Optimal Control and Geodesics on Quadratic Matrix Lie Groups

The purpose of this paper is to extend the symmetric representation of the rigid body equations from the group SO (n) to other groups. These groups are matrix subgroups of the general linear group that are defined by a quadratic matrix identity. Their corresponding Lie algebras include several classical semisimple matrix Lie algebras. The approach is to start with an optimal control problem on these groups that generates geodesics for a left-invariant metric. Earlier work by Bloch, Crouch, Marsden, and Ratiu defines the symmetric representation of the rigid body equations, which is obtained by solving the same optimal control problem in the particular case of the Lie group SO (n). This paper generalizes this symmetric representation to a wider class of matrix groups satisfying a certain quadratic matrix identity. We consider the relationship between this symmetric representation of the generalized rigid body equations and the generalized rigid body equations themselves. A discretization of this symmetric representation is constructed making use of the symmetry, which in turn give rise to numerical algorithms to integrate the generalized rigid body equations for the given class of matrix Lie groups. Dedicated to Professor Arieh Iserles on the Occasion of his Sixtieth Birthday.     

Shapes of Spheres of Special Nonholonomic Left-Invariant Intrinsic Metrics on Some Lie Groups

The exact shapes of spheres are found for some nonholonomic left-invariant metrics on three-dimensional Lie groups: the Heisenberg group H, , and By easy geometric arguments, the task is reduced to solving isoperimetric problems for planes of constant Gaussian curvature.     

Homogeneous Geodesics of Three-Dimensional Unimodular Lorentzian Lie Groups

We consider three-dimensional unimodular Lie groups equipped with a Lorentzian metric and we determine, for all of them, their sets of homogeneous geodesics through a point. Dedicated to the memory of Professor Aldo Cossu Authors supported by funds of M.U.R.S.T., G.N.S.A.G.A. and the University of Lecce.     

Sub-Riemannian Geodesics on the 3-D Sphere

The unit sphere can be identified with the unitary group SU(2). Under this identification the unit sphere can be considered as a non-commutative Lie group. The commutation relations for the vector fields of the corresponding Lie algebra define a 2-step sub-Riemannian manifold. We study sub-Riemannian geodesics on this sub-Riemannian manifold making use of the Hamiltonian formalism and solving the corresponding Hamiltonian system. The first author is partially supported by a research grant from the United State Army Research Office and a Hong Kong RGC competitive earmarked research grant #600607. The second and the third authors have been supported by the grant of the Norwegian Research Council #177355/V30, and by the European Science Foundation Research Networking Programme HCAA.     

End-Point Equations and Regularity of Sub-Riemannian Geodesics

For a large class of equiregular sub-Riemannian manifolds, we show that length-minimizing curves have no corner-like singularities. Our first result is the reduction of the problem to the homogeneous, rank-2 case, by means of a nilpotent approximation. We also identify a suitable condition on the tangent Lie algebra implying existence of a horizontal basis of vector fields whose coefficients depend only on the first two coordinates x ₁, x ₂. Then, we cut the corner and lift the new curve to a horizontal one, obtaining a decrease of length as well as a perturbation of the end-point. In order to restore the end-point at a lower cost of length, we introduce a new iterative construction, which represents the main contribution of the paper. We also apply our results to some examples. Received: July 2006, Revision: October 2006, Accepted: November 2006     

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.     

Geodesics of Sasakian Metrics on Tensor Bundles

On the basis of the phase completion the notion of vertical and horizontal lifts of vector fields is defined in the tensor bundles over a Riemannian manifold. Such a tensor bundle is made into a manifold with a Riemannian structure of special type by endowing it with Sasakian metric. The components of the Levi-Civita and other metric connections with respect to Sasakian metrics on tensor bundles with respect to the adapted frame are presented. This having been done, it is shown that it is possible to study geodesics of Sasakian metrics dealing with geodesics of the base manifolds. Dedicated to the memory of Vladimir Vishnevskii (1929-2007)     

Minimal Geodesics on Manifolds with Discontinuous Metrics

The paper describes some qualitative properties of minimizerson a manifold M endowed with a discontinuous metric. The discontinuityoccurs on a hypersurface disconnecting M. Denote by ₁ and₂ the open subsets of M such that M\ =₁₂. Assume that and are endowed with metrics ·, · ₍₁₎ and ·,·₍₂₎, respectively, such that (i=1, 2) is convex or concave. The existence of a minimizerof the length functional on curves joining two given pointsof M is proved. The qualitative properties obtained allows therefraction law in a very general situation to be described.     

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...     

一类拟凸域的不变Khler度量与曲率

本文对Ｃｍ＋ｎ中的一类有界拟凸域给出了不变Ｋｈｌｅｒ度量及其曲率以及不变调和函数的显表达式     

Minimal Geodesics and Nilpotent Fundamental Groups

Hedlund [18] constructed Riemannian metrics on n-tori, n 3 for which minimal geodesics are very rare. In this paper we construct similar examples for every nilpotent fundamental group. These examples show that Bangert's existence results of minimal geodesics [4] are optimal for nilpotent fundamental groups.     

Sub-Riemannian distance on the Lie group SL(2)

We find the distances between arbitrary elements of the Lie group SL(2) for the left invariant sub-Riemannian metric also invariant with respect to the right shifts by elements of the Lie subgroup SO(2) ? SL(2), in other words, the invariant sub-Riemannian metric on the weakly symmetric space (SL(2) × SO(2))/ SO(2) of Selberg.     

几种特殊曲面上的测地线

具体刻画了柱面、锥面、旋转曲面上测地线的几何特征,所得结果一方面匡正了某些文献关于锥面上测地线的错误断言,一方面推广了现有文献关于旋转曲面上测地线几何性质的描述.     

The Geodesics of a Sub-Riemannian Metric on the Group of Semiaffine Transformations of the Euclidean Plane

Siberian Mathematical Journal - We obtain the parametrized representations of the geodesics of a left-invariant sub-Riemannian metric on the group of semiaffine transformations of the Euclidean...     

Metrics on 4-dimensional unimodular Lie groups

We classify left invariant metrics on the 4-dimensional, simply connected, unimodular Lie groups up to automorphism. When the corresponding Lie algebra is of type (R), this is equivalent to classifying the left invariant metrics up to isometry, but in general the classification up to automorphism is finer than that up to isometry. In the abelian case, all left invariant metrics are isometric. In the nilpotent case, the space of metrics can have dimension 1 or 3. In the solvable case, the dimension can be 2, 4, or 5. There are two non-solvable 4-dimensional unimodular groups, and the space of metrics has dimension 6 in both of these cases.     

一类步数为2(k+1)的次黎曼流形上测地线的研究

构造了一类步数为2(k+1)的次黎曼流形,给出其上连接原点和t轴上一点测地线的条数和相应测地线的长度,同时得到其中最短的测地线.     

Sub-Riemannian Balls on the Heisenberg Groups: An Invariant Volume

The Popp measure of a sub-Riemannian ball is calculated for a left-invariant sub-Riemannian structure on the Heisenberg group.     

关于完备李群与完备李代数

孟道骥等对完备李代数作了系统的研究并已获得很多基本和重要的结果。本文给出完备李群与完备李代数的某些关系。