Round function

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"Round function" may also refer to rounding.

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

Video Round function

For instance

For example, let ${\displaystyle M}$ be the torus. Let

${\displaystyle K=(0,2\pi )\times (0,2\pi ).\,}$

Then we know that a map

${\displaystyle X\colon K\to {\mathbb {R} }^{3}\,}$

given by

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

is a parametrization for almost all of ${\displaystyle M}$. Now, via the projection ${\displaystyle \pi _{3}\colon {\mathbb {R} }^{3}\to {\mathbb {R} }}$ we get the restriction

${\displaystyle G=\pi _{3}|_{M}\colon M\to {\mathbb {R} },(\theta ,\phi )\mapsto \sin \theta \,}$

${\displaystyle G=G(\theta ,\phi )=\sin \theta }$ is a function whose critical sets are determined by

${\displaystyle {\rm {grad}}\ G(\theta ,\phi )=\left({{\partial }G \over {\partial }\theta },{{\partial }G \over {\partial }\phi }\right)\!\left(\theta ,\phi \right)=(0,0),\,}$

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

These two values for ${\displaystyle \theta }$ give the critical sets

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

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

which represent two extremal circles over the torus ${\displaystyle M}$.

Observe that the Hessian for this function is

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

which clearly it reveals itself as rank of ${\displaystyle {\rm {hess}}(G)}$ equal to one at the tagged circles, making the critical point degenerate, that is, showing that the critical points are not isolated.

Maps Round function

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.

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References

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

Source of article : Wikipedia