Conditions for Pullback of Pseudo-Riemannian Metric to be Nondegenerate

Theorem
Let $\struct {\tilde M, \tilde g}$ be a pseudo-Riemannian manifold of signature $\tuple {r, s}$.

Let $M$ be a smooth hypersurface in $\tilde M$.

Let $\iota : M \to \tilde M$ be the inclusion mapping.

Let $N_p M$ be the normal space at $p \in M$.

Then the pullback tensor field $\iota^* \tilde g$ is nondegenerate


 * $\forall p \in M : \forall v \in N_p M : v \ne 0 : \map {\tilde g} {v, v} \ne 0$