Abstract
The relationship between the Schwarzian and Möbius transformations is well-known in dimensions 1 and 2. However, in dimension n ≥ 3, even the “proper” definition of the Schwarzian is not clear. In this paper, we introduce a “natural” generalization of the Schwarzian using the Clifford algebra and show that it vanishes exactly for Möbius transformations. The situation is simplest for non-singular transformations of the Euclidean space although the framework can be applied, with as light modification, to maps as general as immersions between any Riemannian manifolds.
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
References
L. V. Ahlfors, Möbius transformations and Clifford numbers, Differential Geometry and Complex Analysis, dedicated to H. E. Rauch, Springer-Verlag, 1985, pp. 65–73.
L. V. Ahlfors, Old and new in Мöbius groups, Ann. Acad. Sci. Fenn. 9 (1984), 93–105.
L. V. Ahlfors, Cross-ratios and Schwarzian derivatives in ℝ n, Complex Analysis J. Hersch and A. Huber, eds., articles dedicated to Albert Pfluger on the occasion of his 80th birthday, Вirkhäuser, 1988, 1–15.
K. Came, The Schwarzian derivative for conformal maps, J. Reine Angew. Math. 408 (1990), 10–33.
R. Fueter, Sur les groupes improprement discontinus, C. R. Acad. Sсi. Paris 182 (1926), 432–434.
R. Fueter, Über automorphe Funktionen in bezug auf Gruppen, die in der Ebene uneigentlich diskontinuierlich sind, J. Reine Angew. Math. 157 (1927), 66–78.
O. Kobayashi, and M. Wada, Circular geometry and the Schwarzian, preprint.
H. Maass, Automorphe Funktionen von mehreren Veränderlichen und Dirichletsche Reihen, Abh. Math. Sem. Univ. Hamburg 16 (1949), 53–104.
B. Osgood, and D. Stowe, The Schwarzian derivative and conformal mapping of Riemannian manifolds, Duke Math. J. 67 (1992), 57–99.
J. Ryan, Generalized Schwarzian derivatives for generalized fractional linear transformations, Ann. Polon. Math. LVII (1992), 29–44.
K. T. Vahlen, Über Bewegungen und complexe Zahlen, Math. Ann. 55 (1902), 585–593.
M. Wada, Conjugacy invariants of Möbius transformations, Complex Variables 15 (1990), 125–133.
M. Wada, A generalization of the Schwarzian via Clifford numbers, Ann. Acad. Sci. Fenn. 23 (1998), 453–460.
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2000 Springer Science+Business Media New York
About this chapter
Cite this chapter
Wada, M., Kobayashi, O. (2000). The Schwarzian and Möbius Transformations in Higher Dimensions. In: Ryan, J., Sprößig, W. (eds) Clifford Algebras and their Applications in Mathematical Physics. Progress in Physics, vol 19. Birkhäuser, Boston, MA. https://doi.org/10.1007/978-1-4612-1374-1_12
Download citation
DOI: https://doi.org/10.1007/978-1-4612-1374-1_12
Publisher Name: Birkhäuser, Boston, MA
Print ISBN: 978-1-4612-7119-2
Online ISBN: 978-1-4612-1374-1
eBook Packages: Springer Book Archive