Namespaces
Variants
Actions

Difference between revisions of "Cut locus"

From Encyclopedia of Mathematics
Jump to: navigation, search
(Importing text file)
 
(TeX)
 
Line 1: Line 1:
''from a point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c027/c027450/c0274501.png" />''
+
{{TEX|done}}
 +
''from a point $O$''
  
The set of points <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c027/c027450/c0274502.png" /> of a Riemannian manifold <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c027/c027450/c0274503.png" /> on the geodesic rays emanating from <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c027/c027450/c0274504.png" /> for which the ray <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c027/c027450/c0274505.png" /> is not extendable as a geodesic beyond the point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c027/c027450/c0274506.png" />. In the two-dimensional case the cut locus is a one-dimensional graph with no cycles (see [[#References|[2]]]); if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c027/c027450/c0274507.png" /> is analytic of arbitrary dimension, then it is a polyhedron of analytic submanifolds (see [[#References|[3]]]). The cut locus depends continuously on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c027/c027450/c0274508.png" />. The cut locus is defined not only with respect to a point but also with respect to other subsets, for example, the boundary <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c027/c027450/c0274509.png" />, and also in spaces other than Riemannian manifolds, for example, on convex surfaces (see [[#References|[4]]]) and in two-dimensional manifolds of bounded curvature.
+
The set of points $x$ of a Riemannian manifold $W$ on the geodesic rays emanating from $O$ for which the ray $Ox$ is not extendable as a geodesic beyond the point $x$. In the two-dimensional case the cut locus is a one-dimensional graph with no cycles (see [[#References|[2]]]); if $W$ is analytic of arbitrary dimension, then it is a polyhedron of analytic submanifolds (see [[#References|[3]]]). The cut locus depends continuously on $O$. The cut locus is defined not only with respect to a point but also with respect to other subsets, for example, the boundary $\partial W$, and also in spaces other than Riemannian manifolds, for example, on convex surfaces (see [[#References|[4]]]) and in two-dimensional manifolds of bounded curvature.
  
 
====References====
 
====References====
Line 9: Line 10:
  
 
====Comments====
 
====Comments====
"Non-extendable as a geodesic"  means that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c027/c027450/c02745010.png" /> looses the property of minimality after the point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c027/c027450/c02745011.png" />, i.e. <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c027/c027450/c02745012.png" /> is no longer the minimal path from <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c027/c027450/c02745013.png" /> to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c027/c027450/c02745014.png" /> if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c027/c027450/c02745015.png" />.
+
"Non-extendable as a geodesic"  means that $Ox$ looses the property of minimality after the point $x$, i.e. $Ox'$ is no longer the minimal path from $O$ to $x'$ if $Ox\subset Ox'$.

Latest revision as of 10:28, 15 April 2014

from a point $O$

The set of points $x$ of a Riemannian manifold $W$ on the geodesic rays emanating from $O$ for which the ray $Ox$ is not extendable as a geodesic beyond the point $x$. In the two-dimensional case the cut locus is a one-dimensional graph with no cycles (see [2]); if $W$ is analytic of arbitrary dimension, then it is a polyhedron of analytic submanifolds (see [3]). The cut locus depends continuously on $O$. The cut locus is defined not only with respect to a point but also with respect to other subsets, for example, the boundary $\partial W$, and also in spaces other than Riemannian manifolds, for example, on convex surfaces (see [4]) and in two-dimensional manifolds of bounded curvature.

References

[1] D. Gromoll, W. Klingenberg, W. Meyer, "Riemannsche Geometrie im Grossen" , Springer (1968)
[2] S.B. Myers, "Connections between differential geometry and topology. I Simply connected surfaces" Duke Math. J. , 1 (1935) pp. 376–391
[3] M.A. Buchner, "Simplicial structure of the real analytic cut locus" Proc. Amer. Math. Soc. , 64 : 1 (1977) pp. 118–121
[4] J. Kunze, "Der Schnittort auf konvexen Verheftungsflächen" , Springer (1969)


Comments

"Non-extendable as a geodesic" means that $Ox$ looses the property of minimality after the point $x$, i.e. $Ox'$ is no longer the minimal path from $O$ to $x'$ if $Ox\subset Ox'$.

How to Cite This Entry:
Cut locus. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Cut_locus&oldid=12250
This article was adapted from an original article by V.A. Zalgaller (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article