Abstract
In this article, we study the geometry of an infinite-dimensional hyperbolic space. We will consider the group of isometries of the Hilbert ball equipped with the Carathéodory metric and learn about some special subclasses of this group. We will also find some unitary equivalence condition and compute some cardinalities.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Anderson, J.W.: Hyperbolic geometry, 2nd edn. Springer, London (2005)
Baez, J.C.: The octonions. Bull. Amer. Math. Soc .(N.S.) 39(2), 145–205 (2002)
Bhunia, S., Singh, A.: z-classes in groups: a survey. arXiv preprint arXiv:2004.07529 (2020)
Cao, W., Parker, J.R., Wang, X.: On the classification of quaternionic Möbius transformations. Math. Proc. Cambridge Philos. Soc 137(2), 349–361 (2004)
Chen, S.S., Greenberg, L.: Hyperbolic spaces, in Contributions to analysis (a collection of papers dedicated to Lipman Bers), 49–87, Academic Press, New York
Conway, J.B.: A course in functional analysis, Graduate Texts in Mathematics, vol. 96. Springer-Verlag, New York (1985)
Franzoni, T., Vesentini, E.: Holomorphic maps and invariant distances. Notas de Matem\(\acute{a}\)tica [Mathematical Notes], 69. North-Holland Publishing Co., Amsterdam-New York, (1980). viii+226 pp. ISBN: 0-444-85436-3
Gongopadhyay, K., Kulkarni, R.S.: \(z\)-classes of isometries of the hyperbolic space. Conform. Geom. Dyn 13, 91–109 (2009)
Hájek, P., Johanis, M.: Smooth analysis in Banach spaces, vol. 19. De Gruyter Series in Nonlinear Analysis and Applications, De Gruyter, Berlin (2014)
Hayden, T.L., Suffridge, T.J.: Biholomorphic maps in Hilbert space have a fixed point. Pacific J. Math. 38, 419–422 (1971)
Kim, I., Parker, J.R.: Geometry of quaternionic hyperbolic manifolds. Math. Proc. Cambridge Philos. Soc. 135(2), 291–320 (2003)
Kobayashi, S.: Hyperbolic complex spaces, Grundlehren der Mathematischen Wissenschaften, vol. 318. Springer-Verlag, Berlin (1998)
Krantz, S.G.: Geometric function theory. Cornerstones, Birkhäuser Boston Inc., Boston, MA (2006)
Kulkarni, R.S.: Dynamical types and conjugacy classes of centralizers in groups. J. Ramanujan Math. Soc 22, 35–56 (2007)
Markham, S., Parker, J.R.: Jørgensen’s inequality for metric spaces with application to the octonions. Adv. Geom 7(1), 19–38 (2007)
Parker, J. R.: Notes on complex hyperbolic geometry. preprint (2003)
Rudin, W.: Real and complex analysis, 3rd edn. McGraw-Hill Book Co., New York (1987)
Stillwell, J.: Sources of hyperbolic geometry, History of Mathematics, vol. 10. American Mathematical Society, Providence, RI (1996)
Acknowledgements
The research is supported by Council of Scientific and Industrial Research, India (File No. 09/045(1668)/2019-EMR-I).
Author information
Authors and Affiliations
Corresponding author
Additional information
Communicated by Rosihan M. Ali.
Publisher's Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Rights and permissions
About this article
Cite this article
Mishra, M.M., Aggarwal, R. Group of Isometries of Hilbert Ball Equipped with the Carathéodory Metric. Bull. Malays. Math. Sci. Soc. 45, 1945–1954 (2022). https://doi.org/10.1007/s40840-022-01270-8
Received:
Revised:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s40840-022-01270-8