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Nonholonomic Lorentzian geometry on some \(\mathbb H\)-type groups. (English) Zbl 1178.53070

The sub-Riemannian manifolds and the geometry introduced by bracket generating distributions of smoothly varying \(k\)-plains, have applications in control theory, quantum physics and other areas. In the present paper, the authors study some examples of \(H\)-type groups furnished with a Lorentzian metric. More exactly there are studied the Heisenberg group and the quaternion \(H\)-type group, endowed with the Lorentzian metric defining a causal character of the manifold under consideration. Also there are given a description of the set reachable by time-like future directed curves.

MSC:

53C50 Global differential geometry of Lorentz manifolds, manifolds with indefinite metrics
83A05 Special relativity
53C17 Sub-Riemannian geometry
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