Möbius invariant quaternion geometry
HTML articles powered by AMS MathViewer
- by R. Michael Porter
- Conform. Geom. Dyn. 2 (1998), 89-106
- DOI: https://doi.org/10.1090/S1088-4173-98-00032-0
- Published electronically: October 14, 1998
- PDF | Request permission
Abstract:
A covariant derivative is defined on the one point compactification of the quaternions, respecting the natural action of quaternionic Möbius transformations. The self-parallel curves (analogues of geodesics) in this geometry are the loxodromes. Contrasts between quaternionic and complex Möbius geometries are noted.References
- Lars V. Ahlfors, Old and new in Möbius groups, Ann. Acad. Sci. Fenn. Ser. A I Math. 9 (1984), 93–105. MR 752394, DOI 10.5186/aasfm.1984.0901
- Lars V. Ahlfors, Clifford numbers and Möbius transformations in $\textbf {R}^n$, Clifford algebras and their applications in mathematical physics (Canterbury, 1985) NATO Adv. Sci. Inst. Ser. C: Math. Phys. Sci., vol. 183, Reidel, Dordrecht, 1986, pp. 167–175. MR 863437
- Helmer Aslaksen, Quaternionic determinants, Math. Intelligencer 18 (1996), no. 3, 57–65. MR 1412993, DOI 10.1007/BF03024312
- Alan F. Beardon, The geometry of discrete groups, Graduate Texts in Mathematics, vol. 91, Springer-Verlag, New York, 1983. MR 698777, DOI 10.1007/978-1-4612-1146-4
- F. Brackx, Richard Delanghe, and F. Sommen, Clifford analysis, Research Notes in Mathematics, vol. 76, Pitman (Advanced Publishing Program), Boston, MA, 1982. MR 697564
- J. Cnops, Spherical geometry and Möbius transformations, Clifford algebras and their applications in mathematical physics (Deinze, 1993) Fund. Theories Phys., vol. 55, Kluwer Acad. Publ., Dordrecht, 1993, pp. 75–84. MR 1266855
- Peter Greenberg and Michael Porter, A generalization of geodesic flow, Proceedings of the XIXth National Congress of the Mexican Mathematical Society, Vol. 2 (Spanish) (Guadalajara, 1986) Aportaciones Mat. Comun., vol. 4, Soc. Mat. Mexicana, México, 1987, pp. 197–204. MR 990680
- R. C. Gunning, On uniformization of complex manifolds: the role of connections, Mathematical Notes, vol. 22, Princeton University Press, Princeton, N.J., 1978. MR 505691
- F. Reese Harvey, Spinors and calibrations, Perspectives in Mathematics, vol. 9, Academic Press, Inc., Boston, MA, 1990. MR 1045637
- R. Heidrich and G. Jank, On the iteration of quaternionic Moebius transformations, Complex Variables Theory Appl. 29 (1996), no. 4, 313–318. MR 1390616, DOI 10.1080/17476939608814899
- Yves Hellegouarch, Quaternionic homographies: application to Ford hyperspheres, C. R. Math. Rep. Acad. Sci. Canada 11 (1989), no. 5, 171–176. MR 1010923
- Erich Kähler, Die Poincaré-Gruppe, Rend. Sem. Mat. Fis. Milano 53 (1983), 359–390 (1986) (German, with English and Italian summaries). MR 858510, DOI 10.1007/BF02924908
- Koecher, M. and Remmert, R., Hamilton’s quaternions, in J. H. Ewing, ed., Numbers, Graduate Texts in Mathematics 123, Springer–Verlag, New York (1991), 189–220.
- Irwin Kra, Quadratic differentials, Rev. Roumaine Math. Pures Appl. 39 (1994), no. 8, 751–787. MR 1319117
- Irwin Kra and Bernard Maskit, Remarks on projective structures, Riemann surfaces and related topics: Proceedings of the 1978 Stony Brook Conference (State Univ. New York, Stony Brook, N.Y., 1978) Ann. of Math. Stud., vol. 97, Princeton Univ. Press, Princeton, N.J., 1981, pp. 343–359. MR 624824
- Pertti Lounesto and Arthur Springer, Möbius transformations and Clifford algebras of Euclidean and anti-Euclidean spaces, Deformations of mathematical structures (Łódź/Lublin, 1985/87) Kluwer Acad. Publ., Dordrecht, 1989, pp. 79–90. MR 987727, DOI 10.1007/978-94-009-2643-1_{8}
- Pertti Lounesto and Esko Latvamaa, Conformal transformations and Clifford algebras, Proc. Amer. Math. Soc. 79 (1980), no. 4, 533–538. MR 572296, DOI 10.1090/S0002-9939-1980-0572296-4
- Sergio Sispanov, Generalización del teorema de Laguerre, Bol. Mat. 12 (1939), 113–117 (Spanish). MR 3
- R. Michael Porter, Differential invariants in Möbius geometry, J. Natur. Geom. 3 (1993), no. 2, 97–123. MR 1205080
- R. Michael Porter, Quaternionic linear and quadratic equations, J. Natur. Geom. 11 (1997), no. 2, 101–106. MR 1432599
- —, Quaternionic Möbius transformations and loxodromes, Complex Variables, Theory and Applications (to appear).
- John Ryan, Generalized Schwarzian derivatives for generalized fractional linear transformations, Ann. Polon. Math. 57 (1992), no. 1, 29–44. MR 1176799, DOI 10.4064/ap-57-1-29-44
- Shub, M., Global Stability of Dynamical Systems, Springer–Verlag, New York (1987).
- William P. Thurston, Zippers and univalent functions, The Bieberbach conjecture (West Lafayette, Ind., 1985) Math. Surveys Monogr., vol. 21, Amer. Math. Soc., Providence, RI, 1986, pp. 185–197. MR 875241, DOI 10.1090/surv/021/15
- V. S. Varadarajan, Lie groups, Lie algebras, and their representations, Graduate Texts in Mathematics, vol. 102, Springer-Verlag, New York, 1984. Reprint of the 1974 edition. MR 746308, DOI 10.1007/978-1-4612-1126-6
- Masaaki Wada, Conjugacy invariants of Möbius transformations, Complex Variables Theory Appl. 15 (1990), no. 2, 125–133. MR 1058518, DOI 10.1080/17476939008814442
Bibliographic Information
- R. Michael Porter
- Affiliation: Departamento de Matemáticas, Centro de Investigación y de Estudios Avanzados del I.P.N., Apdo. Postal 14-740, 07000 México D. F., Mexico
- Email: mike@math.cinvestav.mx
- Received by editor(s): January 29, 1998
- Received by editor(s) in revised form: August 25, 1998
- Published electronically: October 14, 1998
- Additional Notes: Partially supported by CONACyT grant 211085-5-2585P-E
- © Copyright 1998 American Mathematical Society
- Journal: Conform. Geom. Dyn. 2 (1998), 89-106
- MSC (1991): Primary 53A55; Secondary 53B10, 15A66, 51N30, 20G20
- DOI: https://doi.org/10.1090/S1088-4173-98-00032-0
- MathSciNet review: 1649091