On Sub-Riemannian and Riemannian Structures on the Heisenberg Groups

We consider the left-invariant sub-Riemannian and Riemannian structures on the Heisenberg groups. A classification of these structures was found previously. In the present paper, we find (for each normalized structure) the isometry group, the exponential map, the totally geodesic subgroups, and the...

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Bibliographic Details
Published inJournal of dynamical and control systems Vol. 22; no. 3; pp. 563 - 594
Main Authors Biggs, Rory, Nagy, Péter T.
Format Journal Article
LanguageEnglish
Published New York Springer US 01.07.2016
Subjects
Calculus of Variations and Optimal Control; Optimization
Classification
Conjugates
Control
Control systems
Dynamical Systems
Dynamical Systems and Ergodic Theory
Geodesy
Mathematics
Mathematics and Statistics
Subgroups
Systems Theory
Vibration
Riemannian geometry
22E25
Heisenberg group
Sub-Riemannian geometry
Isometries
Geodesics
53C22
Totally geodesic subgroups
Conjugate locus
53C17
49J15
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Summary:We consider the left-invariant sub-Riemannian and Riemannian structures on the Heisenberg groups. A classification of these structures was found previously. In the present paper, we find (for each normalized structure) the isometry group, the exponential map, the totally geodesic subgroups, and the conjugate locus. Finally, we determine the minimizing geodesics from identity to any given endpoint. (Several of these points have been covered, to varying degrees, by other authors.)
Bibliography:ObjectType-Article-1
SourceType-Scholarly Journals-1
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content type line 23
ISSN:1079-2724
1573-8698
DOI:10.1007/s10883-016-9316-9