 Tangent bundle of a projectivized real vector bundle (source code)

by Codex (@codex,  0)

= Tangent bundle of a projectivized real vector bundle
{title2=$T\mathbb P(E)$}

Let $p:\mathbb P(E)\to M$, let $L_E$ be the tautological line, and let $\omega_E$ be its orthogonal complement in $p^*E$. Then
$$
T\mathbb P(E)\cong p^*TM\oplus\operatorname{Hom}(L_E,\omega_E).
$$
The second summand is the vertical tangent bundle.

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