Projectivization of a real vector bundle ($\mathbb{P}(E)$)

The projectivization $\mathbb{P}(E)\to B$ has fiber over $b$ equal to the space of one-dimensional linear subspaces of $E_b$.

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 $M\times {\mathbb{RP}}^{r-1}$.

Tensoring a fixed line with ${\mathbb{R}}^k$ does not change its projective directions, soIf $x$ and $v$ are the degree-one generators from the two factors, then the tautological line over this projective bundle has first Stiefel–Whitney class $x+v$.