Round function explained

M\to{R

}, over a manifold

, 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.

For instance

For example, let

be the torus. Let

K=(0,2\pi) x (0,2\pi).

Then we know that a map

X\colonK\to{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

. Now, via the projection

\pi_3\colon{R

}^3\towe get the restriction

G=\pi_3|_M\colonM\to{R

}, (\theta,\phi) \mapsto \sin \theta \,

G=G(\theta,\phi)=\sin\theta

is a function whose critical sets are determined by

{\rmgrad} 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\over2}, {3\pi\over2}

These two values for

\theta

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

Observe that the Hessian for this function is

{\rmhess}(G)= \begin{bmatrix}-\sin\theta&0\ 0&0\end{bmatrix}

which clearly it reveals itself as rank of

{\rmhess}(G)

equal to one 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.http://citeseer.ist.psu.edu/287481.html. An update at http://igitur-archive.library.uu.nl/math/2001-0628-161022/1118.pdf