round function

Let $M$ be a manifold. By a round function we a function $M\to ℝ$ 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$X:K\to {ℝ}^3$ given by

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

is a parametrization for almost all of $M$. Now, via the projection ${\pi }_3:{ℝ}^3\to ℝ$we get the restriction $G={{\pi }_3|}_M:M\to ℝ$ whose critical sets are determined by

$$\nabla G(\theta ,\varphi )=\left(\begin{array}{c}\hfill \frac{\partial G}{\partial \theta },\frac{\partial G}{\partial \varphi }\hfill \end{array}\right)(\theta ,\varphi )=(0,0)$$

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

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

$$X\left(\begin{array}{c}\hfill \frac{\pi }{2},\varphi \hfill \end{array}\right)=(2\cos \varphi ,2\sin \varphi ,1)$$

$$X\left(\begin{array}{c}\hfill \frac{3\pi }{2},\varphi \hfill \end{array}\right)=(2\cos \varphi ,2\sin \varphi ,-1)$$

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

Observe that the Hessian for this function iswhich clearly it reveals itself as of $\mathrm{rank}(d^2(G))=1$ at the tagged circles,making the critical point degenerate, that is, showing that the critical points are not isolated.