Real projective space - Definition and Overview

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

As with all projective spaces, RPⁿ is formed by taking the quotient of Rⁿ⁺¹ − {0} under the equivalence relation x ∼ λx for all real numbers λ ≠ 0. For all x in Rⁿ⁺¹ − {0} 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 bounday, ∂Dⁿ = Sⁿ⁻¹, identified.

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. A generator for the fundamental group is the closed curve obtained by projecting any curve connecting antipodal points in Sⁿ down to RPⁿ.

See also

• Complex projective space
• Lens space