Let $\mathbb{R} P^3$ be the real projective space of $\mathbb{R}^4$ and Q be a quadric defined as
$$
Q = \left\{\,\left[\,\left(\begin{smallmatrix} x_1 \\ x_2 \\ x_3 \\ x_4\end{smallmatrix}\right)\,\right] \in \mathbb{R}P^3 \mid x_1^2 + x_2^2 -x_3^2 -x_4^2 = 0 \,\right\}
$$
The associated symetric bilinear form of the the quadratic form mentioned in the quadric above is non-degenerate. By using the polarisations theorem I arrive at the result
$$B(x,y) = x_1y_1 + x_2y_2 - x_3y_3 - x_4y_4 \qquad \forall x,y \in \mathbb{R}^4$$
The polar hyperplane $E \subset \mathbb{R} P^3$ of a point $[\, p\,] \in \mathbb{R} P^3$ is defined as
$$ E = \{\,\left[\, x \,\right] \in \mathbb{R}P^3 \mid B(p,x) = 0 \,\} $$
Show that
- $\mathcal{C} = Q \cap E \neq \emptyset$
- $\mathcal{C}$ is a non-degenerate conic (conic section: circle, ellipse, parabola, hyperbola)
I have tried a couple of things - here is a GeoGebra plot of mine I'm not allowed to post a picture (<>_<>)
With an image, it is somehow clear that the intersection is surely not empty. Here are the problems
- The quadric (even from the projective point of view) is a manifold of codimension 2 and the plane is a 2-dimensional subspace but this is a not enough to conclude the first question.
- I have tried to construct a point that lies in the intersection. I could not find anything
- To prove that the intersection is a conic, I can equivalently show that $|C| \geq 5$. Since $\mathcal{C}$ is not empty, there is at least one point $[\, r\,] \in \mathcal{C}$ so do $-r$. I failed to find another 3 points.
- My idea of how to show, that the conic is non-degenerate is to show, that it does not contain a line. This can be achieved by showing, that no 3 points in $\mathcal{C}$ lie on the same line.
Since I failed at showing 1, 3, I am just out of ideas. Here is the honor mention
For any $[\,x\,] \in \mathcal{C}$ - the line through $[\,p\,], [\,x\,]$ is a tangent of the quadric $Q$ at $[\, x \,]$ and only $[\, x\,]$. What worries me the most is, that none of these ideas involves the bilinear form (except for the last one). What am I missing?