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 $\tuple {r, s - 1}$.

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

Proof

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Sources

• 2018: John M. Lee: Introduction to Riemannian Manifolds (2nd ed.) ... (previous) ... (next): $\S 2$: Riemannian Metrics. Pseudo-Riemannian Metrics