Signature of Embedded Pseudo-Riemannian Submanifold

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

Let $f \in \map {C^\infty} {\tilde M}$ be a smooth mapping.

Let $M = \map {f^{-1}} c$ for some $c \in \R$.

Suppose:


 * $\forall p \in M : \map {\tilde g_p} {\grad f, \grad f} > 0$

Then $M$ is an embedded pseudo-Riemannian submanifold of $\tilde M$ of signature $\tuple {r - 1, s}$.

Otherwise suppose:


 * $\forall p \in M : \map {\tilde g_p} {\grad f, \grad f} < 0$

Then $M$ is an embedded pseudo-Riemannian submanifold of $\tilde M$ of signature $\tuple {r, s - 1}$.

Furthremore, in either case, $\grad f$ is everywhere normal to $M$.