View source for Cut locus

{{one source |date=May 2024}}
[[File:Geodesics and geodesic circles on an oblate ellipsoid.svg|thumb|
[[Geodesics on an ellipsoid]] (blue) from a single point (for flattening {{math|''f'' {{=}} {{frac|10}}}},
latitude {{math|''φ''<sub>1</sub> {{=}} {{val|-30|u=deg}}}}) form a segment of a [[circle of latitude]];
[[geodesic circle]]s are shown in green and the cut locus in red.]]

In [[differential geometry]], the '''cut locus''' of a [[point (geometry)|point]] {{mvar|p}} on a [[manifold (mathematics)|manifold]] is the closure of the set of all other points on the manifold that are connected to {{mvar|p}} by two or more distinct shortest [[geodesic]]s.<ref>{{cite web |title=Cut locus |url=http://encyclopediaofmath.org/index.php?title=Cut_locus&oldid=12250 |website=Encyclopedia of Mathematics |access-date=February 18, 2024}}</ref> More generally, the cut locus of a [[closed set]] {{mvar|X}} on the manifold is the closure of the set of all other points on the manifold connected to {{mvar|X}} by two or more distinct shortest geodesics.

== Examples ==

[[File:Zylinder pfad.svg|thumb|Cut locus C(P) of a point P on the surface of a cylinder. A point Q in the cut locus is shown with two distinct shortest paths <math>\gamma_1, \gamma_2</math> connecting it to P.]]

In the [[Euclidean plane]], a point ''p'' has an empty cut locus, because every other point is connected to ''p'' by a unique geodesic (the line segment between the points).

On the [[sphere]], the cut locus of a point consists of the single [[antipodal point]] diametrically opposite to it.

On an infinitely long [[cylinder (geometry)|cylinder]], the cut locus of a point consists of the line opposite the point.

Let ''X'' be the boundary of a simple [[polygon]] in the Euclidean plane. Then the cut locus of ''X'' in the interior of the polygon is the polygon's [[medial axis]].  Points on the medial axis are centers of disks that touch the polygon boundary at two or more points, corresponding to two or more shortest paths to the disk center.

Let ''x'' be a point on the surface of a convex [[polyhedron]] ''P''.  Then the cut locus of ''x'' on the polyhedron's surface is known as the '''ridge tree''' of ''P'' with respect to ''x''.  This ridge tree has the property that cutting the surface along its edges unfolds ''P'' to a simple planar polygon.  This polygon can be viewed as a [[net (polyhedron)|net]] for the polyhedron.

== Formal definition ==
Fix a point <math>p</math> in a [[complete space|complete]] Riemannian manifold <math>(M,g)</math>, and consider the [[tangent space]] <math>T_pM</math>.  It is a standard result that for sufficiently small <math>v</math> in <math>T_p M</math>, the curve defined by the [[exponential map (Riemannian geometry)|Riemannian exponential map]], <math>\gamma(t) = \exp_p(tv)</math> for <math>t</math> belonging to the interval <math>[0,1]</math> is a [[geodesic|minimizing geodesic]], and is the unique minimizing geodesic connecting the two endpoints.  Here <math>\exp_p</math> denotes the exponential map from <math>p</math>.  The '''cut locus of <math>p</math> in the tangent space''' is defined to be the set of all vectors <math>v</math> in <math>T_pM</math> such that <math>\gamma(t)=\exp_p(tv)</math> is a minimizing geodesic for <math>t \in [0,1]</math> but fails to be minimizing for <math>t = 1 + \varepsilon</math> for every <math>\varepsilon > 0</math>.  Thus the cut locus in the tangent space is the boundary of the set<ref>Cheeger, J., Ebin, D. G., & Ebin, D. G. (1975). Comparison theorems in Riemannian geometry (Vol. 9). Amsterdam: North-Holland publishing company, p. 94.</ref> 
<math display="block">\{ v\in T_pM | d(\exp_pv,p) = \|v\|\}</math>
where <math>d</math> denotes the length metric of <math>M</math>, and <math>\|\cdot\|</math> is the Euclidean norm of <math>T_pM</math>.  The '''cut locus of <math>p</math> in <math>M</math>''' is defined to be image of the cut locus of <math>p</math> in the tangent space under the exponential map at <math>p</math>.  Thus, we may interpret the cut locus of <math>p</math> in <math>M</math> as the points in the manifold where the geodesics starting at <math>p</math> stop being minimizing.

The least distance from ''p'' to the cut locus is the [[injectivity radius]] at ''p''.  On the open ball of this radius, the exponential map at ''p'' is a diffeomorphism from the tangent space to the manifold, and this is the largest such radius.  The global injectivity radius is defined to be the infimum of the injectivity radius at ''p'', over all points of the manifold.

== Characterization ==
Suppose <math>q</math> is in the cut locus of <math>p</math> in <math>M</math>.  A standard result<ref>{{cite book |last=Petersen |first=Peter |title=Riemannian Geometry |edition=1st |publisher=Springer-Verlag |year=1998 |chapter=Chapter 5, Lemma 8.2 }}</ref> is that either (1) there is more than one minimizing geodesic joining <math>p</math> to <math>q</math>, or (2) <math>p</math> and <math>q</math> are [[conjugate point|conjugate]] along some geodesic
which joins them.  It is possible for both (1) and (2) to hold.

== Applications ==
The significance of the cut locus is that the distance function from a point <math>p</math> is smooth, except on the cut locus of <math>p</math> and <math>p</math> itself.  In particular, it makes sense to take the [[gradient]] and [[Hessian matrix|Hessian]] of the distance function away from the cut locus and <math>p</math>.  This idea is used in the [[local Laplacian comparison theorem]] and the [[local Hessian comparison theorem]].  These are used in the proof of the local version of the [[Toponogov theorem]], and many other important theorems in Riemannian geometry.

For the [[Alexandrov's uniqueness theorem|metric space of surface distances on a convex polyhedron]], cutting the polyhedron along the cut locus produces a shape that can be unfolded flat into a plane, the [[source unfolding]].<ref>{{cite book
 | last1 = Demaine | first1 = Erik | author1-link = Erik Demaine
 | last2 = O'Rourke | first2 = Joseph | author2-link = Joseph O'Rourke (professor)
 | contribution = 24.1.1 Source unfolding
 | isbn = 978-0-521-71522-5
 | pages = 359–362
 | publisher = Cambridge University Press
 | title = Geometric Folding Algorithms
 | title-link = Geometric Folding Algorithms
 | year = 2007}}</ref> The unfolding process can be performed continuously, as a [[Blooming (geometry)|blooming]] of the polyhedron.<ref>{{cite journal
 | last1 = Demaine | first1 = Erik D. | author1-link = Erik Demaine
 | last2 = Demaine | first2 = Martin L. | author2-link = Martin Demaine
 | last3 = Hart | first3 = Vi | author3-link = Vi Hart
 | last4 = Iacono | first4 = John
 | last5 = Langerman | first5 = Stefan | author5-link = Stefan Langerman
 | last6 = O'Rourke | first6 = Joseph | author6-link = Joseph O'Rourke (professor)
 | doi = 10.1007/s00373-011-1024-3
 | issue = 3
 | journal = [[Graphs and Combinatorics]]
 | mr = 2787423
 | pages = 363–376
 | title = Continuous blooming of convex polyhedra
 | volume = 27
 | year = 2011| s2cid = 82408 | citeseerx = 10.1.1.150.9715}}</ref> Analogous methods of cutting along the cut locus can be used to unfold higher-dimensional convex polyhedra as well.<ref>{{cite journal
 | last1 = Miller | first1 = Ezra
 | last2 = Pak | first2 = Igor | author2-link = Igor Pak
 | doi = 10.1007/s00454-008-9052-3
 | issue = 1–3
 | journal = [[Discrete & Computational Geometry]]
 | mr = 2383765
 | pages = 339–388
 | title = Metric combinatorics of convex polyhedra: Cut loci and nonoverlapping unfoldings
 | volume = 39
 | year = 2008| doi-access = free
 }}</ref>

== Cut locus of a subset ==
One can similarly define the cut locus of a submanifold of the Riemannian manifold, in terms of its normal exponential map.

==References==
{{reflist}}

[[Category:Mathematical structures]]

[[de:Schnittort]]
[[ru:Множество раздела]]