Oriented Grassmannian - Imported Knowlpedia Content

Definition

Let $V$ be an $n$ -dimensional real vector space and $1\leq k\leq n$ . The oriented Grassmannian

\widetilde{\operatorname{Gr}}_k(V) =\{(W,o):W\in\operatorname{Gr}_k(V),\ o\text{ is an orientation of }W\}

is the smooth manifold of oriented $k$ -dimensional subspaces of $V$ . Forgetting $o$ defines a smooth two-sheeted covering

\widetilde{\operatorname{Gr}}_k(V)\longrightarrow \operatorname{Gr}_k(V).

The target is the ordinary real Grassmannian. The covering transformation reverses the orientation of $W$ . Because the orientation choice is discrete, the oriented Grassmannian has the same dimension $k(n-k)$ as its target.

Homogeneous-space model

Choose an orientation and inner product on $V\cong\mathbb R^n$ . For $0<k<n$ , the special orthogonal group acts transitively, and the stabilizer of a reference oriented plane is $SO(k)\times SO(n-k)$ . Hence

\widetilde{\operatorname{Gr}}_k(\mathbb R^n) \cong SO(n)/(SO(k)\times SO(n-k)).

The corresponding quotient for the ordinary Grassmannian uses $S(O(k)\times O(n-k))$ ; the difference records whether changes of frame preserve the plane's orientation.

Tautological orientation and universal role

The pullback of the tautological rank- $k$ bundle from $\operatorname{Gr}_k(V)$ has fiber $W$ over $(W,o)$ , equipped with the orientation $o$ . It is therefore canonically an oriented vector bundle. In the stable limit, oriented Grassmannians serve as classifying spaces for oriented real vector bundles, and the tautological orientation supports the Euler class and oriented characteristic constructions; see Milnor–Stasheff, §§5 and 9.

Examples and boundary cases

$\widetilde{\operatorname{Gr}}_1(\mathbb R^n)$ is the sphere $S^{n-1}$ : an oriented line is determined by its positive unit vector. The forgetful map to $\mathbb RP^{n-1}$ is the antipodal double cover. At $k=n$ , the underlying subspace is $V$ itself, so the oriented Grassmannian consists of its two possible orientations; the homogeneous-space formula above was intentionally restricted to $0<k<n$ .

Complex Grassmannians are not examples of this double-cover construction. A complex subspace carries a canonical real orientation, so no independent binary orientation choice is required.

References

John W. Milnor and James D. Stasheff, Characteristic Classes, Princeton University Press, 1974. Publisher DOI record. Relevant: §§5–6 on Grassmann manifolds and universal bundles, and §9 on oriented bundles.
Dale Husemoller, Fibre Bundles, 3rd ed., Springer, 1994. Springer DOI record. Relevant: Grassmannians and classifying bundles.