round function

Let M be a manifold. By a round function we a functionMathworldPlanetmath M whose critical pointsDlmfMathworld form connected componentsMathworldPlanetmathPlanetmathPlanetmath, each of which is homeomorphic to the circle S1.

For example, let M be the torus. Let K=]0,2π[×]0,2π[. Then we know that a map X:K3 given by


is a parametrization for almost all of M. Now, via the projection π3:3 we get the restriction G=π3|M:M whose critical sets are determined by


if and only if θ=π2,3π2.

These two values for θ give the critical set


which represent two extremal circles over the torus M.

Observe that the Hessian for this function is d2(G)=(-sinθ000) which clearly it reveals itself as of rank(d2(G))=1 at the tagged circles, making the critical point degenerate, that is, showing that the critical points are not isolated.

Title round function
Canonical name RoundFunction
Date of creation 2013-03-22 15:44:12
Last modified on 2013-03-22 15:44:12
Owner juanman (12619)
Last modified by juanman (12619)
Numerical id 11
Author juanman (12619)
Entry type Definition
Classification msc 57R70
Synonym functions with critical loops
Related topic DifferntiableFunction