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Difference between revisions of "Liouville surface"

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A surface for which the equations of the geodesics admit a quadratic integral $a_{ij} du^i du^j$, where the tensor $a_{ij}$ is distinct from the [[metric tensor]] $g_{ij}$ of the surface. For example, a surface of constant [[Gaussian curvature]] is a Liouville surface. For a surface to admit a [[geodesic mapping]] onto a plane it is necessary and sufficient that it be a Liouville surface (Dini's theorem). See also [[Liouville net]].
 
A surface for which the equations of the geodesics admit a quadratic integral $a_{ij} du^i du^j$, where the tensor $a_{ij}$ is distinct from the [[metric tensor]] $g_{ij}$ of the surface. For example, a surface of constant [[Gaussian curvature]] is a Liouville surface. For a surface to admit a [[geodesic mapping]] onto a plane it is necessary and sufficient that it be a Liouville surface (Dini's theorem). See also [[Liouville net]].
 
 
 
====Comments====
 
 
  
 
====References====
 
====References====
 
<table>
 
<table>
<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">[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>
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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)</TD></TR>
 
</table>
 
</table>

Latest revision as of 15:02, 10 April 2023

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

A surface for which the equations of the geodesics admit a quadratic integral $a_{ij} du^i du^j$, where the tensor $a_{ij}$ is distinct from the metric tensor $g_{ij}$ of the surface. For example, a surface of constant Gaussian curvature is a Liouville surface. For a surface to admit a geodesic mapping onto a plane it is necessary and sufficient that it be a Liouville surface (Dini's theorem). See also Liouville net.

References

[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)
How to Cite This Entry:
Liouville surface. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Liouville_surface&oldid=40018
This article was adapted from an original article by I.Kh. Sabitov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article