Cut Locus Research Articles

On fine differentiability properties of horizons and applications to Riemannian geometry

We study fine differentiability properties of horizons. We show that the set of end points of generators of an n-dimensional horizon H (which is included in an ( n+1)-dimensional space–time M) has vanishing n-dimensional Hausdorff measure. This is proved by showing that the set of end points of generators at which the horizon is differentiable has the same property. For 1≤ k≤ n+1, we show (using deep results of Alberti) that the set of points where the convex hull of the set of generators leaving the horizon has dimension k is “almost a C 2 manifold of dimension n+1− k”: it can be covered, up to a set of vanishing ( n+1− k)-dimensional Hausdorff measure, by a countable number of C 2 manifolds. We use our Lorentzian geometry results to derive information about the fine differentiability properties of the distance function and the structure of cut loci in Riemannian geometry.

The locus-controlling region (LCR) from the cut locus of Drosophila activates the reporter gene lacZ periodically.

The locus-controlling region (LCR) from the cut locus of Drosophila activates the reporter gene lacZ periodically.

The Lipschitz continuity of the distance function to the cut locus

Let N be a closed submanifold of a complete smooth Riemannian manifold M and Uv the total space of the unit normal bundle of N. For each v C Uv, let p(v) denote the distance from N to the cut point of N on the geodesic -y, with the velocity vector tyv(0) = v. The continuity of the function p on Uv is well known. In this paper we prove that p is locally Lipschitz on which p is bounded; in particular, if M and N are compact, then p is globally Lipschitz on Uy. Therefore, the canonical interior metric 6 may be introduced on each connected component of the cut locus of N, and this metric space becomes a locally compact and complete length space. Let N be an immersed submanifold of a complete CC Riemannian manifold M and 7 : Uv -N the unit normal bundle of N. For each positive integer k and vector v C Uv, let a number k (v) denote the parameter value of fy, where 7yv denotes the geodesic for which the velocity vector is v at t = 0, such that 'yv(Ak(v)) is the k-th focal point (conjugate point for the case in which N is a point) of N along -y, counted with focal (or conjugate) multiplicities. A unit speed geodesic segment : [0, a] -+ M emanating from N is called an N-segment if t = d(N, y(t)) on [0, a]. By the first variation formula, an N-segment is orthogonal to N. A point 7y(to) on an N-segment 7y, v e Uv, is called a cut point of N if there is no N-segment properly containing y[O, to]. For each v C Uv, let p(v) denote the distance from N to the cut point on -y of N. Whitehead [27] investigated the structure of the conjugate locus and the cut locus of a point on a real analytic Finsler manifold. He determined the structure of the conjugate locus around a conjugate point for which the conjugate multiplicity is locally constant on its neighborhood (cf. also [25]) and proved the continuity of the function p. In compact symmetric spaces, T. Sakai [19] and M. Takeuchi [23] determined the detailed structure of the cut locus of a point. The detailed structure of the cut locus of a point in a 2-dimensional Riemannian manifold has been investigated by Poincare, Myers, and others [7], [11], [13]. Hartman first tried to show the absolute continuity of the function p when M is 2-dimensional. He proved in [8] that if p is of bounded variation, then p is absolutely continuous. Recently, Hebda [11] and the first named author [13] independently proved Ambrose's problem by showing that p is absolutely continuous on a closed arc on which p is bounded when N is a point in a 2-dimensional Riemannian manifold. Therefore, the cut locus of a point in a compact 2-dimensional Received by the editors October 14, 1998 and, in revised form, April 13, 1999. 2000 Mathematics Subject Classification. Primary 53C22; Secondary 28A78. Supported in part by a Grant-in-Aid for Scientific Research from The Ministry of Education, Science, Sports and Culture, Japan. (2000 American Mathematical Society

Stably Embedded Surfaces of Bounded Integral Curvature

We study the effect of simultaneous bounds on the local L1 norms of the second fundamental form and of the Gauss curvature on the geometry of surfaces Σ embedded in a Riemannian manifold M. Such bounds are natural since (together with an area bound) they amount to a local bound on the area of the manifold of unit normals to Σ, living in the sphere bundle of M. The main technical point is that, given a sequence of surfaces Σi with uniform local bounds of this type converging in the Hausdorff metric topology to a surface Σ0, the length space structures of the Σi converge to that of Σ0 (i.e., distances do not collapse). It follows that under suitable local topological conditions the surface Σ0 is a manifold of bounded integral curvature (MBC), in the sense of Alexandrov, under all smooth complete changes of the metric on M. In particular the boundary of a convex set with nonempty interior in any smooth complete M3 is MBC. Our method is to construct local coordinates for smoothly embedded surfaces Σ satisfying W1, BV bounds based solely on curvature integrals. In the process we demonstrate a regularity property for cut loci of surfaces.

Blaschke's rolling theorem for manifolds¶with boundary

For a complete Riemannian manifold M with compact boundary ∂M denote by $\Cut$ the cut locus of $\f M$ in M. The rolling radius of M is roll(M)≔ dist(∂M, ?∂M). Let Focal(∂M) be the focal distance of ∂M in M. Then conditions are given that imply the equality roll(M)= Focal(∂M). This generalizes Blaschke's rolling theorem from bounded convex domains in Euclidean space to more general Euclidean domains and to Riemannian manifolds with boundary.

The metric projection onto the soul

We study geometric properties of the metric projection r: M S of an open manifold M with nonnegative sectional curvature onto a soul S. ir is shown to be C? up to codimension 3. In arbitrary codimensions, small metric balls around a soul turn out to be convex, so that the unit normal bundle of S also admits a metric of nonnegative curvature. Next we examine how the horizontal curvatures at infinity determine the geometry of M, and stu(dy the structure of Sharafutdinov lines. We conclude with regularity properties of the cut and conjugate loci of M. The resolution of the Soul conjecture of Cheeger and Gromoll by Perelinan showed that the structure of open manifolds with nonnegative sectional curvature is more rigid than expected. Orne of the key results in [14] is that the metric projection wF: M -> S which maps a point p in M to the point 7r(p) in S that is closest to p is a Riemannian submersioni. Perelman also observed that this map is at least of class C1, and later it was shown in [9] that 7w is at least C2, and C? at almnost every point. The existence of such a Riemannian submersion combined with the restriction on the sign of the curvature suggests that the geometry of M inust be special. The purpose of this paper is to illustrate this in several different directions. In fact, we show that most natural geometric objects classically used in the study of these spaces are intrinsically related to the structure of the map -w. The paper is essentially structured as follows: After introducing notation and recalling some basic results in sectioni 1, we establish in section 2 the existence of maps similar to -w for any submanifold of M homologous and isometric to the soul, and use this to prove the existence of convex tubular neighborhoods for them. In particular, this also proves: Theorem 2.5. The unit normal bundle vti(S) admits a metric with nonnegative sectional curvature. As a consequence, open manifolds with nonnegative curvature provide examples of compact manifolds with the same lower curvatuire bound (see also [81). Theorem 2.5 also implies that nontrivial plane bundles over a Bieberbach maanifold do not admit nonnegatively curved metrics, thereby providing a shorter proof of the main result in [13]. We are grateful to the referee for pointing out this fact to us. Properties such as these are established via stanidard submersion techniques, but the fact that 7r is not known to be smooth everywhere prevents us fromi using these techniques to study the geometry of M far away from the soul. One way around Received by the editors August 18, 1997. 1991 Mathematics Subject Classification. Primary 53C20.

Coherent states, line bundles and divisors

For homogeneous simply connected Hodge manifolds it is proved that the set of coherent vectors orthogonal to a given one is the divisor responsible for the homogeneous holomorphic line bundle of the coherent vectors. In particular, for naturally reductive spaces, the divisor is the cut locus.

On the Dido Problem and Plane Isoperimetric Problems

This paper is a continuation of a series of papers, dealing with contact sub-Riemannian metrics on R3. We study the special case of contact metrics that correspond to isoperimetric problems on the plane. The purpose is to understand the nature of the corresponding optimal synthesis, at least locally. It is equivalent to studying the associated sub-Riemannian spheres of small radius. It appears that the case of generic isoperimetric problems falls down in the category of generic sub-Riemannian metrics that we studied in our previous papers (although, there is a certain symmetry). Thanks to the classification of spheres, conjugate-loci and cut-loci, done in those papers, we conclude immediately. On the contrary, for the Dido problem on a 2-d Riemannian manifold (i.e. the problem of minimizing length, for a prescribed area), these results do not apply. Therefore, we study in details this special case, for which we solve the problem generically (again, for generic cases, we compute the conjugate loci, cut loci, and the shape of small sub-Riemannian spheres, with their singularities). In an addendum, we say a few words about: (1) the singularities that can appear in general for the Dido problem, and (2) the motion of particles in a nonvanishing constant magnetic field.

Méthodes géométriques et analytiques pour étudier l'application exponentielle, la sphère et le front d'onde en géométrie sous-riemannienne dans le cas Martinet

Consider a sub-riemannian geometry (U,D,g) where U is a neighborhood of 0 in R 3 , D is a Martinet type distribution identified to ker ω , ω being the 1-form: , q=(x,y,z) and g is a metric on D which can be taken in the normal form : , a=1+yF(q) , c=1+G(q) , . In a previous article we analyze the flat case : a=c=1 ; we describe the conjugate and cut loci , the sphere and the wave front . The objectif of this article is to provide a geometric and computational framework to analyze the general case. This frame is obtained by analysing three one parameter deformations of the flat case which clarify the role of the three parameters in the gradated normal form of order 0 where: , . More generally this analysis provides an explanation of the role of abnormal minimizers in SR-geometry.

Forum domain in Drosophila melanogaster cut locus possesses looped domains inside.

We have studied the relationship between chromosomal forum domains and looped domains in the cut locus of Drosophila melanogaster . Forum domains were earlier detected by separation in pulsed-field gels of 50-150 kb chromosomal DNA fragments obtained after spontaneous non-random degradation of chromosomes. We have localized the boundary region where cleavage sites are scattered between two forum domains in the regulatory region of the cut locus. We have sequenced a 13 kb region spanning few kilobases from distal domain, the boundary region and part of the proximal forum domain where several scaffold associated regions (SARs) were observed. We conclude that forum domains and looped domains are physically different types of domains and belong to different levels of organization in eukaryotic chromosomes. The boundary region between the neighboring forum domains in the cut locus possesses the Doc element insertion and a micro-satellite stretch and thus might remind a small island of heterochromatin and correspond to so-called intercalary heterochromatin that is known to be located in the 7B1-2 band where the major part of the cut locus is reside.

The homeobox gene cut interacts genetically with the homeotic genes proboscipedia and Antennapedia.

The cut locus (ct) codes for a homeodomain protein (Cut) and controls the identity of a subset of cells in the peripheral nervous system in Drosophila. During a screen to identify ct-interacting genes, we observed that flies containing a hypomorphic ct mutation and a heterozygous deletion of the Antennapedia complex exhibit a transformation of mouthparts into leg and antennal structures similar to that seen in homozygous proboscipedia (pb) mutants. The same phenotype is produced with all heterozygous pb alleles tested and is fully penetrant in two different ct mutant backgrounds. We show that this phenotype is accompanied by pronounced changes in the expression patterns of both ct and pb in labial discs. Furthermore, a significant proportion of ct mutant flies that are heterozygous for certain Antennapedia (Antp) alleles have thoracic defects that mimic loss-of-function Antp phenotypes, and ectopic expression of Cut in antennal discs results in ectopic Antp expression and a dominant Antp-like phenotype. Our results implicate ct in the regulation of expression and/or function of two homeotic genes and document a new role of ct in the control of segmental identity.

Cut locus of a separating fractal set in a Riemannian manifold

We study the geometry of the cut locus of a separating fractal set A in a Riemannian manifold. In particular, we prove that every point of A is a limit point of the cut locus C{A) of A, and the Hausdorff dimension of C{A) is greater than or equal to that of A. Furthermore, we study the cut locus of the well-known Koch snowflake, and show the Hausdorff dimension of its cut locus is log 6/log 3 which is greater than the Hausdorff dimension, Iog4/log3, of the Koch snowflake itself. We also give another example for which the Hausdorff dimension of the cut locus stays the same. These two new examples are new fractal objects which are of interest on their own right.

Extreme points of the distance function on convex surfaces

We first see that, in the sense of Baire categories, many convex surfaces have quite large cut loci and infinitely many relative maxima of the distance function from a point. Then we find that, on any convex surface, all these extreme points lie on a single subtree of the cut locus, with at most three endpoints. Finally, we confirm (both in the sense of measure and in the sense of Baire categories) Steinhaus’ conjecture that “almost all" points admit a single farthest point on the surface.

The dimension of a cut locus on a smooth Riemannian manifold

The dimension of a cut locus on a smooth Riemannian manifold

Farthest points on convex surfaces

In the very enjoyable book ([3], p. 44) of Craft, Falconer and Guy we read: “We take... a... convex surface C in R3.... Steinhaus... asked... what can be said qualitatively about the set of all “farthest points” from a point x .” Our aim here is to investigate this question. Let S be the space of all closed convex surfaces (i.e. boundaries of open bounded convex sets) in R3 and denote, for any S ∈ S and x ∈ S , by Fx the set of all farthest points from x and by Cx the set of all points joined with x by at least two segments (i.e. shortest paths). We shall see that to any point x on a closed convex surface we may associate in a natural way a point or a Jordan arc Jx ⊃ Fx lying in the cut locus of x . This will provide the topological characterization of Fx . Further we shall prove an easily stated, remarkable geometric property of Fx : any three of its points (if it contains at least three) form an obtuse or right geodesic triangle. Moreover, the preceding result is shown to hold for a large class of geodesic triangles with vertices in the cut locus. Interestingly, the extreme case of a right triangle does not imply the degeneracy of the surface. For the reader, to be familiar with Aleksandrov’s book [1] (see also [2), [4]) would be of considerable help. The usual intrinsic metric of S ∈ S , induced by the Euclidean distance in R3 will be denoted by ρ. Let x ∈ S , and denote by Ex the cut locus of x , i.e. the set of all endpoints different from x of maximal (by inclusion) segments starting at x . Also, let E be the set of all endpoints of S , i.e. points not interior to any segment of S . The set

Mathematical theory of medial axis transform

The medial axis of a plane domain is dened to be the set of the centers of the maximal inscribed disks. It is essentially the cut loci of the inward unit normal bundle of the boundary. We prove that if a plane domain has nite number of boundary curves each of which consists of nite number of real analytic pieces, then the medial axis is a connected geometric graph in R 2 with nitely many vertices and edges. And each edge is a real analytic curve which can be extended in the C 1 manner at the end vertices. We clarify the relation between the vertex degree and the local geometry of the domain. We also analyze various continuity and regularity results in detail, and show that the medial axis is a strong deformation retract of the domain which means in the practical sense that it retains all the topological informations of the domain. We also obtain parallel results for the medial axis transform.

Some Consequences of the Nature of the Distance Function on the Cut Locus in a Riemannian Manifold

We render the cut locus more accessible to analysis by showing that each non-conjugate cut point has a neighbourhood on which the distance function is the minimum of finitely many smooth ‘radial functions’. This enables us to generalise to an arbitrary complete manifold a number of recent results, mainly from stochastic analysis, which were either limited or not valid on the cut locus because of the lack of differentiability there of the distance function.

A remark on Berezin's quantization and cut locus

The consequences for Berezin's quantization on symmetric spaces of the identity of the set of coherent vectors orthogonal to a fixed one with the cut locus are stated precisely. It is shown that functions expressing the coherent states, the covariant symbols of operators, the diastasis function, the characteristic and the two-point functions are defined when one variable does not belong to the cut locus of the other one.

Burdock, a novel retrotransposon in Drosophila melanogaster, integrates into the coding region of the cut locus

The burdock element is known to be the 2.6-kb insertion into the same region of the cut locus in 12 independently obtained ct-lethal mutants. Here we have determined the complete sequences of this insertion and of the hot spot region. It was found that the burdock is a short retrotransposon with long terminal repeats and a single open reading frame (ORF). The polypeptide encoded by the burdock ORF contains two adjacent regions homologous to the gag and pol polyproteins of the gypsy mobile element. The burdock insertion interrupts the short ORF of the cut locus. The target site sequence of the burdock insertions is similar to the Drosophila topoisomerase II cleavage site.

Cut and conjugate loci in two-step nilpotent Lie groups

We show that cut and conjugate loci coincide for a large class of nilpotent Lie groups with left-invariant metric. One consequently obtains lower bounds for the injectivity radius of nonsingular 2-step nilmanifolds. The Jacobi equations are also used to classify Riemannian foliations of 3-dimensional nilmanifolds.