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

The black circle in one of this critical loops.

For instance

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For example, let   be the torus. Let

 

Then we know that a map

 

given by

 

is a parametrization for almost all of  . Now, via the projection   we get the restriction

 

  is a function whose critical sets are determined by

 

this is if and only if  .

These two values for   give the critical sets

 
 

which represent two extremal circles over the torus  .

Observe that the Hessian for this function is

 

which clearly it reveals itself as rank of   equal to one at the tagged circles, making the critical point degenerate, that is, showing that the critical points are not isolated.

Round complexity

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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

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