I am trying to read the paper *Marginally Trapped Submanifolds in Space Forms with Arbitrary Signature* by Henri Anciaux, but I think that there is a mistake in Lemma $1$, in page $5$:

The second fundamental form $h$ is collinear to $\nu$ if and only if the mean curvature vector $H$ is collinear to $\nu$ and $\nu$ has rank at most $1$.

Here $\Sigma$ is a $n$-dimensional submanifold of $\Bbb R^{n+2}_{p+1}$ such that metric induced in the normal spaces to $\Sigma$ is Lorentzian, and $(\nu,\xi)$ is a null frame normal to $\Sigma$ satisfying $\langle \nu, \xi\rangle = 2$, and $(e_1,\ldots,e_n)$ is an orthonormal local frame tangent to $\Sigma$. The author puts $h^1_{ij} = \langle h(e_i,e_j), \nu\rangle = -\langle d\nu(e_i),e_j\rangle$, so far, so good. Then he states that $$d\nu(e_i) = -\sum_{j=1}^n h^1_{ij}e_j + \frac{1}{2}\langle d\nu(e_i),\xi\rangle \nu.$$

I disagree: by orthonormal expansion, we should have $$d\nu(e_i) = -\sum_{j=1}^n \color{red}{\epsilon_j}h^1_{ij}e_j + \frac{1}{2}\langle d\nu(e_i),\xi\rangle \nu,$$where $\epsilon_j$ denotes the indicator of $e_j$ ($1$ if $e_j$ is spacelike, $-1$ if timelike). His idea was to use that $h^1_{ij}$ is symmetric with rank $1$ to write $h^1_{ij} = c\lambda_i\lambda_j$ for some certain constants, and proceed with the argument. As it is written, his reasoning seems to hold only if $\Sigma$ is itself spacelike.

This detail seems to put the rest of the proof in jeopardy: we'll have that the matrix $(\epsilon_ih^1_{ij})$ has rank $1$ instead of $(h^1_{ij})$, but it is not anymore symmetric, so the argument fails. We'll have that $$c\sum_{i=1}^n\epsilon_i \lambda_i^2 = 0$$instead, and if $c \neq 0$ we can't conclude that $(\lambda_1,\ldots,\lambda_n) = (0,\ldots,0)$.

I don't know if this is fixable. Please help.