Let $(V,\langle\,\cdot\,,\,\cdot\,\rangle)$ be an $(n+1)$-dimensional real vector space, equipped with a nondegenerate symmetric bilinear form of indefinite signature, and denote by $\nu(v):=\langle v,v\rangle$ the corresponding quadratic form.

As a homogeneous polynomial, $\nu$ cuts out the **projective** ``light cone'' (or null space) $\mathcal{N}:=\{[v]\in\mathbb{P}V\mid \nu(v)=0\}$, which is a smooth hypersurface in $\mathbb{P}V\equiv\mathbb{RP}^n$.

PRELIMINARY QUESTION: what is the ``remnant'' of the scalar product $\langle\,\cdot\,,\,\cdot\,\rangle$ on $\mathcal{N}$? I mean: does it induce a metric, a conformal structure, or something even weaker on $\mathcal{N}$?

BIG EDIT:such a ``remnant" is a conformal structure $[g]$, according to Vít Tuček's answer below.

By a **hyperplane section** of $\mathcal{N}$ I simply mean the intersection $\mathcal{N}\cap\pi$ of $\mathcal{N}$ with a projective hyperplane $\pi\in \mathbb{RP}^{n\,\ast}$, and I'm interested in the possibility of **characterising hyperplane sections** by means of $[g]$.

MAIN QUESTION:Given a smooth hypersurface $S\subset \mathcal{N}$, is it possible to tell whether or not $S$ is an hyperplane section, just by using the conformal structure $[g]$ on $\mathcal{N}$?

*Any reference/hint concerning the ``instrinsic geometry'' of this projective light cone will be warmly appreciated!*

**- - - COMMENTS & MOTIVATIONS - - -**

This post is a **major edit** of a previous question, whose "no" answer was given in a very convincing way by Willie Wong (see below). Thanks to his geneorous explanation, I was able to rearrange my thoughts ane reformulate the question in a hopefully more robust way. The original question, which is a sort of "affine version" of the current one, is attached below, to keep sustaining Wong's answer and to provide some motivations as well.

In my previous post, the light cone $\mathcal{N}$ was not projectivised, and I made the mistake of believing that $\langle\,\cdot\,,\,\cdot\,\rangle$ induced a metric $g$ on $\mathcal{N}$. Hence, I considered the Hessiian $\mathrm{hess}^V$ on $V$ (i.e., the one defined by means of the flat metric induced by $\langle\,\cdot\,,\,\cdot\,\rangle$) and the Hessian $\mathrm{hess}^\mathcal{N}$ on $\mathcal{N}$ (which in fact is ill-defined, as $g$ is degenerate). Finally, by ``hyperplane section'' I just meant the intersection of $\mathcal{N}$ with an affine hyperplane of $V$. Denoting smooth functions on $\mathcal{N}$ (resp., $V$) by $f$ (resp., $F$), I formulated the following question, which (if Wille Wong hadn't proved it wrong) would have answered the main question of the present post.

PREVIOUS QUESTION: is it true that $\{f=0\}\subset \mathcal{N}$ is an hyperplane section if and only if $$\mathrm{hess}^\mathcal{N}f\textrm{ is proportional to }g\,\textrm{?} \quad\quad{(*)}$$

*The reasoning below is what originally led me to suspect that hyperplane sections can be indeed characterised by means of formula $(*)$ above.*

The hypersurface $\{F=0\}\subset V$ is an affine hyperplane if and only if $$\mathrm{hess}^VF=0\, ,\quad\quad{(**)}$$ so that it all comes down to show that $(*)$ is the ``restricted version'' of $(**)$.

Allow me to regard $\nu$ as a local coordinate function (normal to the light cone) and to decompose the Hessian of $F$ as follows: $$ \mathrm{hess}^VF=\mathrm{hess}^{\mathcal{N}}(F|_{\mathcal{N}})+F_{\nu}\odot d\nu+F_{\nu\nu}\mathrm{hess}^V\nu\, . $$ By restricting the last expression to the light cone, and observing that $\mathrm{hess}^V\nu=\langle\,\cdot\,,\,\cdot\,\rangle$, one gets $(*)$ for $f=F|_{\mathcal{N}}$.

Conversely, if $(*)$ is true for a **scalar factor** $\lambda$, then I can extend $f$ to a function $F$ on $V$, such that $(**)$ is valid: the idea is to set
$$
F:=\pi_{\mathcal{N}}^*(f)-\frac{1}{\lambda}\nu\, ,
$$
where $=\pi_{\mathcal{N}}$ is the projection of $V\stackrel{\textrm{loc.}}{\equiv}\mathcal{N}\times\mathbb{R}$ onto its first factor.

(The present question is linked to my previous one).