SL(2,R)上的亚黎曼测地线

IF 1.3 3区数学 Q4 AUTOMATION & CONTROL SYSTEMS

Esaim-Control Optimisation and Calculus of Variations Pub Date : 2022-03-09 DOI:10.1051/cocv/2022068

D. D’Alessandro, Gunhee Cho

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引用次数: 1

摘要

我们明确地描述了在SL (2,R)上定义的椭圆型亚黎曼结构的长度最小化测地线。我们的方法使用对称约简，将问题转化为二维商空间上的黎曼问题，在二维商空间上测地线的投影可以很容易地可视化。我们利用李群上三维右不变亚黎曼结构的分类结果，精确地识别出了我们的结果所适用的自同构结构。

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Sub-Riemannian Geodesics on SL(2,R)

We explicitly describe the length minimizing geodesics for a sub-Riemannian structure of the elliptic type defined on SL (2,R). Our method uses a symmetry reduction which translates the problem into a Riemannian problem on a two dimensional quotient space, on which projections of geodesics can be easily visualized. As a byproduct, we obtain an alternative derivation of the characterization of the cut-locus obtained in the literature We use classification results for three dimensional right invariant sub-Riemannian structures on Lie groups to identify exactly automorphic structures on which our results apply.

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来源期刊

Esaim-Control Optimisation and Calculus of Variations Mathematics-Computational Mathematics

自引率

7.10%

发文量

期刊介绍： ESAIM: COCV strives to publish rapidly and efficiently papers and surveys in the areas of Control, Optimisation and Calculus of Variations. Articles may be theoretical, computational, or both, and they will cover contemporary subjects with impact in forefront technology, biosciences, materials science, computer vision, continuum physics, decision sciences and other allied disciplines. Targeted topics include: in control: modeling, controllability, optimal control, stabilization, control design, hybrid control, robustness analysis, numerical and computational methods for control, stochastic or deterministic, continuous or discrete control systems, finite-dimensional or infinite-dimensional control systems, geometric control, quantum control, game theory; in optimisation: mathematical programming, large scale systems, stochastic optimisation, combinatorial optimisation, shape optimisation, convex or nonsmooth optimisation, inverse problems, interior point methods, duality methods, numerical methods, convergence and complexity, global optimisation, optimisation and dynamical systems, optimal transport, machine learning, image or signal analysis; in calculus of variations: variational methods for differential equations and Hamiltonian systems, variational inequalities; semicontinuity and convergence, existence and regularity of minimizers and critical points of functionals, relaxation; geometric problems and the use and development of geometric measure theory tools; problems involving randomness; viscosity solutions; numerical methods; homogenization, multiscale and singular perturbation problems.