Displaying similar documents to “Geodesics in the Heisenberg Group”

On the Heisenberg sub-Lorentzian metric on ℝ³

Marek Grochowski (2004)

Banach Center Publications

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In this paper we study properties of the Heisenberg sub-Lorentzian metric on ℝ³. We compute the conjugate locus of the origin, and prove that the sub-Lorentzian distance in this case is differentiable on some open set. We also prove the existence of regular non-Hamiltonian geodesics, a phenomenon which does not occur in the sub-Riemannian case.

Some properties of Carnot-Carathéodory balls in the Heisenberg group

Roberto Monti (2000)

Atti della Accademia Nazionale dei Lincei. Classe di Scienze Fisiche, Matematiche e Naturali. Rendiconti Lincei. Matematica e Applicazioni

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Using the exact representation of Carnot-Carathéodory balls in the Heisenberg group, we prove that: 1. ∇ H n ⁡ d ⁢ z , t = 1 in the classical sense for all z , t ∈ H n with z ≠ 0 , where d is the distance from the origin; 2. Metric balls are not optimal isoperimetric sets in the Heisenberg group.

Lorentzian geometry in the large

John Beem (1997)

Banach Center Publications

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Lorentzian geometry in the large has certain similarities and certain fundamental differences from Riemannian geometry in the large. The Morse index theory for timelike geodesics is quite similar to the corresponding theory for Riemannian manifolds. However, results on completeness for Lorentzian manifolds are quite different from the corresponding results for positive definite manifolds. A generalization of global hyperbolicity known as pseudoconvexity is described. It has important...