# round function

Let $M$ be a manifold. By a round function we a function $M\to{\mathbb{R}}$ whose critical points form connected components, each of which is homeomorphic to the circle $S^{1}$.

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 $G=\pi_{3}|_{M}\colon M\to{\mathbb{R}}$ whose critical sets are determined by

 $\nabla G(\theta,\phi)=\left(\begin{array}[]{c}{\partial G\over\partial\theta},% {\partial G\over\partial\phi}\end{array}\right)(\theta,\phi)=(0,0)$

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

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

 $X\left(\begin{array}[]{c}{\pi\over 2},\phi\end{array}\right)=(2\cos\phi,2\sin% \phi,1)$
 $X\left(\begin{array}[]{c}{3\pi\over 2},\phi\end{array}\right)=(2\cos\phi,2\sin% \phi,-1)$

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

Observe that the Hessian for this function is $d^{2}(G)=\left(\begin{array}[]{cc}-\sin\theta&0\\ 0&0\end{array}\right)$ which clearly it reveals itself as of ${\rm rank}(d^{2}(G))=1$ at the tagged circles, making the critical point degenerate, that is, showing that the critical points are not isolated.

Title round function RoundFunction 2013-03-22 15:44:12 2013-03-22 15:44:12 juanman (12619) juanman (12619) 11 juanman (12619) Definition msc 57R70 functions with critical loops DifferntiableFunction