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Concept
Real projective space
Résumé
In mathematics, real projective space, denoted \mathbb{RP}^n or \mathbb{P}_n(\R), is the topological space of lines passing through the origin 0 in the real space \R^{n+1}. It is a compact, smooth manifold of dimension n, and is a special case \mathbf{Gr}(1, \R^{n+1}) of a Grassmannian space.
As with all projective spaces, RPn is formed by taking the quotient of Rn+1 ∖ under the equivalence relation x ∼ λx for all real numbers λ ≠ 0. For all x in Rn+1 ∖ one can always find a λ such that λx has norm 1. There are precisely two such λ differing by sign.
Thus RPn can also be formed by identifying antipodal points of the unit n-sphere, Sn, in Rn+1.
One can further restrict to the upper hemisphere of Sn and merely identify antipodal points on the bounding equator. This shows that RPn is also equivalent to the closed n-dimensional disk, Dn, with antipodal points on the boundary, ∂Dn = Sn−1, identified.
RP1 is called the real projective line, which is topologically equivalent to a circle.
RP2 is called the real projective plane. This space cannot be embedded in R3. It can however be embedded in R4 and can be immersed in R3 (see here). The questions of embeddability and immersibility for projective n-space have been well-studied.
RP3 is (diffeomorphic to) SO(3), hence admits a group structure; the covering map S3 → RP3 is a map of groups Spin(3) → SO(3), where Spin(3) is a Lie group that is the universal cover of SO(3).
The antipodal map on the n-sphere (the map sending x to −x) generates a Z2 group action on Sn. As mentioned above, the orbit space for this action is RPn. This action is actually a covering space action giving Sn as a double cover of RPn. Since Sn is simply connected for n ≥ 2, it also serves as the universal cover in these cases. It follows that the fundamental group of RPn is Z2 when n > 1. (When n = 1 the fundamental group is Z due to the homeomorphism with S1). A generator for the fundamental group is the closed curve obtained by projecting any curve connecting antipodal points in Sn down to RPn.
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