The Quaternionic Hardy Space and the Geometry of the Unit Ball

Giulia Sarfatti


The quaternionic Hardy space of slice regular functions H2(B) is a reproducing kernel Hilbert space. In this note we see how this property can be exploited to construct a Riemannian metric on the quaternionic unit ball B and we study the geometry arising from this construction. We also show that, in contrast with the example of the Poincaré metric on the complex unit disc, no Riemannian metric on B is invariant with respect to all slice regular bijective self maps of B.


Hardy space on the quaternionic ball; functions of a quaternionic variable; invariant Riemannian metric

Full Text:

PDF (English)


N. Arcozzi, R. Rochberg, E. Sawyer, B. D. Wick, Distance functions for reproducing kernel Hilbert spaces, Function spaces in modern analysis, 25–53, Contemp. Math., 547, Amer. Math. Soc., Providence, RI, 2011.

N. Arcozzi, G. Sarfatti, Invariant metrics for the quaternionic Hardy space, J. Geom. Anal. 25 (2015), 2028-2059.

C. Bisi, G. Gentili, Möbius transformations and the Poincaré distance in the quaternionic setting, Indiana Univ. Math. J., 58 (2009), 2729–2764.

C. Bisi, C. Stoppato, Regular vs. classical Möbius transformations of the quaternionic unit ball, Advances in hypercomplex analysis, Springer INdAM Ser. 1, Springer, Milan, 2013, 1–13.

C. Bisi, C. Stoppato, The Schwarz-Pick Lemma for slice regular functions, Indiana Univ. Math. J., 61 (2012), 297-317.

C. de Fabritiis, G. Gentili, G. Sarfatti, Quaternionic Hardy spaces, Preprint,, (2013).

M. P. do Carmo, Differential geometry of curves and surfaces, Prentice-Hall, Inc., Englewood Cliffs, N.J., 1976.

R. Fueter, Die Funktionentheorie der Differentialgleichungen ∆u = 0 und ∆∆u = 0 mit vier reellen Variablen, Comment. Math. Helv. 7 (1934),307-330.

R. Fueter, Über Hartogs’schen Satz, Comm. Math. Helv. 12 (1939), 75-80.

G. Gentili, C. Stoppato, D. C. Struppa, Regular functions of a quaternionic variable, Springer Monographs in Mathematics, Springer, Berlin-Heidelberg, 2013.

G. Gentili, D. C. Struppa, A new approach to Cullen-regular functions of a quaternionic variable, C. R. Math. Acad. Sci. Paris, 342 (2006), 741–744.

G. Gentili, D. C. Struppa, A new theory of regular functions of a quaternionic variable, Adv. Math., 216 (2007), 279–301.

R. Ghiloni, V. Moretti, A. Perotti, Continuous slice functional calculus in quaternionic Hilbert spaces, Rev. Math. Phys. 25 (2013).

M. Gromov, Metric structures for Riemannian and non-Riemannian spaces, Based on the 1981 French original, translated from the French by Sean Michael Bates. Progress in Mathematics, 152, Birkhäuser Boston, Inc., Boston, MA, 1999.

B. O’Neill, Semi-Riemannian geometry, with applications to relativity, Pure and Applied Mathematics, 103. Academic Press, Inc., New York, 1983.

C. Stoppato, Regular Moebius transformations of the space of quaternions, Ann. Global Anal. Geom. 39 (2011), 387–401.

DOI: 10.6092/issn.2240-2829/5893


  • There are currently no refbacks.

Copyright (c) 2015 Giulia Sarfatti

Creative Commons License
This work is licensed under a Creative Commons Attribution 3.0 Unported License.