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This mathematical physics paper contains a result (Lemma B.1(a) in Appendix B) about vectors in $\mathbb C^3$ that square to zero. It says, among other things, that if $\mathbf s$ is a nonzero vector in $\mathbb C^3$ such that $\mathbf s^2 := \mathbf s \cdot \mathbf s = 0$, then one can write $$\mathbf s = s(\mathbf n_1 + i \mathbf n_2)$$ with $s \in \mathbb C$ and $\mathbf n_1$, $\mathbf n_2$ orthogonal unit vectors in $\mathbb R^3$, and this representation of $\mathbf s$ is unique up to the transformations $$(s, \mathbf n_1, \mathbf n_2) \mapsto (s e^{i\alpha}, \mathbf n_1 \cos \alpha + \mathbf n_2 \sin \alpha, -\mathbf n_1 \sin \alpha + \mathbf n_2 \cos \alpha), \quad \alpha \in \mathbb R.$$

Now, it seems very unlikely that this is the first time this result has been derived, so I wonder if it rings any bells for anyone here. Maybe it is a consequence of some well-known more general result in geometry?

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    $\begingroup$ I think this is just basic arithmetic. The first assertion follows easily if you write $\mathbf s$ as $\mathbf x+i\mathbf y$ for some real vectors $\mathbf x$ and $\mathbf y$. When $\mathbf s\ne0$, the second assertion follows directly from the fact that $e^{-i\alpha}(\mathbf n_1+i\mathbf n_2)=(\mathbf n_1\cos\alpha+\mathbf n_2\sin\alpha)+(-\mathbf n_1\sin\alpha+\mathbf n_2\cos\alpha)$. $\endgroup$
    – user1551
    Commented Jul 10, 2022 at 16:43
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    $\begingroup$ Can't we just take $s\in \Bbb{R}$ in this representation? $\endgroup$
    – Bob Dobbs
    Commented Jul 10, 2022 at 16:57
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    $\begingroup$ Like user1551 writes, this is almost immediate. Another way to view the punctured null cone in $\Bbb C^3$ is as a quotient of $\Bbb C^* \times SO(3)$. If we map $(s, \pmatrix{{\bf n}_1 & {\bf n}_2 & {\bf n}_1 \times {\bf n}_2)}$ to $s({\bf n}_1 + i {\bf n}_2)$, the preimage of a fixed element, say, ${\bf e}_1 + i {\bf e}_2$ is exactly the circle of elements parameterized by $\alpha$. As Mehmet hints in his own comment, we get a unique representation if we further impose $s \in \Bbb R^+$. $\endgroup$ Commented Jul 10, 2022 at 19:56

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This question sent me back in time to 1993-94, the days when I was writing my MsC thesis.

I don't know how much geometric but the solution space of the equation $\Bbb{s}\cdot\Bbb{s}=0$ in $\Bbb{C}^3$ is connected with the Stiefel Variety (or Manifold) $V_2(\Bbb{R}^3)$ of 2-frames in $\Bbb{R}^3$. This maniold is homeomorf to $SO(3)$ so that the solution space is homeomorf to the sapce $\Bbb{R}SO(3)$.

I think we can generalize: The solution space of the equation $\Bbb{s}\cdot\Bbb{s}=0$ in $\Bbb{C}^n$ is homeomorf to $\Bbb{R}V_2(\Bbb{R}^n)$ whose dimension is $1+2n-3=2n-2$.

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