Namespaces
Variants
Actions

Difference between revisions of "Liouville surface"

From Encyclopedia of Mathematics
Jump to: navigation, search
(Importing text file)
 
(Tex done)
Line 1: Line 1:
A surface for which the equations of the geodesics admit a quadratic integral <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059670/l0596701.png" />, where the tensor <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059670/l0596702.png" /> is different from the [[Metric tensor|metric tensor]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059670/l0596703.png" /> of the surface. For example, a surface of constant [[Gaussian curvature|Gaussian curvature]] is a Liouville surface. For a surface to admit a [[Geodesic mapping|geodesic mapping]] onto a plane it is necessary and sufficient that it be a Liouville surface (Dini's theorem). See also [[Liouville net|Liouville net]].
+
{{TEX|done}}{{MSC|53A05}}
 +
 
 +
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]].
  
  
Line 7: Line 9:
  
 
====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>
+
<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>

Revision as of 19:57, 15 December 2016

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.


Comments

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