In topology and in calculus, a round function is a scalar function, over a manifold, whose critical pointss form one or several connected componentss, each homeomorphic to the circle, also called critical loops. Morse-Bott functions are special cases of round functions.

For instance

be the torus. Let

Then we know that a map

given by

we get the restriction

is a function whose critical sets are determined by

.

give the critical sets

.

Observe that the Hessian for this function is

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