The sphere of dimension 15.


Octonionic Hopf fibration

The 15-sphere participates in the octonionic Hopf fibration, the analog of the complex Hopf fibration with the field of complex numbers replaced by the division ring of octonions 𝕆\mathbb{O}.

S 7 S 15 p S 8 \array{ S^7 &\hookrightarrow& S^15 \\ && \downarrow^\mathrlap{p} \\ && S^8 }

Here the idea is that S 15S^{15} can be construed as {(x,y)𝕆 2:|x| 2+|y| 2=1}\{(x, y) \in \mathbb{O}^2: {|x|}^2 + {|y|}^2 = 1\}, with pp mapping (x,y)(x, y) to x/yx/y as an element in the projective line 1(𝕆)S 8\mathbb{P}^1(\mathbb{O}) \cong S^8, with each fiber a torsor parametrized by octonionic scalars λ\lambda of unit norm (so λS 7\lambda \in S^7).

Other properties

  • S 15S^{15} is the only sphere that admits three homogeneous Einstein metrics.
  • It is the only sphere that appears as a regular orbit in three cohomogeneity one actions on projective spaces, namely of SU(8)SU(8), Sp(4)Sp(4) and Spin(9)Spin(9) on P 8\mathbb{C}P^8, P 4\mathbb{H}P^4 and 𝕆P 2\mathbb{O}P^2, respectively (OPPV, p. 1)


  • Liviu Ornea, Maurizio Parton, Paolo Piccinni, Victor Vuletescu, Spin(9) geometry of the octonionic Hopf fibration, (arXiv:1208.0899)

Last revised on November 27, 2020 at 14:17:03. See the history of this page for a list of all contributions to it.