# Show that the set of lines in $\mathbb{R}^n$ is a (smooth) manifold of dimension $2(n-1)$

I was recently made aware of the result in the title. It's easy to show for $$\mathbb{R}^2$$, but I'm having trouble coming up with a generalization for $$\mathbb{R}^n$$.

There are a couple of ways to represent the set of lines in $$\mathbb{R}^n$$. One way is to let $$X=\mathbb{R}^n \times (\mathbb{R}^n \setminus \lbrace 0\rbrace)$$ with the usual topology, viewed as the space of points and direction vectors defining lines through those points. Then define a relation $$\sim$$ on $$X$$ by $$(p_1,v_1)\sim (p_2,v_2) \iff \text{v_1 \parallel v_2 and p_1 - p_2 \parallel v_1}$$. So $$Y=X/\sim$$ with the quotient topology is the space of lines in $$\mathbb{R}^n$$, up to homeomorphism.

We can also let $$Y=\mathbb{R}^n \times \mathbb{R}P^{n-1}$$ with the product topology, viewed as the set of translated lines through the origin.

Clearly both definitions of $$Y$$ the set of lines are well-defined and homeomorphic, but how do we come up with a smooth atlas of charts onto $$\mathbb{R}^{2(n-1)}$$? I tried considering $$n$$ open sets $$\lbrace U_i \rbrace_{1\leq i\leq n}$$ where $$U_i$$ is the set of all the lines not parallel to $$e_i$$ the $$i$$-th standard basis vector, but could not get very far.

• If you know about Grassmannians, there's a natural way to think of the set of affine lines in $\Bbb R^n$ as a large open subset of $G(2,n+1)$, the Grassmannian of $2$-planes in $\Bbb R^{n+1}$. – Ted Shifrin Mar 14 at 19:47
• @TedShifrin I've been trying to work up to them, thanks :) – terrygarcia Mar 14 at 19:50
• @Arthur as jmerry answered, I believe it is $2(n-1)$. I think I forgot to impose a quotient on $Y$ that accounts for distinct translations producing equivalent lines – terrygarcia Mar 14 at 19:51

I tried considering $$n$$ open sets $$\lbrace U_i \rbrace_{1\leq i\leq n}$$ where $$U_i$$ is the set of all the lines not parallel to $$e_i$$ the $$i$$-th standard basis vector, but could not get very far.
You have to exclude more than a single point for this to work cleanly; it's best to exclude a subspace of codimension $$1$$.
The atlas: Let $$U_i$$ be the set of all lines not contained in a hyperplane $$x_i=c$$ parallel to the $$i$$th coordinate hyperplane. For coordinates on $$U_i$$, associate each such line to the ordered pair $$(u,v)$$ of its intersections with the hyperplanes $$x_i=0$$ and $$x_i=1$$ respectively.
The dimension is $$2(n-1)$$, of course.