Sard's theorem and regular values

For a C¹ map from an m-manifold to an n-manifold, a source point is critical when its derivative is not surjective. A critical value is the image of at least one critical point. A regular value has no critical preimages, including values with no preimages at all. If m<n, every source point is critical, because no linear map from a smaller-dimensional space can be onto the target tangent space. Sard then says that the entire image has target measure zero under its hypotheses.

When m≥n, a critical set can still be large. A constant smooth map to a positive-dimensional target has every source point critical but only one critical value. Thus the theorem does not say that almost every source point has full rank. Nor does it guarantee that a regular value belongs to the image. To use the regular-value theorem for an interesting fiber, first find a regular target value and separately establish that its preimage is nonempty.

For smooth maps between finite-dimensional smooth manifolds without boundary, Sard's theorem states that the critical values have measure zero in the target. More generally a Cʳ map satisfies this conclusion when the positive integer r is greater than max(m−n,0), equivalently r≥max(1,m−n+1). This is a sufficient sharp general threshold, not permission to use a C¹ theorem in every dimension. Our standing second-countability assumption permits countable chart reductions, while smooth coordinate changes preserve null sets locally.

Measure zero is interpreted in target coordinate charts; no global Euclidean volume form is needed to define the null-set property. Every nonempty target open set has positive coordinate measure, so regular values are dense. Density does not imply that the critical values form a finite or empty set, and without additional conditions it does not by itself make the set of regular values open. Compactness or properness can supply useful stability statements, but those are extra ingredients beyond the measure-zero conclusion.

Sard's theorem is often used by introducing parameters and applying it to a projection from a solution manifold. That is how regular-value arguments lead to transversality results. One must first prove that the parameterized solution set is a manifold and check the projection's dimensions and differentiability. Declaring a perturbation generic without that setup skips the mechanism that makes the claim valid. The next lesson makes the tangent-space condition explicit.

For a practical example, calculate the derivative, solve its rank-failure equations, map those solutions into the target, and distinguish the resulting critical values from other levels. A critical value does not force every point of its fiber to be critical, and a critical fiber might still happen to be a manifold. The theorem classifies which values enjoy the regularity guarantee; it does not classify every exceptional fiber or turn a plotted near-zero derivative into an exact rank certificate.