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Let $S^3$ be the standard sphere (with the metric induced from $\mathbb{R}^4$) and $\pi\colon S^3\rightarrow \mathbb{C}P^1$ the hopf fibration. Equip $\mathbb{C}P^1$ with the Fubini-Study metric so that the hopf fibration turns into a riemannian submersion.

Let $(X_1,X_2)$ be a positively oriented local orthonormal frame on $\mathbb{C}P^1$ and denote $(\bar{X}_1,\bar{X}_2)$ its horizontal lift to $S^3$ via the hopf fibration $\pi$.

Claim: The lie bracket $[\bar{X}_1,\bar{X}_2]$ of the horizontal lifts is independend of the choice of the positively oriented local orthonormal frame $(X_1,X_2)$.

Is that claim true? If yes I'd appreciate if someone would explain to me why it is true.

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  • $\begingroup$ A horizontal lift is not unique unless you fix a connection on the fibration. $\endgroup$ – Eric O. Korman Oct 27 '14 at 1:42

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