Round function

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In topology and in calculus, a round function is 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 S1, also called critical loops.

The black circle in one of this critical loops.
The black circle in one of this critical loops.

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

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

These two values for θ 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 rankHess(G) = 1 at the tagged circles, making the critical point degenerate, that is, showing that the critical points are not isolated.

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.

  • Siersma and Khimshiasvili, On minimal round functions, [1]

Preprint 1118, Department of Mathematics, Utrecht University, 1999, pp. 18.

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