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

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 $\mathrm{Hom}(\ell ,E_x/\ell )$. Thus there is a canonical short exact sequence of vector bundlesA fiber metric identifies $p^{\ast }E/L$ with the orthogonal complement ${\omega }_E$. A chosen connection on a vector bundle defines horizontal lifts and splits the short exact sequence, givingThe 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 $M\times {\mathbb{RP}}^{r-1}$.