Connected Components of a Quadric in RP(3)

Sep 26—2025

This post is based upon a question that I have encountered in Arnold’s ODE a while ago:

How many connected components does $X^2 - Y^2 - Z^2 = \text{c}W^2$ have in $\mathbb{RP}^3$ ?

Today, we will address the specific case of when c = 0, i.e.:

$X^2 - Y^2 - Z^2 - W^2 = 0$ (*).

Let’s first consider this in $\mathbb{R}^4$ . After learning a bit about Lorentzian geometry in my General Relativity course, it is now easy for me to see the geometric object defined by (*). This is the zero set (or the null cone) of the Minkowski inner product on $\mathbb{R}^4$ . This surface in $\mathbb{R}^4$ is a cone (more specifically, 2 cones) with its vertex at the origin. I will attach a picture of the null-cone of the 3-dimensional Minkowski space:

Screenshot taken from Desmos 3D Graphing Calculator

$\mathbb{RP}^3$ is the space of lines passing through the origin of $\mathbb{R}^4$ . Geometrically, the curve for this equation in $\mathbb{RP}^3$ is given by the set of lines passing through the origin intersecting with the null-cone (including the points at infinity on the cone). So it seems intuitive (at least in $\mathbb{R}^3$ ) that this curve will have only one connected component.

Let’s make this more rigorous.

Claim: the curve defined by (*) in $\mathbb{RP}^3$ is diffeomorphic to $S^2$ .

Let $v = [X, Y, Z, W]$ be a solution to (*). Equivalently, we can consider $v$ to be a vector (up to scaling) of the null-cone of the 4-dim. Minkowski space. We take for granted the following fact:

$v$ can be decomposed as $v = v_{+} + v_{-}$ where $v_{+}$ is a space-like vector and $v_{-}$ is a time-like vector and both are non-zero.

This decomposition is geometrically intuitive as the time-like vectors lie in the time-cones, which are inside the null-cones, and the space-like vectors like outside the null-cones. But it seems there are infinitely many ways to decompose, how do we define the most natural decomposition?

Since $v$ is defined up to scaling, $v_{+}, v_{-}$ are both defined up to scaling. Thus, we can normalize the 2 vectors; we denote the two new vectors to be $u, w$ , respectively. The dimension of the time-like subspace is 1 and the dimension of the space-like subspace is 3 because the Minkowski product has signature (1, 3), and so: