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 if and only if

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

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