What is the moduli space of lines in $\mathbb R^3$?

Question

If we restrict to those lines passing through the origin, we of course get $\mathbb{R}P^2$. Is there a good topological description of the space that we get when we remove the restriction that they pass through the origin? Is there a name for this space?

Answer 1

To every line in $\mathbb R^3$ there corresponds a line in $\mathbb P^3(\mathbb R)$ obtained by adding a point at infinity to it.
This way you obtain all lines in $\mathbb P^3(\mathbb R)$, except those lines completely included in the plane at infinity.
The lines in $\mathbb P^3(\mathbb R)$ form the grassmannian $\mathbb G(1,3)$ and have as moduli space under the Plücker embedding a smooth $4$-dimensional quadric $Q\subset \mathbb P^5(\mathbb R)$ (the Klein quadric).
The lines of $\mathbb P^3(\mathbb R)$ lying completely at infinity correspond to a plane $P\subset \mathbb P^5(\mathbb R)$ included in the Klein quadric $Q$: $P\subset Q$

Conclusion
The moduli space of lines in $\mathbb R^3$ is a quadric hypersurface in $Q\subset \mathbb P^5(\mathbb R)$ minus a plane included in it.
This moduli space is thus quasi-projective, but neither projective nor affine.

Answer 2

One way to think about it is this. Call your space $U$.

If we think about $\mathbf R^3$ as an open submanifold of $\mathbf RP^3$, then $U$ is an open submanifold of the space of lines in $\mathbf RP^3$, which is a Grassmannian variety $G(2,4)$. (This shows that $U$ is 4-dimensional, as it should be.)

The complement $V$ of $U$ inside $G(2,4)$ consists of lines that are contained in the boundary $\mathbf RP^3 \setminus \mathbf R^3 = \mathbf RP^2$. So $V$ is an embedded copy of $G(2,3)$ inside $G(2,4)$: the standard name for a submanifold of this kind is a Schubert cycle $\sigma_{1,1}$.

So $U$ is the complement of a $\sigma_{1,1}$ inside $G(2,4)$.

Answer 3

Just for kicks here is another "bundle" description of the moduli space.

It is not hard to see that this is the "orthogonal complement" of the canonical bundle over $\mathbb{RP}^2$, that is it is the space

$$ \{ (l,v) \in \mathbb{RP}^2 \times \mathbb{R}^3 | v \perp w \mbox{ for any }w \in l \}$$

after all all you need to describe an arbitrary line is its slope and a point, and you can choose this point to be the one closest to the origin.