Geodesic
On $\mathrm{SL}(3,\mathbb{R})$-actions on $4$-spheres
Fundamentalʹnaâ i prikladnaâ matematika, Tome 11 (2005) no. 5, pp. 99-105.

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We construct a natural, continuous $\mathrm{SL}(3,\mathbb{R})$-action on $S^{4}$ which is an extension of the $\mathrm{SO}(3)$-action $\psi$ of Uchida. The construction is based on the Kuiper theorem asserting that the quotient space of $\mathbb{C}P(2)$ by complex conjugation is $S^{4}$. We also give a new proof of the Kuiper theorem. 
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Sh. Kuroki. On $\mathrm{SL}(3,\mathbb{R})$-actions on $4$-spheres. Fundamentalʹnaâ i prikladnaâ matematika, Tome 11 (2005) no. 5, pp. 99-105. http://geodesic.mathdoc.fr/item/FPM_2005_11_5_a8/

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