Nihonkai Mathematical Journal

A confirmation by hand calculation that the Möbius ball is a gyrovector space

Keiichi Watanabe

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Abstract

We give a confirmation that the Möbius ball of any real inner product space is a gyrovector space by using only elementary hand calculation. Some remarks to [2] will also be made.

Note

This work is partly supported by Grant-in-Aid for Scientific Research (C) Number 23540190, Japan Society for the Promotion of Science.

Article information

Source
Nihonkai Math. J., Volume 27, Number 1-2 (2016), 99-115.

Dates
Received: 12 January 2016
Revised: 4 April 2016
First available in Project Euclid: 14 September 2017

Permanent link to this document
https://projecteuclid.org/euclid.nihmj/1505419744

Mathematical Reviews number (MathSciNet)
MR3698244

Zentralblatt MATH identifier
1382.51014

Subjects
Primary: 51M10: Hyperbolic and elliptic geometries (general) and generalizations
Secondary: 20N05: Loops, quasigroups [See also 05Bxx] 46C99: None of the above, but in this section 51P05: Geometry and physics (should also be assigned at least one other classification number from Sections 70-86)

Keywords
Möbius gyrogroups Möbius gyrovector spaces

Citation

Watanabe, Keiichi. A confirmation by hand calculation that the Möbius ball is a gyrovector space. Nihonkai Math. J. 27 (2016), no. 1-2, 99--115. https://projecteuclid.org/euclid.nihmj/1505419744


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References

  • M. Ferreira and G. Ren, M${\ddot o}$bius gyrogroups: A Clifford algebra approach, J. Algebra 328 (2011), 230–253.
  • A. A. Ungar, Analytic Hyperbolic Geometry and Albert Einstein's Special Theory of Relativity, World Scientific Publishing Co. Pte. Ltd., Singapore, 2008.