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]]. | ||
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====References==== | ====References==== | ||
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| − | <TR><TD valign="top">[a1]</TD> <TD valign="top"> | + | <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"> | + | <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
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