Embedded Smooth Hypersurface from Regular Points of Smooth Function

Theorem

Let $\struct {M, g}$ be a Riemannian manifold.

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

Let $R \subseteq M$ be the set of regular points of $f$.

Let $c \in \R$.

Let $M_c$ be a set such that:

$M_c = \map {f^{-1} } c \cap R$

Suppose $M_c$ is non-empty.

Then $M_c$ is an embedded smooth hypersurface in $M$.

Furthermore, the gradient $\grad f$ is everywhere normal to $M_c$.

Proof

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Sources

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