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Spin(9) geometry of the octonionic Hopf fibration

Abstract

We deal with Riemannian properties of the octonionic Hopf fibration S 15S 8, in terms of the structure given by its symmetry group Spin(9). In particular, we show that any vertical vector field has at least one zero, thus reproving the non-existence of S 1 subfibrations. We then discuss Spin(9)-structures from a conformal viewpoint and determine the structure of compact locally conformally parallel Spin(9)-manifolds. Eventually, we give a list of examples of locally conformally parallel Spin(9)-manifolds.

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Correspondence to Liviu Ornea.

Additional information

L. O. and V. V. were partially supported by CNCS UEFISCDI, project number PNII-ID-PCE-2011-3-0118 and by the INdAM-GNSAGA visiting program.

M. P. and P. P. were supported by the MIUR under the 2010–11 PRIN Project “Varietà reali e complesse: geometria, topologia e analisi armonica”.

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Ornea, L., Parton, M., Piccinni, P. et al. Spin(9) geometry of the octonionic Hopf fibration. Transformation Groups 18, 845–864 (2013). https://doi.org/10.1007/s00031-013-9233-x

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Keywords

  • Conformal Class
  • Vectorial Type
  • Vertical Vector
  • Parallel Spin
  • Vector Bundle Versus