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In topology and in calculus, a round function is a scalar function $M\to ℝ$, over a manifold $M$, whose critical points form one or several connected components, each homeomorphic to the circle $S^1$, also called critical loops. They are special cases of Morse-Bott functions.

File:Critical-loop.PNG

The black circle in one of this 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

$X:K\to {ℝ}^3$

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

$G={\pi }_3{|}_M:M\to ℝ,(\theta ,\varphi )\mapsto \sin \theta$

$G=G(\theta ,\varphi )=\sin \theta$ is a function whose critical sets are determined by

$\mathrm{\nabla }G(\theta ,\varphi )=(\frac{\partial G}{\partial \theta },\frac{\partial G}{\partial \varphi })(\theta ,\varphi )=(0,0),$

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

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

$X(\pi /2,\varphi )=(2\cos \varphi ,2\sin \varphi ,1)$

$X(3\pi /2,\varphi )=(2\cos \varphi ,2\sin \varphi ,-1)$

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

Observe that the Hessian for this function is

$\mathrm{H}\mathrm{e}\mathrm{s}\mathrm{s}(G)=\left[\begin{array}{cc}-\sin \theta & 0\\ 0& 0\end{array}\right]$

which clearly it reveals itself as of $\mathrm{r}\mathrm{a}\mathrm{n}\mathrm{k}\mathrm{H}\mathrm{e}\mathrm{s}\mathrm{s}(G)=1$ at the tagged circles, making the critical point degenerate, that is, showing that the critical points are not isolated.

Round complexity

Mimicking the L-S category theory one can define the round complexity asking for whether or not exist round functions on manifolds and/or for the minimum number of critical loops.

References

• Siersma and Khimshiasvili, On minimal round functions, Preprint 1118, Department of Mathematics, Utrecht University, 1999, pp. 18.[1]. An update at [2]