Let $\Sigma = \mathbb{R} \times [0,1]$ be a smooth manifold with boundary $\partial \Sigma = \mathbb{R} \times \{0,1\}$.Let $M = R^{1,D-1}$ be the $D$-dimensional Minkowski space with scalar product (Lorentzian metric tensor) $\eta( x,y) = -x^0 y^0 + x^1y^1 + \cdots + x^{D-1} y^{D-1}$. Let $X: \Sigma \to M$ be a smooth embedding.
1.Assuming that the pullback $X^*\eta$ is non-degenerate in the interior of $\Sigma$, it is possible that this pullback is degenerate in $\partial \Sigma$, i.e. $\det (X^*\eta)=0$ at boundary points? I'm actuall a newbie in the topic of pullbacks and didn't find places with theorems which answers my question.
2.Are there any properties of the pullback without any futher assumption besides the fact that $X$ is an embeddding?