Talk:Cut locus

I have added a special case definition, along with its reference. Moreover, I made some editing changes.

Should the "STUB" be removed now?

How do you find the example? Is it a good one?

Drorata (talk) 17:08, 29 October 2008 (UTC)

This article seems to take "metric space" to mean a topological space for which lengths of curves are defined. However, the standard definition is simply a set with a distance function -- i.e., a general metric space does not have enough structure to define the length of curves. Maybe it's best to merge this article with the one on the cut locus for a Riemannian manifold. — Preceding unsigned comment added by 66.57.43.129 (talk) 00:18, 29 April 2010 (UTC)

{{discussion top|result=The result of this discussion was merge. TRL (talk) 01:55, 2 April 2024 (UTC)}}{{anchor|Proposed_merge_of_Cut_locus_(Riemannian_manifold)_into_Cut_locus}}

• Support merging the two articles, as their definitions seem identical; but the final article would no longer need the parenthetical disambiguation, thus would be titled simply "Cut locus". fgnievinski (talk) 06:50, 25 September 2023 (UTC)
• While I support merging both of the articles, I would like to note that there is a notion of curve length in metric spaces, the supremum over the lengths of all polygonal approximations of the curve (some curves have infinite length though). This is hinted at in arc length (ctrl-F supremum) but not explained in detail. —Kusma (talk) 13:33, 6 November 2023 (UTC)
• Support. The articles do seem to be about the same topic. Novellasyes (talk) 14:11, 18 February 2024 (UTC)

: FWIW, the Cut locus (Riemannian manifold) article is (only) in :Category:Riemannian geometry, while this article is only in :Category:Mathematical structures and :Category:Mathematics stubs. Novellasyes (talk) 14:14, 18 February 2024 (UTC){{discussion bottom}}

:{{merge done}} Klbrain (talk) 11:34, 23 June 2024 (UTC)

::@Klbrain both articles are about the exact same subject, so the merger needs to merge them to top level, not just insert one as a section of the other. –jacobolus (t) 16:44, 23 June 2024 (UTC)

:::Feel free to fix in situ. No objection from me! Klbrain (talk) 18:50, 23 June 2024 (UTC)

A very brief literature search turns up http://www.numdam.org/item/CM_1978__37_1_103_0.pdf which says:

: Historically, the cut locus was first introduced by Poincare [10] for compact simply connected surfaces of positive curvature. In 1935, Summer B. Meyers [7], [8], and J.H.C. Whitehead [15] both studied the cut locus. Whitehead showed that any compact n-dimensional Riemannian manifold decomposes into the cut locus and an open cell with the cut locus as the cell boundary. Meyers showed that forcompact analytic surfaces the cut locus is a graph, for simply connected surfaces the graph is a tree, the end points of which are conjugate to the origin of the cut locus and are cusps of the locus of first conjugate points.

: {{hr}}

: [7] S.B. MYERS: Connections between differential geometry and topology: I. Simply connected surfaces. Duke Math. J., 1 (1935) 376-391.

: [8] S.B. MYERS: Connections between differential geometry and topology: II. Closed surfaces. Duke Math. J., 2 (1936) 95-102.

: [10] H. POINCARÉ: Trans. Amer. Math. Soc., 6 (1905) 243.

: [15] J.H.C. WHITEHEAD: On the covering of a complete space by the geodesics through a point. Ann. of Math., 36 (1935) 679-704.

This paper also has a definition:

: If {{tmath|\alpha}} is a metric on {{tmath|M}} then {{tmath|C(p, \alpha)}} will denote the cut locus in {{tmath|M}} with respect to {{tmath|p}} and the metric {{tmath|\alpha}} i.e. the set of those points {{tmath|q}} in {{tmath|M}} which are joined to {{tmath|p}} by a length minimizing geodesic which fails to minimize the length to points beyond {{tmath|q}} on the geodesic.

Another paper that turned up recommends the survey paper:

: Shoshichi Kobayashi. On conjugate and cut loci. In S.-S. Chern, editor, Studies in global geometry and analysis, number 4 in Studies in Mathematics, pages 96–122. MAA, Englewood Cliffs, NJ, 1967.

–jacobolus (t) 17:43, 23 June 2024 (UTC)