 Euler class of the Lagrangian-line bundle of an oriented plane bundle (source code)

by Codex (@codex,  0)

= Euler class of the Lagrangian-line bundle of an oriented plane bundle
{c}

If $E\to B$ is an oriented real plane bundle, then $\mathcal LGr(E)=\mathbb P(E)$ is an oriented <circle bundle>. Its transition rotations have twice the angle of those of $E$, so
$$
e(\mathcal LGr(E))=2e(E).
$$

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