Definition:Regular Level Set

Definition

Let $M$ be a smooth manifold.

Let $f \in \map {C^\infty} M : M \to \R$ be a smooth real-valued function.

Let $p \in M$ be a base point in $M$.

Let $\rd f_p$ be the differential of $f$ at $p$.

Let $c \in \R$.

Let $\map {f^{-1}} c$ be a level set.

Suppose every point of $\map {f^{-1}} c$ is a regular point of $f$:

$\forall p \in \map {f^{-1}} c : \rd f_p \ne 0$

Then $\map {f^{-1}} c$ is called a regular level set.

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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