Round function
In topology and in calculus, a round function is a scalar function formula_1, over a manifold formula_2, whose critical points form one or several connected components, each homeomorphic to the circle formula_3, also called critical loops. They are special cases of Morse-Bott functions. For instance. For example, let formula_2 be the torus. Let Then we know that a map given by is a parametrization for almost all of formula_2. Now, via the projection formula_9 we get the restriction formula_11 is a function whose critical sets are determined by this is if and only if formula_13. These two  values for formula_14 give the critical sets which represent two extremal circles over the torus formula_2. Observe that the Hessian for this function is which clearly it reveals itself as rank of formula_19 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.