The Transition Function of G₂ over S⁶

Abstract

We obtain explicit formulas for the trivialization functions of the SU(3) principal bundle G 2 → 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 π 5 (SU (3)).Let U 1 = S 6 \{S} and U 2 = S 6 \{N } be two affine charts on S 6 , where S and N are the south and north poles. Here we consider S 6 as the unit six-spherein the 7-dimensional vector space of purely imaginary octonions, such that S = (0, −1, 0, . . . , 0) and N = (0, 1, 0, . . . , 0). Our first result is an explicit formula for the trivialization functions ψ 1 : p −1 (U 1 ) → U 1 × SU(3) and ψ 2 : p −1 (U 2 ) → U 2 × SU(3) deduced in Propositions 3.3 and 3.4 below.