Oriented Grassmannians and the tautological oriented bundle

For $0\le n\le N$, the oriented Grassmannian is

${\mathrm{Gr}}_n^{+}({\mathbb{R}}^N)=V_n({\mathbb{R}}^N)\mathrm{/}\mathrm{SO}(n)\mathrm{.}$

An ordered orthonormal frame determines its span together with the orientation it transports from ${\mathbb{R}}^n$, and the $\mathrm{SO}(n)$ quotient identifies exactly the frames inducing the same oriented plane. The associated standard bundle is the tautological bundle ${\gamma }_n^{+,N}$; its fiber over $(W,o)$ is $W$ with orientation $o$, in the sense of Oriented real bundles and oriented frame bundles.

The stable coordinate inclusions from Stiefel spaces, Grassmannians, and tautological bundles define

${\mathrm{Gr}}_n^{+}({\mathbb{R}}^{\mathrm{\infty }})={\bigcup }_{N\ge n}{\mathrm{Gr}}_n^{+}({\mathbb{R}}^N),$

with the weak direct-limit topology. This chosen model is denoted $B\mathrm{SO}(n)$.

For $n\ge 1$, forgetting the orientation is the double cover ${\mathrm{Gr}}_n^{+}\to {\mathrm{Gr}}_n$, since a positive-dimensional real vector space has exactly two orientations. After forgetting orientation, ${\gamma }_n^{+}$ is the pullback of ${\gamma }_n$ along this cover. For $n=0$, both Grassmannians are points and the tautological bundle has its canonical rank-zero orientation.