Let $1 \leq \nu \leq n+1$ and $M^n \subseteq \Bbb S^{n+2}\_\nu$ be a non-degenerate submanifold. Assume that $\renewcommand{\vec}[1]{{\bf #1}} \vec{L}\_0 \in \Bbb R^{n+3}\_\nu$ is lightlike and $M \subseteq \vec{L}\_0^\perp \cap \Bbb S^{n+2}\_\nu$. 



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> Is is true that $M$'s Second Fundamental Form is always lightlike?
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Since $\vec{L}\_0$ is normal to $M$ at every point, we have that ${\rm coind}\_{\Bbb R^{n+3}\_\nu}(M) = 1$ or $2$. Since the orthogonal complement of $T\_pM$ with respect to $\Bbb R^{n+3}\_\nu$ is the orthogonal direct sum of the orthogonal complement with respect to $\Bbb S^{n+2}\_\nu$ and the line spanned by $p$ (which is spacelike), we know that ${\rm coind}\_{\Bbb S^{n+2}\_\nu}(M)=1$ or $2$.


I'm having trouble checking that ${\rm coind}\_{\Bbb S^{n+2}\_\nu}(M) = 2$ cannot happen (this would forbid ${\rm II}$ being lightlike).


(Context: I've already proven that if $M^n \subseteq \Bbb S^{n+2}\_\nu$ has lightlike Second Fundamental Form, then every point in $M$ admits a neighbourhood in $M$ contained in a intersection of the form $\vec{L}\_0^\perp \cap \Bbb S^{n+2}\_\nu$. I'm interested in the converse. Supposedly it is proven in [theorem 1 here](https://arxiv.org/pdf/1309.3875.pdf),
 but I can't follow it.
 I've also had [previous trouble](https://mathoverflow.net/questions/283417/possible-mistake-in-classification-of-marginally-trapped-submanifolds-of-bbb-r) with this paper, so I'm skeptic about it.)




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If $(\overline{M}, \langle\cdot,\cdot\rangle)$ is pseudo-Riemannian, and $M\subseteq \overline{M}$ is non-degenerate, by ${\rm coind}\_{\overline{M}}(M)$ I mean the index of the metric of $\overline{M}$ restricted to the normal spaces of $M$, following the decomposition $T\_p\overline{M} = T\_pM \oplus T^\perp\_pM$.




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Update: I have proven that if $M^2\subseteq \Bbb S^4\_1\cap \vec{L}\_0^\perp$, then $M$ is spacelike, by brute-forcing my way through it. Applying $\Lambda \in {\rm O}\_1 (4,\Bbb R)$ if necessary, we can assume that $\vec{L}\_0 =(1,0,0,0,1)$. Then parametrize using $$\varphi(u,v)=(f(u,v),\cos g(u,v)\cos h(u,v), \cos g(u,v)\sin h(u,v),\sin g(u,v),f(u,v)), $$compute $(g\_{ij})\_{i,j=1}^2$ and apply Sylvester's criterion. But I'd prefer avoiding this approach in the general case.



Does every submanifold of $\Bbb S^{n+2}_\nu$ contained in a lightlike hyperplane have lightlike Second Fundamental Form?
Might as well answer it now. The point is that $\vec{L}\_0$ is not tangent to $\Bbb S^{n+2}\_\nu$ at every point, but it is tangent at the points of $\vec{L}\_0^\perp \cap \Bbb S^{n+2}\_\nu$. Because of this, $\vec{L}\_0$ being normal to $M$ ensures that the metric induced in the normal spaces to $M$ relative to $\Bbb S^{n+2}\_\nu$ is Lorentz.


