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created a minimum at projective bundle
I can add some observation about how not every projective bundle is the projectivisation of a vector bundle, etc. in the coming days.
Thanks. When you do that, you should also state the definition (I suppose fiber bundle with a projective space as the typical fiber) because at the moment the entry only talks about projectivisation of vector bundles.
Will do.
I think also the complete splitting could be executed in one shot by pulling back to a “flag bundle” associated to a vector bundle. I’m not sure what terminology people use though.
Sure, as in the reference given. Discussion of that might naturally go to a page flag bundle.
spelled out (here) the example of complex projective bundles of tautological quaternionic line bundles:
$\array{ P_{\mathbb{C}} \big( \mathcal{L}_{{}_{\mathbb{H}P^n}} \big) && \simeq && \mathbb{C}P^{2n+1} \\ & \searrow && \swarrow_{ \mathrlap{ v \cdot \mathbb{C}^\times \,\mapsto\, v \cdot \mathbb{H}^\times } } \\ && \mathbb{H}P^n }$1 to 8 of 8