Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)

SIGMA 15 (2019), 078, 16 pages      arXiv:1811.03613      https://doi.org/10.3842/SIGMA.2019.078

The Transition Function of $G_2$ over $S^6$

Ádám Gyenge
Mathematical Institute, University of Oxford, UK

Received May 23, 2019, in final form September 26, 2019; Published online October 09, 2019

Abstract
We obtain explicit formulas for the trivialization functions of the ${\rm SU}(3)$ principal bundle $G_2 \to S^6$ over two affine charts. We also calculate the explicit transition function of this fibration over the equator of the six-sphere. In this way we obtain a new proof of the known fact that this fibration corresponds to a generator of $\pi_{5}({\rm SU}(3))$.

Key words: $G_2$; six-sphere; octonions; fibration; transition function.

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