In topology and in calculus , a round function is a scalar function M → R {\displaystyle M\to {\mathbb {R} }} , over a manifold M {\displaystyle M} , whose critical points form one or several connected components , each homeomorphic to the circle S 1 {\displaystyle S^{1}} , also called critical loops. They are special cases of Morse-Bott functions .
The black circle in one of this critical loops. For instance [ edit ] For example, let M {\displaystyle M} be the torus . Let
K = ( 0 , 2 π ) × ( 0 , 2 π ) . {\displaystyle K=(0,2\pi )\times (0,2\pi ).\,} Then we know that a map
X : K → R 3 {\displaystyle X\colon K\to {\mathbb {R} }^{3}\,} given by
X ( θ , ϕ ) = ( ( 2 + cos θ ) cos ϕ , ( 2 + cos θ ) sin ϕ , sin θ ) {\displaystyle X(\theta ,\phi )=((2+\cos \theta )\cos \phi ,(2+\cos \theta )\sin \phi ,\sin \theta )\,} is a parametrization for almost all of M {\displaystyle M} . Now, via the projection π 3 : R 3 → R {\displaystyle \pi _{3}\colon {\mathbb {R} }^{3}\to {\mathbb {R} }} we get the restriction
G = π 3 | M : M → R , ( θ , ϕ ) ↦ sin θ {\displaystyle G=\pi _{3}|_{M}\colon M\to {\mathbb {R} },(\theta ,\phi )\mapsto \sin \theta \,} G = G ( θ , ϕ ) = sin θ {\displaystyle G=G(\theta ,\phi )=\sin \theta } is a function whose critical sets are determined by
g r a d G ( θ , ϕ ) = ( ∂ G ∂ θ , ∂ G ∂ ϕ ) ( θ , ϕ ) = ( 0 , 0 ) , {\displaystyle {\rm {grad}}\ 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 θ = π 2 , 3 π 2 {\displaystyle \theta ={\pi \over 2},\ {3\pi \over 2}} .
These two values for θ {\displaystyle \theta } give the critical sets
X ( π / 2 , ϕ ) = ( 2 cos ϕ , 2 sin ϕ , 1 ) {\displaystyle X({\pi /2},\phi )=(2\cos \phi ,2\sin \phi ,1)\,} X ( 3 π / 2 , ϕ ) = ( 2 cos ϕ , 2 sin ϕ , − 1 ) {\displaystyle X({3\pi /2},\phi )=(2\cos \phi ,2\sin \phi ,-1)\,} which represent two extremal circles over the torus M {\displaystyle M} .
Observe that the Hessian for this function is
h e s s ( G ) = [ − sin θ 0 0 0 ] {\displaystyle {\rm {hess}}(G)={\begin{bmatrix}-\sin \theta &0\\0&0\end{bmatrix}}} which clearly it reveals itself as rank of h e s s ( G ) {\displaystyle {\rm {hess}}(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 [ edit ] 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 [ edit ] Siersma and Khimshiasvili, On minimal round functions , Preprint 1118, Department of Mathematics, Utrecht University, 1999, pp. 18.[1] . An update at [2]