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.