# Complex projective space CP3 (Twistor space), as bundle space with base CP1, and fiber 4-D Minkowski space-time?

Twistor space, as complex projective space $CP3$, is related to Minkowski 4-D space time (metric $1, -1,-1,-1)$, by the incidence relation.

Let $Z = (v_a, u^{\dot{a}})$ a point in $CP3$, where $v_a$ and $u^{\dot{a}}$ are 2-complex components spinors.

Let $x_{a\dot{a}} = \sum x^\mu (\sigma_\mu)_{a\dot{a}}$, where $\sigma_\mu$ are the Pauli matrices, and $x^\mu$ a point in Minkowski 4-D space-time.

The incidence relation is then $v_a = x_{a\dot{a}} u^{\dot{a}}$

The representation of a space-time point, in the twistor space $CP3$, is a complex line $CP1$.

So, $CP3$ may be seen as a bundle space, with base Minkowski space-time, and fiber $CP1$

But, is the inverse true, that is : could we see CP3 as a bundle space, with base CP1, and as fiber the 4-D Minkowski space-time ?

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