Conjugate Loci Research Articles

Randers metrics on two-spheres of revolution with simple cut locus

<abstract><p>In the present paper, we study the Randers metric on two-spheres of revolution in order to obtain new families of Finsler of Randers type metrics with simple cut locus. We determine the geodesics behavior, conjugate and cut loci of some families of Finsler metrics of Randers type whose navigation data is not a Killing field and without sectional or flag curvature restrictions. Several examples of Randers metrics whose cut locus is simple are shown.</p></abstract>

Zermelo navigation problems on surfaces of revolution and geometric optimal control

In this article, the historical study from Carathéodory-Zermelo about computing the quickest nautical path is generalized to Zermelo navigation problems on surfaces of revolution, in the frame of geometric optimal control. Using the Maximum Principle, we present two methods dedicated to analyzing the geodesic flow and to compute the conjugate and cut loci. We apply these calculations to investigate case studies related to applications in hydrodynamics, space mechanics and geometry.

Ct: control toolbox – Numerical tools and examples in optimal control*

Combining direct and indirect methods to have the best of both worlds is an efficient method to solve numerically optimal control problems. A direct solver will typically provide information on the structure of the optimal control, allowing an educated guess for indirect shooting. The control toolbox ct offers such possibilities and is presented on two examples. The first example has a bang-singular solution and is solved by chaining direct and indirect solvers. The second one consists in computing conjugate and cut loci on an ellipsoid of revolution, which is performed using a more advanced combination of indirect methods with differential continuation.

The Geometry of a Randers Rotational Surface with an Arbitrary Direction Wind

In the present paper, we study the global behaviour of geodesics of a Randers metric, defined on Finsler surfaces of revolution, obtained as the solution of the Zermelo’s navigation problem. Our wind is not necessarily a Killing field. We apply our findings to the case of the topological cylinder R×S1 and describe in detail the geodesics behaviour, the conjugate and cut loci.

Local contact sub-Finslerian geometry for maximum norms in dimension 3

<p style='text-indent:20px;'>The local geometry of sub-Finslerian structures in dimension 3 associated with a maximum norm is studied in the contact case. A normal form is given. The short extremals, the local switching, conjugate and cut loci, and the small spheres are described in the generic case.

The Structure of the Conjugate Locus of a General Point on Ellipsoids and Certain Liouville Manifolds

It is well known since Jacobi that the geodesic flow of the ellipsoid is "completely integrable", which means that the geodesic orbits are described in a certain explicit way. However, it does not directly indicate that any global behavior of the geodesics becomes easy to see. In fact, it happened quite recently that a proof for the statement "The conjugate locus of a general point in two-dimensional ellipsoid has just four cusps" in Jacobi's Vorlesungen \"uber dynamik appeared in the literature. In this paper, we consider Liouville manifolds, a certain class of Riemannian manifolds which contains ellipsoids. We solve the geodesic equations; investigate the behavior of the Jacobi fields, especially the positions of the zeros; and clarify the structure of the conjugate locus of a general point. In particular, we show that the singularities arising in the conjugate loci are only cuspidal edges and $D_4^+$ Lagrangian singularities, which would be the higher dimensional counterpart of Jacobi's statement.

Hamiltonian Boundary Value Problems, Conformal Symplectic Symmetries, and Conjugate Loci

In this paper we continue our study of bifurcations of solutions of boundary-value problems for symplectic maps arising as Hamiltonian diffeomorphisms. These have been shown to be connected to catastrophe theory via generating functions and ordinary and reversal phase space symmetries have been considered. Here we present a convenient, coordinate free framework to analyse separated Lagrangian boundary value problems which include classical Dirichlet, Neumann and Robin boundary value problems. The framework is then used to prove the existence of obstructions arising from conformal symplectic symmetries on the bifurcation behaviour of solutions to Hamiltonian boundary value problems. Under non-degeneracy conditions, a group action by conformal symplectic symmetries has the effect that the flow map cannot degenerate in a direction which is tangential to the action. This imposes restrictions on which singularities can occur in boundary value problems. Our results generalise classical results about conjugate loci on Riemannian manifolds to a large class of Hamiltonian boundary value problems with, for example, scaling symmetries.&#x0D;

The conjugate locus on convex surfaces

The conjugate locus of a point on a surface is the envelope of geodesics emanating radially from that point. In this paper we show that the conjugate loci of generic points on convex surfaces satisfy a simple relationship between the rotation index and the number of cusps. As a consequence we prove the ‘vierspitzensatz’: the conjugate locus of a generic point on a convex surface must have at least four cusps. Along the way we prove certain results about evolutes in the plane and we extend the discussion to the existence of ‘smooth loops’ and geodesic curvature.

Central conjugate locus of 2-step nilpotent Lie groups

We compute the conjugate locus with multiplicities of a geodesic that lies in the center of a 2-step nilpotent Lie group N with a left invariant metric. We compare the isometry types of two such nilpotent groups whose conjugate loci for central geodesics are the same in a suitable sense. This comparison is related to work of Gordon and Wilson (J Differ Geom 47(3):504–529, 1997).

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.

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.

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.

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.

Cut loci and conjugate loci on Liouville surfaces

In the earlier paper (Itoh and Kiyohara, Manuscr Math 114:247–264, 2004), we showed that the cut locus of a general point on two-dimensional ellipsoid is a segment of a curvature line and proved “Jacobi’s last geometric statement” on the singularities of the conjugate locus. In the present paper, we show that a wider class of Liouville surfaces possess such simple cut loci and conjugate loci. The results include the determination of cut loci and the set of poles on two-sheeted hyperboloids and elliptic paraboloids.

Sub-Riemannian and sub-Lorentzian geometry on $SU(1,1)$ and on its universal cover

We study sub-Riemannian and sub-Lorentzian geometry on the Lie group $SU(1,1)$ and on its universal cover SU~(1,1). In the sub-Riemannian case we find the distance function and completely describe sub-Riemannian geodesics on both $SU(1,1)$ and SU~(1,1),connecting two fixed points. In particular, we prove that there is a strong connection between the conjugate loci and the number of geodesics. In the sub-Lorentzian case, we describe the geodesics connecting two points on SU~(1,1), and compare them with Lorentzian ones. It turns out that the reachable sets for Lorentzian and sub-Lorentzian normal geodesics intersect but are not included one to the other. A description ofthe timelike future is obtained and compared in the Lorentzian and sub-Lorentzain cases.

Normal forms and reachable sets for analytic martinet sub-Lorentzian structures of hamiltonian type

In this paper, we construct normal forms for sub-Lorentzian structures (H, g), where H is an analytic Martinet-type distribution on ?3 and g is an analytic Lorentzian metric on H, under the assumption that abnormal curves foliating the Martinet surface for H are timelike Hamiltonian geodesics. As an application, we compute reachable sets from a point for such structures. It turns out that such sets are described by four analytic functions and, consequently, they are semi-analytic. We also compute future null conjugate and cut loci, and the image under the exponential mapping for above-mentioned structures.

Curvature and conjugate points in Lorentz symmetric spaces

We give some relations between conjugate points and curvature in a locally symmetric Lorentzian manifold. In the compact case, we show that the sectional curvature of timelike planes is non positive, and the lightlike sectional curvature of null planes is non negative. We also compute the lightlike conjugate loci of Cahen-Wallach manifolds, which are an important family of symmetric Lorentzian spaces.

Regularity properties of the distance functions to conjugate and cut loci for viscosity solutions of Hamilton-Jacobi equations and applications in Riemannian geometry

Given a continuous viscosity solution of a Dirichlet-type Hamilton-Jacobi equation, we show that the distance function to the conjugate locus which is associated to this problem is locally semiconcave on its domain. It allows us to provide a simple proof of the fact that the distance function to the cut locus associated to this problem is locally Lipschitz on its domain. This result, which was already an improvement of a previous one by Itoh and Tanaka [Trans. Amer. Math. Soc. 353 (2001) 21–40], is due to Li and Nirenberg [Comm. Pure Appl. Math. 58 (2005) 85–146]. Finally, we give applications of our results in Riemannian geometry. Namely, we show that the distance function to the conjugate locus on a Riemannian manifold is locally semiconcave. Then, we show that if a Riemannian manifold is a C 4 -deformation of the round sphere, then all its tangent nonfocal domains are strictly uniformly convex.

Conjugate and cut loci of a two-sphere of revolution with application to optimal control

The objective of this article is to present a sharp result to determine when the cut locus for a class of metrics on a two-sphere of revolution is reduced to a single branch. This work is motivated by optimal control problems in space and quantum dynamics and gives global optimal results in orbital transfer and for Lindblad equations in quantum control.

Invariant Carnot–Caratheodory Metrics on $S^3$, $SO(3)$, $SL(2)$, and Lens Spaces

In this paper we study the Carnot–Caratheodory metrics on $SU(2)\simeq S^3$, $SO(3)$, and $SL(2)$ induced by their Cartan decomposition and by the Killing form. Besides computing explicitly geodesics and conjugate loci, we compute the cut loci (globally), and we give the expression of the Carnot–Caratheodory distance as the inverse of an elementary function. We then prove that the metric given on $SU(2)$ projects on the so-called lens spaces $L(p,q)$. Also for lens spaces, we compute the cut loci (globally). For $SU(2)$ the cut locus is a maximal circle without one point. In all other cases the cut locus is a stratified set. To our knowledge, this is the first explicit computation of the whole cut locus in sub-Riemannian geometry, except for the trivial case of the Heisenberg group.