Signature of Pseudo-Riemannian Submanifold

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

Suppose:


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

Then $M$ is a pseudo-Riemannian submanifold of signature $\tuple {r - 1; s}$.

Otherwise, suppose:


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

Then $M$ is a pseudo-Riemannian submanifold of signature $\tuple {r; s - 1}$.