Original title: Sub-Riemannovy geometrie na ortogonálních Lieových grupách
Translated title: Sub-Riemannian geometries on orthogonal Lie groups
Authors: Nguyen, Duy Long ; Slovák, Jan (referee) ; Návrat, Aleš (advisor)
Document type: Master’s theses
Year: 2026
Language: eng
Publisher: Vysoké učení technické v Brně. Fakulta strojního inženýrství
Abstract: This thesis studies sub-Riemannian geometry on orthogonal Lie groups, with a focus on the groups SO(3), SO(2,1), and SO(3,1). After developing the necessary foundations in differential geometry, sub-Riemannian structures, and Hamiltonian mechanics on the cotangent bundle, we exploit the left-trivialization of T*G to reduce the geodesic problem to the Lie-Poisson equations on the dual of the Lie algebra. For semisimple groups admitting a d+s decomposition, the vertical dynamics decouple and geodesics admit a closed-form reconstruction via the matrix exponential. We carry out this programme explicitly for SO(3), SO(2,1), and SO(3,1), comparing eigenvalue regimes, periodicity properties, and the role of the Killing form's definiteness. In particular, we identify abnormal extremals on SO(3,1) — a case not covered in the standard literature — and compute sub-Riemannian distances numerically.
Keywords: abnormal extremals; differential geometry; geodesics; Hamiltonian mechanics; Killing form; Lie groups; orthogonal groups; Pontryagin maximum principle; sub-Riemannian geometry; abnormal extremals; differential geometry; geodesics; Hamiltonian mechanics; Killing form; Lie groups; orthogonal groups; Pontryagin maximum principle; sub-Riemannian geometry

Institution: Brno University of Technology (web)
Document availability information: Fulltext is available in the Brno University of Technology Digital Library.
Original record: http://hdl.handle.net/11012/259851

Permalink: http://www.nusl.cz/ntk/nusl-775452


The record appears in these collections:
Universities and colleges > Public universities > Brno University of Technology
Academic theses (ETDs) > Master’s theses