= Tangent bundle of a projectivized real vector bundle
{title2=$T\mathbb P(E)$}
Let $p:\mathbb P(E)\to M$ be the <projectivization of a real vector bundle>, using one-dimensional <linear subspaces>, and let $L$ be its <tautological bundle>. At a line $\ell\subset E_x$, differentiating the graph of a nearby line identifies the vertical <tangent space> with $\operatorname{Hom}(\ell,E_x/\ell)$. Thus there is a canonical <short exact sequence> of <vector bundles>
$$
0\longrightarrow\operatorname{Hom}(L,p^*E/L)\longrightarrow T\mathbb P(E)\xrightarrow{dp}p^*TM\longrightarrow0.
$$
A <fiber metric> identifies $p^*E/L$ with the orthogonal complement $\omega_E$. A chosen <connection on a vector bundle> defines horizontal lifts and splits the <short exact sequence>, giving
$$
T\mathbb P(E)\cong p^*TM\oplus\operatorname{Hom}(L,\omega_E).
$$
The vertical term is intrinsic, while this horizontal direct-sum identification depends on the chosen <connection on a vector bundle>. For a <trivial vector bundle> with its product connection, it reduces to the usual <tangent bundle> splitting of $M\times\mathbb{RP}^{r-1}$.
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