Regular value

A value is regular when the derivative is surjective at every point of its preimage.

Every value of a submersion is regular, but a map can have some regular values without being a submersion everywhere.

The preimage of a regular value of a smooth map between manifolds is a smooth submanifold whose codimension equals the dimension of the target.

A nonempty regular level set $f^{-1}(0)$ for a real smooth function has both positive and negative values immediately beside any of its points, by the inverse function theorem. On its complement the function never vanishes, so the positive and negative regions are disjoint nonempty open sets partitioning that complement. Hence the complement is disconnected. This is a global necessary condition beyond local embeddedness or coorientation.

The coordinate hyperplane is an embedded submanifold, as standard projective charts flatten its defining homogeneous coordinate. Its complement is the affine chart ${\mathbb{R}}^n$ and is connected. The lemma regular zero hypersurfaces have disconnected complements therefore excludes its being the zero set of a smooth real function with zero a regular value.

For a regular value $q$ of a smooth map $f:M\to N$, let $F=f^{-1}(q)$. The differential of a smooth map identifies the quotient vector bundle $TM{|}_F/TF$ with the fixed target tangent space $T_{q}N$ over every point of $F$. After choosing a Riemannian metric, that quotient identifies with the normal bundle. Hence the normal bundle is trivial even when the target manifold has no global orientation.

Choose an orientation of a vector space on $T_{q}N$. The quotient identification for a regular value gives a smoothly oriented normal bundle along $F=f^{-1}(q)$. An oriented normal frame followed by a frame of $TF$ defines the latter's orientation through the ambient orientation. Changing normal lifts adds only tangent components, so the determinant sign is unchanged. No orientation on all of $N$ is required.

For a smooth map between smooth manifolds, the set of critical values has measure zero in the target. In particular, regular values are dense.