n-Dimensional Real Projective Spaces as Manifolds

# n-Dimensional Real Projective Spaces as Manifolds

Recall from the n-Dimensional Toruses as Manifolds page that the $n$-dimensional torus in $\mathbb{R}^{n+1}$can be defined as the following product

(1)
\begin{align} \quad T^n = \prod_{i=1}^{n} S_i = S_1 \times S_2 \times ... \times S_n \end{align}

We will now look at another class of manifolds called the $n$-dimensional real projective spaces.

 Definition: The $n$-Dimensional Real Projective Space denoted $\mathbb{R}P^n$ is the set of all lines passing through the origin of $\mathbb{R}^{n+1}$.

The $1$-dimensional real projective space looks like:

Each line in $\mathbb{R}P^1$ passes through two points of $S^1$ and so $\mathbb{R}P^1$ can be defined by identifying an equivalence relation on antipodal points of $S^1$.

Similarly, each line in $\mathbb{R}P^2$ passes through two points in $S^2$ and so $\mathbb{R}P^2$ can be defined by identifying an equivalence relation on antipodal points of $S^2$.