Relation between Euler’s elasticae and sub-Riemannian geodesics on SE(2)

In this note we describe a relation between Euler’s elasticae and sub-Riemannian geodesics on SE(2). Analyzing the Hamiltonian system of the Pontryagin maximum principle, we show that these two curves coincide only in the case when they are segments of a straight line.     

Left-invariant Riemannian problems on the groups of proper motions of hyperbolic plane and sphere

On the Lie groups PSL₂(?) and SO₃ we consider left-invariant Riemannian metrics with two equal eigenvalues. The global optimality of geodesics is investigated. We find the parametrization of geodesics, the cut locus and the equations for the cut time. When the third eigenvalue of a metric tends to the infinity the cut locus and the cut time converge to the cut locus and the cut time of the sub-Riemannian problem.     

Cut Locus in the Sub-Riemannian Problem on Engel Group

Doklady Mathematics - This paper considers the left-invariant sub-Riemannian problem on the Engel group. The stratification of the cut locus and the structure of optimal synthesis are described.     

The timelike cut locus and conjugate points in Lorentz 2-step nilpotent Lie groups

We investigate the timelike cut locus and the locus of conjugate points in Lorentz 2-step nilpotent Lie groups. For these groups with a timelike center, we give some criteria for the existence of conjugate points along timelike geodesics. We show for instance that a nonsingular timelike geodesic which is translated by an element of the group has a conjugate point. For those that we will call of GH-type, and for those which are globally hyperbolic with a timelike center and a one-dimensional derived subgroup, we show that if in addition the derived subgroup is spacelike, then nonsingular timelike geodesics maximize distance up to and including the first conjugate point. Other related results and corollaries are also obtained.Mathematics Subject Classification (2000): 53C50, 53C22, 22E25     

Conjugate points and closed geodesic arcs on convex surfaces

This paper discusses conjugate points on the geodesics of convex surfaces. It establishes their relationship with the cut locus. It shows the possibility of having many geodesics with conjugate points at very large distances from each other. It also shows that on many surfaces there are arbitrarily many closed geodesic arcs originating and ending at a common point. To achieve these goals, Baire category methods are employed.     

Geodesics and shortest arcs of a special sub-Riemannian metric on the Lie group <Emphasis Type="Italic">SL</Emphasis>(2)

The authors found geodesics, shortest arcs, cut loci, and conjugate sets for some leftinvariant sub-Riemannian metric on the Lie group SL(2) that is right-invariant relative to the Lie subgroup SO(2) ? SL(2) (in other words, for invariant sub-Riemannian metric on weakly symmetric space (SL(2) × SO(2))/SO(2)).     

Extremal Paths in the Nilpotent sub-Riemannian Problem on the Engel Group (Subcritical Case of Pendulum Oscillations)

We consider a left-invariant sub-Riemannian problem on an Engel group. This problem arises as a nilpotent approximation of nonholonomic systems in the four-dimensional space with twodimensional control (e.g., the system describing the motion of a mobile robot with a trailer). For the sub-Riemannian problem on the Engel group, abnormal extremal paths are calculated. The subsystem for conjugate variables of normal Hamiltonian system of Pontryagin’s maximum principle is reduced to the pendulum equation. Normal extremal paths corresponding to subcritical pendulum oscillations were calculated.     

Homogeneous Sub-Riemannian Geodesics on a Group of Motions of the Plane

Differential Equations - Homogeneous sub-Riemannian geodesics are described for the standard sub-Riemannian structure on the group $${\mathrm {SE}}(2)$$ of proper motions of the plane. It is shown...     

On a Step 2(k + 1) sub-Riemannian manifold

We compute the sub-Riemannian distance for a Step 2(k + 1) sub-Riemannian manifold from the origin to any given point. We characterize the number of sub-Riemannian geodesics between the origin and any other point.     

Conjugate points in nilpotent sub-Riemannian problem on the Engel group

The left-invariant sub-Riemannian problem on the Engel group is considered. This problem is very important as nilpotent approximation of nonholonomic systems in four-dimensional space with two-dimensional control, for instance of a system which describes motion of mobile robot with a trailer. We study local optimality of extremal trajectories and estimate conjugate time in this article.     

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 on a Certain Step 2 Sub-Riemannian Manifold

The goal of this paper is to consider a step 2 sub-Riemannian manifold where the connectivity bynormal geodesics between distant points fails.     

The sub‐Riemannian cut locus of H‐type groups

In the present paper we prove that the sub‐Riemannian cut locus at the origin of a wide class of nilpotent groups of step two, called H‐type groups, corresponds to the center of the group. We obtain this result by completely describing the sub‐Riemannian geodesics in the group, and using these to obtain three disjoint sets of points in the group determined by the number of geodesics joining them to the origin.     

On extensions of sub-Riemannian structures on Lie groups

We define the extension of a left-invariant sub-Riemannian structure in terms of an extension of the underlying Lie group and compatibility of the respective distributions and metrics. We show that geodesics of a structure can be lifted to geodesics of any extension of the structure. In the case of central extensions, we show that the normal geodesics of the minimal extension are the projection (in a sense) of the normal geodesics of any other compatible extension. Several illustrative examples are discussed.     

On the conjugate locus of pseudo-Riemannian manifolds

Let expm :T_mM → M be the exponential map of a Riemannian manifold M at a point m ∈ M. Warner proved that in any neighbourhood of a conjugate point in T_mM, the map exp_m is not injective. Moreover, he described the exponential map in a suitable coordinate system in a neighbourhood of a regular conjugate point, these points build an open dense set in the conjugate locus. We will investigate in the pseudo-Riemannian case such subsets, where the results of Warner generalize. For the definition of these subsets of the conjugate locus we use a bilinear form on ker(T_v exp_m), where v is a conjugate point, which will defined by the geodesic flow and the pseudo-Riemannian metric tensor.     

Sub-Riemannian geometry of the coefficients of univalent functions

We consider coefficient bodies Mn for univalent functions. Based on the Löwner-Kufarev parametric representation we get a partially integrable Hamiltonian system in which the first integrals are Kirillov's operators for a representation of the Virasoro algebra. Then Mn are defined as sub-Riemannian manifolds. Given a Lie-Poisson bracket they form a grading of subspaces with the first subspace as a bracket-generating distribution of complex dimension two. With this sub-Riemannian structure we construct a new Hamiltonian system to calculate regular geodesics which turn to be horizontal. Lagrangian formulation is also given in the particular case M₃.     

Aubry‐Mather theory and periodic solutions of the forced Burgers equation

Consider a Hamiltonian system with Hamiltonian of the form H(x, t, p) where H is convex in p and periodic in x, and t and x ∈ ℝ¹. It is well‐known that its smooth invariant curves correspond to smooth Z²‐periodic solutions of the PDE u_t + H(x, t, u)_x = 0. In this paper, we establish a connection between the Aubry‐Mather theory of invariant sets of the Hamiltonian system and Z²‐periodic weak solutions of this PDE by realizing the Aubry‐Mather sets as closed subsets of the graphs of these weak solutions. We show that the complement of the Aubry‐Mather set on the graph can be viewed as a subset of the generalized unstable manifold of the Aubry‐Mather set, defined in (2.24). The graph itself is a backward‐invariant set of the Hamiltonian system. The basic idea is to embed the globally minimizing orbits used in the Aubry‐Mather theory into the characteristic fields of the above PDE. This is done by making use of one‐ and two‐sided minimizers, a notion introduced in [12] and inspired by the work of Morse on geodesics of type A [26]. The asymptotic slope of the minimizers, also known as the rotation number, is given by the derivative of the homogenized Hamiltonian, defined in [21]. As an application, we prove that the Z²‐periodic weak solution of the above PDE with given irrational asymptotic slope is unique. A similar connection also exists in multidimensional problems with the convex Hamiltonian, except that in higher dimensions, two‐sided minimizers with a specified asymptotic slope may not exist. © 1999 John Wiley & Sons, Inc.     

Hypoelliptic Heat Kernel Over 3-Step Nilpotent Lie Groups

In this paper, we provide explicitly the connection between the hypoelliptic heat kernel for some 3-step sub-Riemannian manifolds and the quartic oscillator. We study the left-invariant sub-Riemannian structure on two nilpotent Lie groups, namely, the (2,3,4) group (called the Engel group) and the (2,3,5) group (called the Cartan group or the generalized Dido problem). Our main technique is noncommutative Fourier analysis, which permits us to transform the hypoelliptic heat equation into a one-dimensional heat equation with a quartic potential.     

Length minimizing geodesics and the length spectrum of Riemannian two-step nilmanifolds

In the first part of this article, we prove an explicit lower bound on the distance to the cut point of an arbitrary geodesic in a simply connected two-step nilpotent Lie group G with a lieft invariant metric. As a result, we obtaine a lower bound on the injectivity radius of a simply connected two-step nilpotent Lie group with a left invariant metric. We use this lower bound to determine the form of certain length minimizing geodesics from the identity to elements in the center of G. We also give an example of a two-step nilpotent Lie group G such that along most geodesics in this group, the cut point and the first conjugate point do not coincide. In the second part of this article, we examine the relation between the Laplace spectrum and the length spectrum on nilmanifolds by showing that a method developed by Gordon and Wilson for constructing families of isospectral two-step nilmanifolds necessarily yields manifolds with the same length spectrum. As a consequence, all known methods for constructing families of isospectral two-step nilmanifolds necessarily yield manifolds with the same length spectrum. In memory of Robert Brooks     

Universal methods of the search of normal geodesics on Lie groups with left-invariant sub-Riemannian metric

We obtain two versions of ODEs for the control function of normal geodesics for left-invariant sub-Riemannian metrics on Lie groups, involving only the structure of the Lie algebras of these groups. The first version is applicable to all Lie groups, while the second, to all matrix Lie groups; both versions are different invariant forms of the Hamiltonian system of the Pontryagin maximum principle for a left-invariant time-optimal problem on a Lie group. Basing on the first version, we find sufficient conditions for the normality of all geodesics of a given sub-Finslerian metric on a Lie group; in particular, we show that all three-dimensional Lie groups possess this property. The proofs use simple techniques of linear algebra.