Real projective space

In mathematics, real projective space, or RP n is the projective space of lines in R n +1. The case n = 1 gives the real projective line which is topology equivalent to a circle. This case n = 2 is called the real projective plane, RP2. The space RP n is a compact space, smooth manifold of dimension n . It is a special case of a Grassmannian.

As with all projective spaces, RP n is formed by taking the quotient space of R n +1 − {0} under the equivalence relation x ∼ λ x for all real numbers λ ≠ 0. For all x in R n +1 − {0} one can always find a λ such that λ x has norm 1. There are precisely two such λ differing by sign. Thus RP n can also be formed by identifying antipodal points of the unit n -sphere, S n , in R n +1. One can further restrict to the upper hemisphere of S n and merely identify antipodal points on the bounding equator. This shows that RP n is also equivalent to the closed n -dimensional disk, D n , with antipodal points on the boundary, ∂ D n = S n −1, identified.

The antipodal map on the n -sphere (the map sending x to − x ) generates a cyclic group group action on S n . As mentioned above, the orbit space for this action is RP n . This action is actually a covering space action giving S n as a double cover of RP n . Since S n is simply connected for n ≥ 2, it also serves as the universal cover in these cases. It follows that the fundamental group of RP n is Z2. A generator for the fundamental group is the closed curve obtained by projecting any curve connecting antipodal points in S n down to RP n .

= See also =

*Complex projective space *Lens space