Complex Analysis and Operator Theory

, Volume 4, Issue 3, pp 589–618 | Cite as

Geodesics on \({\mathbb{H}}\) -type Quaternion Groups with Sub-Lorentzian Metric and Their Physical Interpretation



We study the existence and cardinality of normal geodesics of different causal types on \({\mathbb {H}(eisenberg)}\) -type quaternion group equipped with the sub-Lorentzian metric. We present explicit formulas for geodesics and describe reachable sets by geodesics of different causal character. We compare results with the sub-Riemannian quaternion group and with the sub-Lorentzian Heisenberg group, showing that there are similarities and distinctions. We show that the geodesics on \({\mathbb{H}}\) -type quaternion groups with the sub-Lorentzian metric satisfy the equations describing the motion of a relativistic particle in a constant homogeneous electromagnetic field.


Quaternion H-type group Sub-Lorentzian metric Electromagnetic field Special relativity 

Mathematics Subject Classification (2000)

53C50 53B30 53C17 


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© Springer Basel AG 2010

Authors and Affiliations

  1. 1.Department of MathematicsUniversity of BergenBergenNorway

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