Abstract
The bounded symmetric spaces naturally associated with the Poincaré and Beltrami-Klein models of hyperbolic geometry on the open unit ball B in \({\mathbb{R}^n}\) and with the automorphism group of biholomorphic maps on the open ball in \({\mathbb{C}^n}\) give rise by a standard construction to specialized loop structures (nonassociative groups), which we use to define canonical metrics, called rapidity metrics. We show that this rapidity metric agrees with the classical Poincaré metric resp. the Cayley-Klein metric resp. the Bergman metric. We introduce the Lorentz boost of vectors in B, which turns out to be a loop isomorphism. It induces a similarity of metrics between the rapidity metric of the Einstein or Möbius loop and the trace metric on positive definite matrices restricted to the Lorentz boosts.
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References
Bhatia R., Holbrook J.: Riemannian geometry and matrix geometric means. Linear Algebra Appl. 413, 594–618 (2006)
Friedman Y., Scarr T.: Physical Applications of Homogeneous Balls. Birkhäuser, Basel (2005)
Glauberman G.: On loops of odd order I. J. Algebra 1, 374–396 (1964)
Glauberman G.: On loops of odd order II. J. Algebra 8, 393–414 (1968)
Karzel H.: Recent developments on absolute geometries and algebraization by K-loops. Discrete Math. 208(209), 387–409 (1999)
Kiechle H.: Theory of K-Loops. Lecture Notes in Mathematics, vol. 1778. Springer, Berlin (2002)
Kreuzer A.: Beispiele endlicher und unendlicher K-loops. Results Math. 23, 355–362 (1993)
Lawson J., Lim Y.: Symmetric sets with midpoints and algebraically equivalent theories. Results Math. 46, 37–56 (2004)
Loos, O.: Symmetric Spaces I: General Theory. Benjamin, New York
Nagy P., Strambach K.: Loops in Group Theory and Lie Theory. Expositions in Math., vol. 35. de Gruyter, Berlin (2002)
Ratcliffe J.: Foundations of Hyperbolic Manifolds. Springer, Berlin (2005)
Rudin W.: Function Theory in the Unit Ball of \({\mathbb{C}^n}\) . Springer, Berlin (1980)
Sabinin L.V., Sabinina L.L., Sbitneva L.V.: On the notion of a gyrogroup. Aequ. Math. 56, 11–17 (1998)
Ungar A.A.: Analytic Hyperbolic Geometry and Albert Einstein’s Special Theory of Relativity. World Scientific, Singapore (2008)
Zhu K.: Spaces of Holomorphic Functions on the Unit Ball. Graduate Texts in Math., vol. 226. Springer, New York (2005)
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Kim, S., Lawson, J. Unit Balls, Lorentz Boosts, and Hyperbolic Geometry. Results. Math. 63, 1225–1242 (2013). https://doi.org/10.1007/s00025-012-0265-7
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DOI: https://doi.org/10.1007/s00025-012-0265-7