I was thinking about a problem and a bivector differential equation appeared, not sure how to start solving the issue.
I want to find parameterization of the n-sphere, we know the implicit equation of the sphere:
$$|x|^2 = 1$$
And for the 1-sphere (embeded in $\mathbb{R}^2$) it is easy to find a parameterization:
$$\frac{d}{dt}|x|^2 = 0$$ $$\left < \dot x, x \right > = 0$$
From this equation we found that $\dot x \bot x$, using Hodge dual operator $I=-e_1e_2$ we can find an orthogonal space for $x$ and its parametric equation:
$$\dot x = I x$$
Note that in $\mathbb R^2$ and $Ix = -e_1e_2 (x^1 e_1 + x^2e_2) = -x^2e_1 + x^1e_2 =\begin{pmatrix} 0& -1\\ 1& 0 \end{pmatrix}x $
We can find a solution:
$$x(t) = e^{It}x(0)$$
I am trying to find the 2-sphere similarly, using a bivector differential equation:
$$\partial_1 x \wedge \partial_2 x = I x $$
How could I start solving this?