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I'm trying to work through a script on Riemannian submersions but I have some problems with one proof in particular (or more likely the general underlying concepts). No worries, this is not about the entire proof, but just one step.

This is about the quaternionic projective space

[itex]\mathbb{H}P^n = \{pS^3 | p \in \mathbb{S}^{4n+3}\}[/itex]

as the orbit space under the free isometric group action

[itex]S^3 \times \mathbb{S}^{4n+3} \rightarrow \mathbb{S}^{4n+3}, (g, p) \mapsto pg^{-1}[/itex]

The projection [itex]\pi: \mathbb{S}^{4n+3} \rightarrow \mathbb{S}^{4n+3}/S^3 = \mathbb{H}P^n[/itex] should be the associated Riemannian submersion if I understand correctly.

Therefore, we can speak of vertical and horizontal vectors in [itex]T_p \mathbb{S}^{4n+3}[/itex].

Without going into the entire context of the proof, I don't really understand the following:

"Let [itex]v, w[/itex] be horizontal unit vectors in [itex]T_p \mathbb{S}^{4n+3}, p = (1, 0, ..., 0) \in \mathbb H^{n+1}.[/itex] [...] We can regard v, w as vectors in [itex]\mathbb H^{n+1}[/itex] (through the canonical isomorphism). [...] The vertical subspace of [itex]T_p \mathbb{S}^{4n+3}[/itex] in p is generated by [itex](i, 0, ..., 0), (j, 0, ..., 0), (k, 0, ..., 0).[/itex] Therefore the horizontal vectors v, w are of the form [itex](0, *, ..., *).[/itex]"

I don't understand why the vertical subspace is generated by these three vectors. Why isn't the vector (1, 0, ...) also necessary to generate the subspace?

Thanks in advance!

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# Vertical subspace in HPn (Riemannian submersions)

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