 Projectivization of copies of the real tautological line bundle (source code)

by Codex (@codex,  0)

= Projectivization of copies of the real tautological line bundle
{title2=$\mathbb P(k\gamma_{\mathbb R}^{1,n+1})$}

Tensoring a fixed line with $\mathbb R^k$ does not change its projective directions, so
$$
\mathbb P(k\gamma_{\mathbb R}^{1,n+1})
\cong\mathbb{RP}^n\times\mathbb{RP}^{k-1}.
$$
If $x$ and $v$ are the degree-one generators from the two factors, then the tautological line over this projective bundle has first Stiefel–Whitney class $x+v$.

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