×

Cut locus of a left invariant Riemannian metric on \(\mathrm{SO}_3\) in the axisymmetric case. (English) Zbl 1352.53044

Summary: We consider a left invariant Riemannian metric on \(\mathrm{SO}_3\) with two equal eigenvalues. We find the cut locus and the equation for the cut time. We find the diameter of such metric and describe the set of all most distant points from the identity. Also we prove that the cut locus and the cut time converge to the cut locus and the cut time in the sub-Riemannian problem on \(\mathrm{SO}_3\) as one of the metric eigenvalues tends to infinity.

MSC:

53C30 Differential geometry of homogeneous manifolds
53C20 Global Riemannian geometry, including pinching
53C17 Sub-Riemannian geometry
53C22 Geodesics in global differential geometry
49J15 Existence theories for optimal control problems involving ordinary differential equations
PDFBibTeX XMLCite
Full Text: DOI arXiv

References:

[1] Landau, L. D.; Lifshitz, E. M., Mechanics, Vol. 1 (1976), Butterworth-Heinemann · Zbl 0081.22207
[2] Bates, L.; Fassò, F., The conjugate locus for the Euler top. I. The axisymmetric case, Int. Math. Forum, 2, 43, 2109-2139 (2007) · Zbl 1151.53348
[3] Sakai, T., Cut loci of Berger’s sphere, Hokkaido Math. J., 10, 143-155 (1981) · Zbl 0469.53041
[4] Berger, M., A Panoramic View of Riemannian Geometry (2002), Springer
[5] Sachkov, Yu. L., Complete description of the Maxwell strata in the generalized Dido problem, Sb. Math., 197, 6, 901-950 (2006) · Zbl 1148.53022
[6] Sachkov, Yu. L., Maxwell strata in the Euler elastic problem, J. Dyn. Control Syst., 14, 2, 169-234 (2008) · Zbl 1203.49004
[7] Berestovskii, V. N.; Zubareva, I. A., Shapes of spheres of special nonholonomic left-invariant intrinsic metrics on some Lie groups, Sib. Math. J., 42, 4, 613-628 (2001) · Zbl 0996.53036
[8] Boscain, U.; Rossi, F., Invariant Carnot-Caratheodory metrics on \(S^3, S O(3), S L(2)\) and Lens Spaces, SIAM Journal on Control and Optimization, 47, 1851-1878 (2008) · Zbl 1170.53016
[9] Pontryagin, L. S.; Boltyanskii, V. G.; Gamkrelidze, R. V.; Mishchenko, E. F., The Mathematical Theory of Optimal Processes (1962), Interscience Publishers John Wiley & Sons, Inc.: Interscience Publishers John Wiley & Sons, Inc. New York, London · Zbl 0102.32001
[10] Agrachev, A. A.; Sachkov, Yu. L., Control Theory from The Geometric Viewpoint (2004), Springer-Verlag: Springer-Verlag Berlin · Zbl 1062.93001
[11] Krantz, S. G.; Parks, H. R., The Implicit Function Theorem: History, Theory and Applications (2001), Birkauser
[12] Agrachev, A. A.; Barilari, D., Sub-Riemannian structures on 3D Lie groups, J. Dyn. Control Syst., 18, 21-44 (2012) · Zbl 1244.53039
[13] Jurdjevic, V., Optimal control, geometry and mechanics, (Bailleu, J.; Willems, J. C., Mathematical Control Theory (1999), Springer), 227-267 · Zbl 1047.93506
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.