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Jupp and Kent’s cubics in Lie groups. (English) Zbl 1314.51001

Summary: We study a generalization of the cubic polynomial to Riemannian manifolds and other affine connection spaces, modifying the differential equation \(x^{(4)} = 0\) by replacing higher derivatives with covariant derivatives. In matrix groups, with two particular choices of a left-invariant connection, we can convert the equation into a system of first order linear differential equations. We give asymptotics in a generic case in \(\operatorname{GL}(n)\) and in the n-sphere.{
©2011 American Institute of Physics}

MSC:

51A15 Linear incidence geometric structures with parallelism
53C20 Global Riemannian geometry, including pinching
53B05 Linear and affine connections
22E15 General properties and structure of real Lie groups
20G20 Linear algebraic groups over the reals, the complexes, the quaternions
53A04 Curves in Euclidean and related spaces
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[1] Camarinha, M.; Leite, F.; Crouch, P., Splines of class \documentclass[12pt]{minimal}\( \begin{document}C^k\end{document}\) on non-Euclidean spaces, IMA J. Math. Control Inf., 12, 4, 399 (1995) · Zbl 0860.58013 · doi:10.1093/imamci/12.4.399
[2] Coddington, E. A.; Levinson, N., Theory of Ordinary Differential Equations (1955) · Zbl 0064.33002
[3] Gabriel, S.; Kajiya, J., State of the Art in Image Synthesis, 11, 1-14 (1985)
[4] Helgason, S., Differential Geometry, Lie Groups, and Symmetric spaces, 34 (2001) · Zbl 0993.53002
[5] Jupp, P. E.; Kent, J. E., Fitting smooth paths to spherical data, J. Roy. Statist. Soc. Ser. C, 36, 1, 34 (1987) · Zbl 0613.62086
[6] Krakowski, K. L. (2002)
[7] Larsen, J. C., The Jacobi map, J. Geom. Phys., 20, 1, 54 (1996) · Zbl 0877.53028 · doi:10.1016/0393-0440(95)00045-3
[8] Noakes, L., Null cubics and Lie quadratics, J. Math. Phys., 44, 3, 1436 (2003) · Zbl 1062.53007 · doi:10.1063/1.1537461
[9] Noakes, L., Non-null Lie quadratics in \documentclass[12pt]{minimal}\( \begin{document}E^3\end{document} \), J. Math. Phys., 45, 11, 4334 (2004) · Zbl 1064.53004 · doi:10.1063/1.1803609
[10] Noakes, L., Lax constraints in semisimple Lie groups, Q. J. Math., 57, 4, 527 (2006) · Zbl 1143.37042 · doi:10.1093/qmath/hal002
[11] Noakes, L.; Heinzinger, G.; Paden, B., Cubic splines on curved spaces, IMA J. Math. Control Inf., 6, 4, 465 (1989) · Zbl 0698.58018 · doi:10.1093/imamci/6.4.465
[12] Pauley, M., Parabolic curves in Lie groups, J. Math. Phys., 51, 5, 053526 (2010) · Zbl 1310.41006 · doi:10.1063/1.3427421
[13] Popiel, T., Higher order geodesics in Lie groups, Math. Control, Signals, Syst., 19, 3, 235 (2007) · Zbl 1163.53028 · doi:10.1007/s00498-007-0012-x
[14] 14.Postnikov, M. M., Geometry VI, Encyclopaedia of Mathematical Sciences Vol. 91 (Springer-Verlag, Berlin, 2001) [Vakhrameev, S. A., Riemannian Geometry (Faktorial, Moscow, 1998) (in Russian)].
[15] Wasow, W., Asymptotic expansions for ordinary differential equations, XIV (1965) · Zbl 0169.10903
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