Difference between revisions of "Liouville net"
From Encyclopedia of Mathematics
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ds^2 = (U+V)(du^2 + dv^2) | ds^2 = (U+V)(du^2 + dv^2) | ||
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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 J. Liouville in 1846 (see [[#References|[1]]], Prop. 3). | + | 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==== | ====References==== | ||
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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> | <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> | <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> | <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> | + | <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> |
</table> | </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=40019
Liouville net. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Liouville_net&oldid=40019
This article was adapted from an original article by V.T. Bazylev (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article