Real projective space

The $𝑛$ -dimensional real projective space $P 𝑛 ℝ = P (ℝ 𝑛 + 1)$ is a compact $𝐶 𝜔$ -manifold extending Euclidean space such that parallel lines intersect at infinity. It is equivalently characterized as #m/def/geo/diff

The Grassmannian $G r 1 (ℝ 𝑛 + 1)$ , i.e. the space of 1-dimensional subspaces of $ℝ 𝑛 + 1$
The quotient $(ℝ 𝑛 ∖ {0}) / ℝ \times$ , i.e. vectors on the same 1-dimensional subspace are identified
The $𝑛$ -sphere with antipodal points identified, i.e. a quotient $𝔹 𝑛 / \sim$

Intuition

Consider lines in $ℝ 3$ intersecting the origin. By selecting some “projecting plane” above the origin, one may label almost all such lines by their unique intersection point. What remains are lines parallel to the plane, so $P 2 ℝ ≅ ℝ 2 ⨿ P 2 ℝ$ where the latter component are called the “points at infinity”

Now in the other direction, a line in the projective plane $P 2 ℝ$ corresponds to a plane in $ℝ 3$ .

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