Subjects: Differential Geometry
Links: Smooth Manifolds, Quotient Topology, Grassmannian Spaces, Embedded Smooth Submanifolds

We define an equivalence relation on ${\mathbb{R}}^{n+1}\setminus \{0\}$ by $$x \sim y \iff \exists t \in \Bbb R(y = tx)$$where $x,y\in {\mathbb{R}}^{n+1}\setminus \{0\}$. The real projective space ${\mathbb{R}\mathbb{P}}^n$ or ${\mathbb{P}}_n(\mathbb{R})$ is the quotient space of ${\mathbb{R}}^{n+1}\setminus \{0\}$ by this equivalence relation. We denote the equivalence class of a point $(a^0,\dots ,a^n)\in {\mathbb{R}}^{n+1}\setminus \{0\}$ by $[a^0,\dots ,a^n]$ and let $\pi :{\mathbb{R}}^{n+1}\setminus \{0\}\to {\mathbb{R}\mathbb{P}}^n$ be the projection. We call $[a^0,\dots ,a^n]$ homogeneous coordinates on ${\mathbb{RP}}^n$.

Geometrically, two nonzero points in ${\mathbb{R}}^{n+1}$ are equivalent iff thy line in the same line through the origin, so ${\mathbb{RP}}^n$ can be interpreted as the set of all lines through the origin in ${\mathbb{R}}^{n+1}$. Each line through the origin in ${\mathbb{R}}^{n+1}$ meets the unit sphere ${\mathbb{S}}^n$ in a pair of antipodal points, and conversely, a pair of antipodal points on ${\mathbb{S}}^n$ determines a unique line through the origin. This suggest that we can define an equivalence relation $\sim$ on ${\mathbb{S}}^n$ by identifying antipodal points: $$x\sim y \iff x = \pm y, \qquad x, y\in \Bbb S^n$$We have the bijection ${\mathbb{RP}}^n\leftrightarrow {\mathbb{S}}^n/\sim$.

Then we have that ${\mathbb{RP}}^n$ is compact.

Prop: The equivalence relation $\sim$ on ${\mathbb{R}}^{n+1}\setminus \{0\}$ in the definition of ${\mathbb{RP}}^n$ is an open equivalence relation. This can be seen as the Continuous Actions of Groups (scalar multiplication) on the topological space ${\mathbb{R}}^{n+1}\setminus \{0\}$.

Cor: The real projective space ${\mathbb{RP}}^n$ is second countable

Prop: The real projective space ${\mathbb{RP}}^n$ is $T_2$.

The Standard Smooth Atlas on a Real Projective Space

Let $[a^0,\dots ,a^n]$ be homogeneous coordinates on the projective space ${\mathbb{RP}}^n$. $a^0$ is not a well-defined function ${\mathbb{RP}}^n$, the condition $a^0\ne 0$ is independent of the choice of representative for $[a^0,\dots ,a^n]$. With this condition, we can define: $$U_0 := {[a^0, \dots, a^n]\in \Bbb {RP}^n \mid a^0\neq 0}$$Similarly, for each $i\in \{1,\dots ,n\}$, let $$U_i := {[a^0, \dots, a^n]\in \Bbb {RP}^n \mid a^i \neq 0}$$
We define the map: ${\varphi }_i:U_i\to {\mathbb{R}}^n$ by $$[a^0, \dots, a^n] \mapsto \left(\frac{a^0}{a^i}, \dots, \hat{\frac{a^i}{a^i}}, \dots \frac{a^n}{a^i}\right)$$where the caret sign $\hat{}$ over $a^i/a^i$ means that entry is to be omitted. This proves that ${\mathbb{RP}}^n$ is locally Euclidean with the charts $(U_i,{\varphi }_i)$ as charts.

Meaning that ${\mathbb{RP}}^n$ is a topological manifold of dimension $n$.

It is easy to show that $\{(U_i,{\varphi }_i)\mid i\in \{0,\dots ,n\}\}$ is a ${\mathcal{C}}^{\mathrm{\infty }}$ atlas for ${\mathbb{RP}}^n$, called the standard atlas. Meaning that ${\mathbb{RP}}^n$ is a smooth manifold.

As CW Complex

If we consider the the usual inclusion ${\mathbb{R}}^{k+1}\subseteq {\mathbb{R}}^{n+1}$ for $k<n$, this allows us to consider ${\mathbb{RP}}^k$ as a subspace of ${\mathbb{RP}}^n$. Then ${\mathbb{RP}}^n$ has a CW decomposition with one cell in each dimension $0,\dots ,n$ such that the $k$-skeleton is ${\mathbb{RP}}^k$ for $0<k<n$.

As Quotient of a Zero Dimensional Lie Group

The two-element group $\{\pm 1\}$ acts on ${\mathbb{S}}^n$ by multiplication. This action is obviously smooth and free, and it is proper because the group is compact. This defines a smooth structure on ${\mathbb{S}}^n/\{\pm 1\}$. In fact, this quotient manifold is diffeomorphic to ${\mathbb{RP}}^n$ with smooth structure we defined above. Let $p:{\mathbb{S}}^n\to {\mathbb{RP}}^n$ be the smooth covering map obtained by restricting the canonical projection ${\mathbb{R}}^{n+1}\setminus \{0\}\to {\mathbb{RP}}^n$ to the sphere. This map makes the same identification as the quotient map $\pi :{\mathbb{S}}^n\to {\mathbb{S}}^n/\{\pm 1\}$. This ${\mathbb{S}}^n/\{\pm 1\}$ is diffeomorphic to ${\mathbb{RP}}^n$, and ${\mathbb{RP}}^n$ is a compact manifold.

Real Projective Varieties

On the projective space ${\mathbb{RP}}^n$ a homogeneous polynomial $F(x_0,\dots ,x_n)$ of degree $k$ is not a function, since its value at a point $[a_0,\dots ,a_n]$ is not unique. However, the zero set in ${\mathbb{RP}}^n$ of a homogeneous polynomial $F(x_0,\dots ,x_n)$ is well defined, since $F(a_0,\dots ,a_n)=0$ if $$F(ta_0, \dots, ta_n) = t^kF(a_0, \dots, a_n) = 0, \qquad \forall t\in \Bbb R ^\times$$The zero set of finitely many homogeneous polynomials in % is called a real projective variety. A projective variety defined by a single homogeneous polynomial of degree $k$ is called a hypersurface of degree $k$.

Covering Spaces

Since $p:{\mathbb{S}}^n\to {\mathbb{RP}}^n$ be the smooth covering map by restricting the canonical projection ${\mathbb{R}}^{n+1}\setminus \{0\}\to {\mathbb{RP}}^n$ to the sphere. We see that ${\mathbb{S}}^n$ to be the universal covering space for ${\mathbb{RP}}^n$ for $n\ge 2$. In the case of $n=1$, we know that ${\mathbb{S}}^1\cong {\mathbb{RP}}^1$, which has a universal covering space of $\mathbb{R}$.

For $n\ge 1$, consider the smooth covering map $q:{\mathbb{S}}^n\to {\mathbb{RP}}^n$. The only nontrivial covering automorphism of $q$ is the antipodal map $\alpha (x):=-x$. We see that $\alpha$ is orientation-preserving iff $n$ is odd, so it follows that ${\mathbb{RP}}^n$ is orientable iff $n$ is odd.