Namespaces
Variants
Actions

Difference between revisions of "Quasi-geodesic line"

From Encyclopedia of Mathematics
Jump to: navigation, search
(Importing text file)
 
(TeX)
 
Line 1: Line 1:
 +
{{TEX|done}}
 
''quasi-geodesic''
 
''quasi-geodesic''
  
 
A curve on a surface on any segment of which the right and left rotations have the same sign (see [[Swerve of a curve|Swerve of a curve]]). For example, the edge of a lens is a quasi-geodesic line.
 
A curve on a surface on any segment of which the right and left rotations have the same sign (see [[Swerve of a curve|Swerve of a curve]]). For example, the edge of a lens is a quasi-geodesic line.
  
The class of quasi-geodesic lines substantially amplifies the class of geodesic lines (cf. [[Geodesic line|Geodesic line]]), making its families (bounded in length and position) compact. In a two-dimensional manifold <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076510/q0765101.png" /> of bounded curvature there passes through each point at least one quasi-geodesic in each direction; it can always be extended. Segments of quasi-geodesics (at the ends of which there are no points of curvature <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076510/q0765102.png" /> on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076510/q0765103.png" />) are limits of geodesics lying on smooth surfaces properly converging to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076510/q0765104.png" />.
+
The class of quasi-geodesic lines substantially amplifies the class of geodesic lines (cf. [[Geodesic line|Geodesic line]]), making its families (bounded in length and position) compact. In a two-dimensional manifold $M$ of bounded curvature there passes through each point at least one quasi-geodesic in each direction; it can always be extended. Segments of quasi-geodesics (at the ends of which there are no points of curvature $2\pi$ on $M$) are limits of geodesics lying on smooth surfaces properly converging to $M$.
  
 
====References====
 
====References====

Latest revision as of 20:24, 3 May 2014

quasi-geodesic

A curve on a surface on any segment of which the right and left rotations have the same sign (see Swerve of a curve). For example, the edge of a lens is a quasi-geodesic line.

The class of quasi-geodesic lines substantially amplifies the class of geodesic lines (cf. Geodesic line), making its families (bounded in length and position) compact. In a two-dimensional manifold $M$ of bounded curvature there passes through each point at least one quasi-geodesic in each direction; it can always be extended. Segments of quasi-geodesics (at the ends of which there are no points of curvature $2\pi$ on $M$) are limits of geodesics lying on smooth surfaces properly converging to $M$.

References

[1] A.D. Aleksandrov, Yu.D. Burago, "Quasigeodesics" Proc. Steklov Inst. Math. , 76 (1965) pp. 58–76 Trudy Mat. Inst. Steklov. , 76 (1965) pp. 49–63


Comments

References

[a1] A.V. Pogorelov, "Extrinsic geometry of convex surfaces" , Amer. Math. Soc. (1972) (Translated from Russian)
How to Cite This Entry:
Quasi-geodesic line. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Quasi-geodesic_line&oldid=17208
This article was adapted from an original article by V.A. Zalgaller (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article