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

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