Critical point of a smooth map - Imported Knowlpedia Content

Definition

Let $f:M\to N$ be a smooth map of finite-dimensional smooth manifolds. A point $p\in M$ is a critical point of $f$ if the differential

df_p:T_pM\longrightarrow T_{f(p)}N

is not surjective. Equivalently, $p$ is critical when the rank of $f$ at $p$ is strictly less than $\dim N$ . A point at which $df_p$ is surjective is a regular point. This definition is intrinsic: coordinate changes multiply a Jacobian on the left and right by invertible matrices, so they do not change its rank or surjectivity.

Scalar-valued maps

For $f:M\to\mathbb R$ , the target is one-dimensional, so $p$ is critical exactly when $df_p=0$ . In local coordinates this means that all first partial derivatives vanish. After choosing a Riemannian metric, the same condition can be written $\operatorname{grad}f(p)=0$ , but the metric and gradient are not needed for the definition.

Dimension and examples

If $\dim M<\dim N$ , no differential $T_pM\to T_{f(p)}N$ can be surjective, so every point is critical under this convention. For $f:\mathbb R^2\to\mathbb R$ , $f(x,y)=x^2+y^2$ , only the origin is critical. For the projection $M\times F\to M$ , every point is regular because the differential is surjective.

Conventions and scope

“Singular point” is often used synonymously with critical point. In immersion theory, however, an author may instead call a point singular when $df_p$ fails to be injective. The convention here is the one used for regular values and Sard's theorem: critical means failure of surjectivity. Compare Guillemin–Pollack, Chapter 1.

References

John M. Lee, Introduction to Smooth Manifolds, 2nd ed., Springer, 2012. DOI record. Relevant: chapters on submersions, regular points, and Sard's theorem.
Victor Guillemin and Alan Pollack, Differential Topology, AMS Chelsea Publishing, 2010 reprint. DOI record. Relevant: Chapter 1, critical points and regular values.