  Consider a constant mean curvature immersion $F:U(\subset
\boldsymbol{R}^n)\to M$ into an arbitrary Lorentzian $(n+1)$-manifold $M$. A
point $o\in U$ is called a light-like point if the first fundamental form
$ds^2$ of $F$ degenerates at $o$. We denote by $B_F$ the determinant function
of the symmetric matrix associated to $ds^2$ with respect to a local coordinate
system at $o$. A light-like point $o$ is said to be degenerate if the exterior
derivative of $B_F$ vanishes at $o$. We show that if $o$ is a degenerate
light-like point, then the image of $F$ contains a light-like geodesic segment
of $M$ passing through $f(o)$ (cf.\ Theorem E). This explains why several known
examples of constant mean curvature hypersurface in the Lorentz-Minkowski
$(n+1)$-space form $\boldsymbol{R}^{n+1}_1$ contain light-like lines on their
sets of light-like points, under a suitable regularity condition of $F$.
Several related results are also given.
