapted from an original article by V.T. Bazylev (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article