Round function

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. They are special cases of Morse-Bott functions.

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

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

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

$G=\pi_3|_M\colon M\to{\mathbb{R}}, (\theta,\phi) \mapsto \sin \theta \,$

G = G(θ,ϕ) = sin θ is a function whose critical sets are determined by

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

this is 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.

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]