Round function

In topology and in calculus, by a round function we mean a scalar function $M\to {\mathbb {R} }$, over a manifold $M$, whose critical points form one or several connected components, each homeomorphic to the circle $S^{1}$, also called critical loops.

For instance

For example, let $M$ be the torus. Let $K=]0,2\pi [\times ]0,2\pi [$. Then we know that a map $X\colon K\to {\mathbb {R} }^{3}$ given by

$X(\theta ,\phi )=((2+\cos \theta )\cos \phi ,(2+\cos \theta )\sin \phi ,\sin \theta )$

is a parametrization for almost all of $M$. Now, via the projection $\pi _{3}\colon {\mathbb {R} }^{3}\to {\mathbb {R} }$ we get the restriction $G=\pi _{3}|_{M}\colon M\to {\mathbb {R} },(\theta ,\phi )\mapsto \sin \theta$ whose critical sets are determined by

$\nabla G(\theta ,\phi )=({{\partial }G \over {\partial }\theta },{{\partial }G \over {\partial }\phi })(\theta ,\phi )=(0,0),$

if and only if $\theta ={\pi \over 2},\ {3\pi \over 2}$.

These two values for $\theta$ give the critical sets

$X({\pi /2},\phi )=(2\cos \phi ,2\sin \phi ,1)$

$X({3\pi /2},\phi )=(2\cos \phi ,2\sin \phi ,-1)$

which represent two extremal circles over the torus $M$.

Observe that the Hessian for this function is

${\rm {Hess}}(G)=$${\begin{bmatrix}-\sin \theta &0\\0&0\end{bmatrix}}$

which clearly it reveals itself as of ${\rm {rankHess}}(G)=1$ at the tagged circles, making the critical point degenerate, that is, showing that the critical points are not isolated.

Reference

Siersma and Khimshiasvili, On minimal round functions, [1] Preprint 1118, Department of Mathematics, Utrecht University, 1999, pp. 18.