Möbius structures and timed causal spaces on the circle

Buyalo, S.

doi:10.1090/spmj/1513

Möbius structures and timed causal spaces on the circle

Author: S. Buyalo
Translated by: THE AUTHOR
Original publication: Algebra i Analiz, tom 29 (2017), nomer 5.
Journal: St. Petersburg Math. J. 29 (2018), 715-747
MSC (2010): Primary 51B10, 53C50
DOI: https://doi.org/10.1090/spmj/1513
Published electronically: July 26, 2018
MathSciNet review: 3724637
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: A conjectural duality is discussed between hyperbolic spaces on one hand and spacetimes on the other, living on the opposite sides of the common absolute. This duality goes via Möbius structures on the absolute, and it is easily recognized in the classical case of symmetric rank one spaces. In the general case, no trace of such duality is known. As a first step in this direction, it is shown how numerous Möbius structures on the circle, including those that stem from hyperbolic spaces, give rise to 2-dimensional spacetimes, which are axiomatic versions of de Sitter 2-space, and vice versa. The paper has two Appendices, one of which is written by V. Schroeder.

References

Lars Andersson, Thierry Barbot, Riccardo Benedetti, Francesco Bonsante, William M. Goldman, François Labourie, Kevin P. Scannell, and Jean-Marc Schlenker, Notes on: “Lorentz spacetimes of constant curvature” [Geom. Dedicata 126 (2007), 3–45; MR2328921] by G. Mess, Geom. Dedicata 126 (2007), 47–70. MR 2328922, DOI https://doi.org/10.1007/s10711-007-9164-6
Martin R. Bridson and André Haefliger, Metric spaces of non-positive curvature, Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 319, Springer-Verlag, Berlin, 1999. MR 1744486
H. Busemann, Timelike spaces, Dissertationes Math. (Rozprawy Mat.) 53 (1967), 52. MR 220238
S. V. Buyalo, Möbius and sub-Möbius structures, Algebra i Analiz 28 (2016), no. 5, 1–20 (Russian, with Russian summary); English transl., St. Petersburg Math. J. 28 (2017), no. 5, 555–568. MR 3637585, DOI https://doi.org/10.1090/S1061-0022-2017-01463-6

---, Deformation of Möbius structures, preprint.

Sergei Buyalo and Viktor Schroeder, Elements of asymptotic geometry, EMS Monographs in Mathematics, European Mathematical Society (EMS), Zürich, 2007. MR 2327160

Sergei Buyalo and Viktor Schroeder, Möbius characterization of the boundary at infinity of rank one symmetric spaces, Geom. Dedicata 172 (2014), 1–45. MR 3253769, DOI https://doi.org/10.1007/s10711-013-9906-6

Sergei Buyalo and Viktor Schroeder, Incidence axioms for the boundary at infinity of complex hyperbolic spaces, Anal. Geom. Metr. Spaces 3 (2015), no. 1, 244–277. MR 3399374, DOI https://doi.org/10.1515/agms-2015-0015

---, Möbius structures on the circle. (in preparation)

Thomas Foertsch and Viktor Schroeder, Metric Möbius geometry and a characterization of spheres, Manuscripta Math. 140 (2013), no. 3-4, 613–620. MR 3019142, DOI https://doi.org/10.1007/s00229-012-0555-0

C. Gerhardt, Pinching estimates for dual flows in hyperbolic and de Sitter space, arXiv: 1510.03747 [math.DG] (2015)

G. Mess, Lorentz spacetimes of constant curvature, Geom. Dedicata 126 (2007), no. 1, 3–45. MR 2010a:53154

A. Papadopoulos and S. Yamada, Timelike Hilbert and Funk geometries, arXiv:1602.07072 [math.DG] (2016).

B. A. Rozenfel′d, Neevklidovy prostranstva, Izdat. “Nauka”, Moscow, 1969 (Russian). MR 0253136

M. Spradlin, A. Strominger, and A. Volovich, Les Houches lectures on de Sitter space, arXiv: hep-th/0110007 (2001).

H. Yu, Dual flows in hyperbolic space and de Sitter space, arXiv:1604.02369 [math.DG] (2016).