Consider the one-sheeted hyperbola $X= \{ x^2 -y^2 -z^2=-1 \}$ with quadratic form $q(x,y,z) = x^2 -y^2 -z^2$. I want to understand when $q$ restricted to the tangent plane $T_pX$ for $p \in X$ is degenerate. And I am hoping to have an approach that will generalize to (anti-)de sitter space.
So here is my attempt to work out the details, but I am struggling with this: Let $p \in X$, $p= (x_0, y_0, z_0)$. Then the normal to the tangent plane at that point is $(2x_0, -2y_0, -2z_0)$. And so $T_pX$ is the plane $2x_0(x-x_0) -2y_0(y-y_0) -2z_0(z-z_0) =0$. And so I want to determine what values of $(x,y,z)$ on $T_pX$ also make the quadratic form 0.
Plugging in equations would be really messy and I don't think give a good picture or lend itself to a generalization, and so I was thinking alternatively, I can think about it more geometrically: by considering the null cone, $C= \{ x^2 -y^2 -z^2=0 \}$. If one of the basis vectors of $T_pX$ lives in the null cone, (centered at $p$), then $q$ should be degenerate. (Right?). So then, one way this could occur, is if $T_pX$ and $T_pC$ have the same normal vectors. Is this a correct approach?
In addition, I am wondering if someone can give a particular example of where $q$ is degenerate, and an intuitive/geometric description of understanding where/why this happens?
And a final question is if there is a better way to do this, one that will lend itself easily toward classifying where on $X$ the quadratic form vanishes?
And finally finally, what is actually happening geometrically when this happens? Why do we often times want to rule this out?