Real projective space

For $n\ge0$ , real projective $n$ -space is the quotient

\mathbb{RP}^n=(\mathbb R^{n+1}\setminus\{0\})/\mathbb R^\times,

where nonzero real scalars act by multiplication and the quotient has the quotient topology. Equivalently, its points are one-dimensional real linear subspaces of $\mathbb R^{n+1}$ . The open sets $U_i=\{[x_0:\cdots:x_n]:x_i\ne0\}$ have affine coordinate charts given by the ratios $x_j/x_i$ for $j\ne i$ . Their transition maps are smooth rational maps, so these charts define a smooth manifold of real dimension $n$ .

Quotient descriptions

Every real line contains exactly two unit vectors, so normalization gives a diffeomorphism

\mathbb{RP}^n\cong S^n/\{x\sim -x\}.

The sphere map $S^n\to\mathbb{RP}^n$ is a two-sheeted covering for $n\ge1$ . These quotient descriptions show that $\mathbb{RP}^n$ is compact and connected for $n\ge1$ .

Homogeneous-space descriptions

The orthogonal group acts transitively on real lines, and the stabilizer of the first coordinate line is $O(1)\times O(n)$ . Hence

\mathbb{RP}^n\cong O(n+1)/(O(1)\times O(n))

as a homogeneous space. The projective general linear group $\operatorname{PGL}_{n+1}(\mathbb R)$ also acts transitively; the stabilizer of a line is a projective parabolic subgroup.

Geometry and topology

The case $\mathbb{RP}^1$ is diffeomorphic to a circle. For $n\ge2$ , the sphere covering is universal and $\pi_1(\mathbb{RP}^n)\cong\mathbb Z/2$ . For $n\ge1$ , $\mathbb{RP}^n$ is orientable exactly when $n$ is odd.

References

John M. Lee, Introduction to Smooth Manifolds, 2nd ed., Springer, 2012. Publisher record. Relevant: quotient manifolds, projective spaces, and covering maps.
John W. Milnor and James D. Stasheff, Characteristic Classes, Princeton University Press, 1974. Publisher record. Relevant: §§4–5, real projective spaces and tautological bundles.