Cut Locus Research Articles

The cut locus and Voronoi diagram of a finite set of points in a surface

The edges of the Voronoi diagram for a finite set P of points in a compact Riemannian surface are a subset of the cut locus of P. From this point of view we study when the Voronoi cells are homeomorphic to disks, using the mini-type distance function for P and its critical points. We will see some relations between the topological structure and the Voronoi diagram.

Triangulations of 3–Manifolds with essential edges

We define essential and strongly essential triangulations of 3–manifolds, and give four constructions using different tools (Heegaard splittings, hierarchies of Haken 3–manifolds, Epstein-Penner decompositions, and cut loci of Riemannian manifolds) to obtain triangulations with these properties under various hypotheses on the topology or geometry of the manifold.

Integrability methods in the time minimal coherence transfer for Ising chains of three spins

The objective of this article is to analyze the integrability properties of extremal solutions of Pontryagin Maximum Principle in the time minimal control of a linear spin system with Ising coupling in relation with conjugate and cut loci computations. Restricting to the case of three spins, the problem is equivalent to analyze a family of almost-Riemannian metrics on the sphere $S^{2}$, with Grushin equatorial singularity. The problem can be lifted into a SR-invariant problem on $SO(3)$, this leads to a complete understanding of the geometry of the problem and to an explicit parametrization of the extremals using an appropriate chart as well as elliptic functions. This approach is compared with the direct analysis of the Liouville metrics on the sphere where the parametrization of the extremals is obtained by computing a Liouville normal form. Finally, an algebraic approach is presented in the framework of the application of differential Galois theory to integrability.

The Structure Theorem for the Cut Locus of a Certain Class of Cylinders of Revolution I

We generalize Toponogov's maximal diameter sphere theorem from the radial curvature geometry's standpoint. As a corollary to our main theorem, we prove that for a complete connected Riemannian $n$-manifold $M$ having radial sectional curvature at a point bounded from below by the radial curvature function of an ellipsoid of prolate type, the diameter of $M$ does not exceed the diameter of the ellipsoid. Furthermore if the diameter of such an $M$ equals that of the ellipsoid, then $M$ is isometric to the $n$-dimensional ellipsoid of revolution.

Geodesic bifurcation on smooth surfaces

Within Riemannian geometry the geodesic exponential map is an essential tool for various distance-related investigations and computations. Several natural questions can be formulated in terms of its preimages, usually leading to quite challenging non-linear problems. In this context we recently proposed an approach for computing multiple geodesics connecting two arbitrary points on two-dimensional surfaces in situations where an ambiguity of these connecting geodesics is indicated by the presence of focal curves. The essence of the approach consists in exploiting the structure of the associated focal curve and using a suitable curve for a homotopy algorithm to collect the geodesic connections. In this follow-up discussion we extend those constructions to overcome a significant limitation inherent in the previous method, i.e. the necessity to construct homotopy curves artificially. We show that considering homotopy curves meeting a focal curve tangentially leads to a singularity that we investigate thoroughly. Solving this so-called geodesic bifurcation analytically and dealing with it numerically provides not only theoretical insights, but also allows geodesics to be used as homotopy curves. This yields a stable computational tool in the context of computing distances. This is applicable in common situations where there is a curvature induced non-injectivity of the exponential map. In particular we illustrate how applying geodesic bifurcation approaches the distance problem on compact manifolds with a single closed focal curve. Furthermore, the presented investigations provide natural initial values for computing cut loci using the medial differential equation which directly leads to a discussion on avoiding redundant computations by combining the presented concepts to determine branching points.

Differential geometric methods for examining the dynamics of slow-fast vector fields

In this work we present computational methods for examining dynamical systems. We focus on those systems being characterized by slow---fast vector fields or corresponding differential algebraic equations that commonly occur in physical applications. In the latter ones scientists usually consider a manifold of admissible physical states and a vector field describing the time evolution of the physical system. The manifold is typically implicitly defined within a higher-dimensional space by a system of equations. Certain physical systems, such as relaxation oscillators, perform sudden jumps in their state evolution when they are forced into an unstable state. The main contribution of the present work is to model the dynamical evolution incorporating the jumping behavior from a perspective of computational geometry which not only provides a qualitative analysis but also produces quantitative results. We use geodesic polar coordinates (GPC) to numerically obtain explicit parametrizations of the implicitly defined manifold and of the relevant jump and hit sets. Moreover, to deal with the possibly high co-dimension of the considered implicitly defined manifold we sketch how GPC in combination with the cut locus concept can be used to numerically obtain an essentially injective global parametrization. This allows us to parametrize and visualize the dynamical evolution of the system including the aforementioned jump phenomena. As main tools we use homotopy approaches in conjunction with concepts from differential geometry. We discuss how to numerically realize and how to apply them to several examples from mechanics, electrical engineering and biology.

The front of increasing concentric balls and cut locus in a surface

Let M be a compact Riemannian 2-manifold without boundary and o∈M. We introduce an index for critical points of the distance function to o and establish the relation among the index, the properties of critical points and the Euler characteristic of M.

On the measure of the cut locus of a Fréchet mean

One of the fundamental differences between the Central Limit Theorem for empirical Frechet means obtained in [Kendall and Le, ‘Limit theorems for empirical Frechet means of independent and non‐identically distributed manifold‐valued random variables’, Braz. J. Probab. Stat. 25 (2011) 323–352] and that for empirical Euclidean means lies on the assumption that the probability measure of the cut locus of the true Frechet mean is zero. In [Hotz and Huckemann, ‘Intrinsic means on the circle: uniqueness, locus and asymptotics’, Preprint, 2011, arXiv:1108.2141v1], the authors show that, in the case of a circle, this assumption holds automatically. This paper shows that this holds for any complete and connected Riemannian manifold, assuming that there are at least two minimal geodesics between the Frechet mean and any point in its cut locus and that, in fact, it can also be generalized to local minima of the

Sub-Riemannian Geometry of Stiefel Manifolds

In the paper we consider the Stiefel manifold $V_{n,k}$ as a principal $U(k)$-bundle over the Grassmann manifold and study the cut locus from the unit element. We give the complete description of this cut locus on $V_{n,1}$ and present a sufficient condition on the general case. At the end, we study the complement to the cut locus of $V_{2k,k}$.

RecA bundles mediate homology pairing between distant sisters during DNA break repair

DNA double-strand break (DSB) repair by homologous recombination (HR) has evolved to maintain genetic integrity in all organisms1. Although many reactions that occur during HR are known1-3, it is unclear where, when and how they occur in cells is lacking. Here, by using conventional and super-resolution microscopy we describe the progression of DSB repair in live Escherichia coli. Specifically, we investigate whether HR can occur efficiently between distant sister loci that have segregated to opposite halves of an E. coli cell. We show that a site-specific DSB in one sister can be repaired efficiently using distant sister homology. After RecBCD processing of the DSB, RecA is recruited to the cut locus, where it nucleates into a bundle that contains many more RecA molecules than can associate with the two ssDNA regions that form at the DSB. Mature bundles extend along the cell long axis in the space between the bulk nucleoid and the inner membrane. Bundle formation is followed by pairing in which the two ends of the cut locus relocate at the periphery of the nucleoid and together move rapidly towards the homology of the uncut sister. After sister locus pairing, RecA bundles disassemble and proteins that act late in HR are recruited to give viable recombinants 1-2 generation time equivalents after formation of the initial DSB. Mutated RecA proteins that do not form bundles are defective in sister pairing and in DSB-induced repair. The work reveals an unanticipated role of RecA bundles in channeling the movement of the DNA DSB ends, thereby facilitating the long-range homology search that occurs before the strand invasion and transfer reactions.

Some Properties of Cut Locus of a Flat Torus

Some Properties of Cut Locus of a Flat Torus

On Means and Their Asymptotics: Circles and Shape Spaces

We survey some effects that singular strata may have in the positive curvature context of circles and shape spaces when conducting (semi-)intrinsic statistical analyses. Here, the analysis of data on a stratified space is based on statistical descriptors defined in a possibly different stratified space. E.g. in geodesic principal component analysis for shape spaces, shape data are described by generalized geodesics which naturally form a shape space of their own, different from the original one. In a general context, if the descriptors are obtained as generalized Fréchet means, under rather general circumstances, a strong law of large numbers is valid. If furthermore the descriptors are sufficiently well behaved, a classical central limit theorem can be adopted. One of the crucial conditions is that hitting of singular strata as well as of cut loci, if present, must be controlled. We review the statistical role of the cut locus of intrinsic means for circles as well as that of singular strata for shape spaces (occurring where the group action is degenerate) and conclude with an identification of potential research directions.

A new notion of angle between three points in a metric space

Abstract We give a new notion of angle in general metric spaces; more precisely, given three points p,x,q in a metric space (X,d), we introduce the notion of angle cone ∠ 𝐩𝐱𝐪 ${{\angle _{\mathbf {pxq}}}}$ as being an interval ∠ 𝐩𝐱𝐪 : = [ ∠ p x q - , ∠ p x q + ] ${{\angle _{\mathbf {pxq}}}:=[\angle ^-_{pxq},\angle ^+_{pxq}]}$ , where the quantities ∠ p x q ± ${\angle ^\pm _{pxq}}$ are defined in terms of the distance functions from p and q via a duality construction of differentials and gradients holding for locally Lipschitz functions on a general metric space. Our definition in the Euclidean plane gives the standard angle between three points and in a Riemannian manifold coincides with the usual angle between the geodesics if x is not in the cut locus of p or q. We show that in general the angle cone is not single-valued (even in case the metric space is a smooth Riemannian manifold, if x is in the cut locus of p or q), but if we endow the metric space with a positive Borel measure 𝔪 obtaining the metric measure space (X,d,𝔪), then under quite general assumptions (which include many fundamental examples as Riemannian manifolds, finite dimensional Alexandrov spaces with curvature bounded from below, Gromov–Hausdorff limits of Riemannian manifolds with Ricci curvature bounded from below, and normed spaces with strictly convex norm), fixed p,q ∈ X, the angle cone at x is single-valued for 𝔪-a.e. x ∈ X. We prove some basic properties of the angle cone (such as the invariance under homotheties of the space) and we analyze in detail the case (X,d,𝔪) is a measured-Gromov–Hausdorff limit of a sequence of Riemannian manifolds with Ricci curvature bounded from below, showing the consistency of our definition with a recent construction of Honda [“A weakly second order differential structure on rectifiable metric measure spaces”, preprint 2012].

Metrics with equatorial singularities on the sphere

Motivated by optimal control of affine systems stemming from mechanics, metrics on the two-sphere of revolution are considered; these metrics are Riemannian on each open hemisphere, whereas one term of the corresponding tensor becomes infinite on the equator. Length-minimizing curves are computed, and structure results on the cut and conjugate loci are given, extending those in Bonnard et al. (Ann Inst H Poincaré Anal Non Linéaire 26(4):1081–1098, 2009). These results rely on monotonicity and convexity properties of the quasi-period of the geodesics; such properties are studied on an example with elliptic transcendency. A suitable deformation of the round sphere allows to reinterpretate the equatorial singularity in terms of concentration of curvature and collapsing of the sphere onto a two-dimensional billiard.

Small time heat kernel asymptotics at the cut locus on surfaces of revolution

In this paper we investigate the small time heat kernel asymptotics on the cut locus on a class of surfaces of revolution, which are the simplest two-dimensional Riemannian manifolds different from the sphere with non-trivial cut-conjugate locus. We determine the degeneracy of the exponential map near a cut-conjugate point and present the consequences of this result to the small time heat kernel asymptotics at this point. These results give a first example where the minimal degeneration of the asymptotic expansion at the cut locus is attained.

Conjugate-cut loci and injectivity domains on two-spheres of revolution

In a recent article [B. Bonnard, J.-B. Caillau, R. Sinclair and M. Tanaka, Ann. Inst. Henri Poincare, Anal. Non Lineaire 26 (2009) 1081–1098], we relate the computation of the conjugate and cut loci of a family of metrics on two-spheres of revolution whose polar form is g = d ϕ 2 + m ( ϕ )d θ 2 to the period mapping of the ϕ -variable. One purpose of this article is to use this relation to evaluate the cut and conjugate loci for a family of metrics arising as a deformation of the round sphere and to determine the convexity properties of the injectivity domains of such metrics. These properties have applications in optimal control of space and quantum mechanics, and in optimal transport.

Normal forms and invariants for 2-dimensional almost-Riemannian structures

2-Dimensional almost-Riemannian structures are generalized Riemannian structures on surfaces for which a local orthonormal frame is given by a Lie bracket generating pair of vector fields that can become collinear. Generically, there are three types of points: Riemannian points where the two vector fields are linearly independent, Grushin points where the two vector fields are collinear but their Lie bracket is not, and tangency points where the two vector fields and their Lie bracket are collinear and the missing direction is obtained with one more bracket.In this paper we consider the problem of finding normal forms and functional invariants at each type of point. We also require that functional invariants are “complete” in the sense that they permit to recognize locally isometric structures.The problem happens to be equivalent to the one of finding a smooth canonical parameterized curve passing through the point and being transversal to the distribution.For Riemannian points such that the gradient of the Gaussian curvature K is different from zero, we use the level set of K as support of the parameterized curve. For Riemannian points such that the gradient of the curvature vanishes (and under additional generic conditions), we use a curve which is found by looking for crests and valleys of the curvature. For Grushin points we use the set where the vector fields are parallel.Tangency points are the most complicated to deal with. The cut locus from the tangency point is not a good candidate as canonical parameterized curve since it is known to be non-smooth. Thus, we analyse the cut locus from the singular set and we prove that it is not smooth either. A good candidate appears to be a curve which is found by looking for crests and valleys of the Gaussian curvature. We prove that the support of such a curve is uniquely determined and has a canonical parametrization.

Small-time heat kernel asymptotics at the sub-Riemannian cut locus

For a sub-Riemannian manifold provided with a smooth volume, we relate the small-time asymptotics of the heat kernel at a point y of the cut locus from $x$ with roughly "how much" $y$ is conjugate to $x$. This is done under the hypothesis that all minimizers connecting $x$ to $y$ are strongly normal, i.e. all pieces of the trajectory are not abnormal. Our result is a refinement of the one of Leandre $4t \log p_t(x, y) \to −d^2(x, y)$ for $t \to 0$, in which only the leading exponential term is detected. Our results are obtained by extending an idea of Molchanov from the Riemannian to the sub-Riemannian case, and some details we get appear to be new even in the Riemannian context. These results permit us to obtain properties of the sub-Riemannian distance starting from those of the heat kernel and vice versa. For the Grushin plane endowed with the Euclidean volume, we get the expansion $p_t(x, y) \sim t^{−5/4} \exp(−d^2(x, y)/4t)$ where $y$ is reached from a Riemannian point $x$ by a minimizing geodesic which is conjugate at $y$.

The Cut Loci, Conjugate Loci and Poles in a Complete Riemannian Manifold

Let M be a complete Riemannian manifold. We first prove that there exist at least two geodesics connecting p and every point in M if the tangent cut locus of \({p \in M}\) is not empty and does not meet its tangent conjugate locus. It follows from this that if M admits a pole and \({p \in M}\) is not a pole, then the tangent conjugate and tangent cut loci of p have a point in common. Here we say that a point q in M is a pole if the exponential map from the tangent space T q M at q onto M is a diffeomorphism. Using this result, we estimate the size of the set of all poles in M having a pole whose sectional curvature is pinched by those of two von Mangoldt surfaces of revolution, meaning that their Gaussian curvatures are monotone and nonincreasing with respect to the distances to their vertices.

Geodesics in infinite dimensional Stiefel and Grassmann manifolds

Let V be a separable Hilbert space, possibly infinite dimensional. Let St(p,V) be the Stiefel manifold of orthonormal frames of p vectors in V, and let Gr(p,V) be the Grassmann manifold of p-dimensional subspaces of V. We study the distance and the geodesics in these manifolds, by reducing the matter to the finite dimensional case. We then prove that any two points in those manifolds can be connected by a minimal geodesic, and characterize the cut locus.