Regular level set - Knowlpedia

Let $f:M\to N$ be a smooth map between finite-dimensional smooth manifolds, and let $y\in N$ be a regular value of $f$ . The subset

S=f^{-1}(y)

is called a regular level set, regular fiber, or regular preimage of $f$ . By the regular-value theorem, $S$ is an embedded submanifold of $M$ ; when nonempty, it has codimension $\dim N$ , and for every $x\in S$ ,

T_xS=\ker(df_x).

The empty preimage is regular under the standard vacuous convention. When $N=\mathbb R$ , this is the usual level set of a smooth function at a regular value.

Local normal form

For each $x\in S$ , the submersion theorem supplies coordinates near $x$ and $y$ in which

f(u_1,\ldots,u_m)=(u_{m-n+1},\ldots,u_m).

In these coordinates the level set is the coordinate slice on which the last $n=\dim N$ coordinates are constant. This both gives the induced smooth structure and explains the tangent-space formula.

Examples and a near miss

For $f:\mathbb R^n\to\mathbb R$ , $f(x)=\|x\|^2$ , each $r>0$ is a regular value and $f^{-1}(r)$ is a sphere of dimension $n-1$ . The value $0$ is critical because $df_0=0$ , although its level set $\{0\}$ happens to be a submanifold. Thus “is a submanifold” does not imply “is a regular level set for the displayed defining map.”

For a submersion $f:M\to N$ , every fiber is a regular level set.

Conventions and scope

Some authors use “level set” only for real-valued functions and “fiber” for general targets. This knowl allows an arbitrary finite-dimensional target manifold. If $M$ or $N$ has boundary or corners, a clean embedded-submanifold conclusion can require additional boundary compatibility or transversality hypotheses; the core states the boundaryless theorem.

References

John M. Lee, Introduction to Smooth Manifolds, 2nd ed., Graduate Texts in Mathematics 218, Springer, 2012. Springer DOI record. Relevant: Chapter 5, especially Theorem 5.12, the regular level set theorem.
Victor Guillemin and Alan Pollack, Differential Topology, AMS Chelsea Publishing, 2010 reprint. AMS DOI record. Relevant: Chapter 1, submersions and the preimage theorem.