# Critical points

General mappings $${\Phi}$$ between manifolds classify points according to how they transform, and can be used to extract information about the manifolds:

• Regular point: $${p\in M}$$ such that $${\mathrm{d}\Phi_{p}}$$ maps $${T_{p}M}$$ onto $${T_{\Phi(p)}N}$$; if the map is not onto, $${p}$$ is called a critical point
• Regular value: $${q\in N}$$ such that $${\Phi^{-1}\left(q\right)}$$ consists of all regular points or is empty; if $${\Phi^{-1}\left(q\right)}$$ includes a critical point, $${q}$$ is called a critical value

The above depicts the critical points of the height function $${\Phi}$$ mapping a hollow bullet to its vertical component. The flat top and tip only have horizontal tangents, so that $${\mathrm{d}\Phi}$$ is not onto. At a regular point, the tangent to a curve $${C\in M}$$ is mapped to the tangent of the mapped curve $${\Phi\left(C\right)}$$ via the Jacobian.

Sard’s theorem states that if $${\Phi}$$ is sufficiently differentiable, almost all values are regular (we will not elaborate on “sufficient” and “almost” here). Morse theory uses these concepts to extract cell complex structures and homological information from a given manifold.