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In [[topology]] and in [[calculus]],  a '''round function''' is a [[scalar function]] <math>M\to{\mathbb{R}}</math>, 
over a [[manifold]] <math>M</math>, whose [[critical point (mathematics)|critical point]]s form one or several [[connected space|connected component]]s, each [[homeomorphic]] to the [[circle]]
<math>S^1</math>, also called critical loops. They are special cases of [[Morse-Bott function]]s.
 
[[Image:Critical-loop.PNG|right|thumb|300px|The black circle in one of this critical loops.]]
 
==For instance==
For example, let <math>M</math> be the [[torus (mathematics)|torus]]. Let
 
:<math>K=(0,2\pi)\times(0,2\pi).\,</math>
 
Then we know that a map
 
:<math>X\colon K\to{\mathbb{R}}^3\,</math>
 
given by
 
:<math>X(\theta,\phi)=((2+\cos\theta)\cos\phi,(2+\cos\theta)\sin\phi,\sin\theta)\,</math>
 
is a parametrization for almost all of <math>M</math>. Now, via the projection <math>\pi_3\colon{\mathbb{R}}^3\to{\mathbb{R}}</math>
we get the restriction
 
:<math>G=\pi_3|_M\colon M\to{\mathbb{R}}, (\theta,\phi) \mapsto \sin \theta \,</math>
 
<math>G=G(\theta,\phi)=\sin\theta</math> is a function whose critical sets are determined by
:<math>\nabla G(\theta,\phi)=
\left({{\partial}G\over {\partial}\theta},{{\partial}G\over {\partial}\phi}\right)\!\left(\theta,\phi\right)=(0,0),\,</math>
 
this is if and only if <math>\theta={\pi\over 2},\ {3\pi\over 2}</math>.
 
These two values for <math>\theta</math> give the critical sets
 
:<math>X({\pi/2},\phi)=(2\cos\phi,2\sin\phi,1)\,</math>
:<math>X({3\pi/2},\phi)=(2\cos\phi,2\sin\phi,-1)\,</math>
 
which represent two extremal circles over the torus <math>M</math>.
 
Observe that the [[Hessian matrix|Hessian]] for this function  is
 
:<math>{\rm Hess}(G)=
\begin{bmatrix}
-\sin\theta & 0 \\ 0 & 0 \end{bmatrix}
</math>
 
which clearly it reveals itself as of <math>{\rm rank Hess}(G)=1</math>
at the tagged circles, making the critical point degenerate, that is, showing that the critical points are not isolated.
 
==Round complexity==
Mimicking the [[Lyusternik-Schnirelmann category|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.&nbsp;18.[http://citeseer.ist.psu.edu/287481.html]. An update at [http://igitur-archive.library.uu.nl/math/2001-0628-161022/1118.pdf]
 
[[Category:Differential geometry]]
[[Category:Geometric topology]]
[[Category:Types of functions]]

Latest revision as of 14:49, 20 July 2013

In topology and in calculus, a round function is a scalar function M, over a manifold M, whose critical points form one or several connected components, each homeomorphic to the circle S1, also called critical loops. They are special cases of Morse-Bott functions.

The black circle in one of this critical loops.

For instance

For example, let M be the torus. Let

K=(0,2π)×(0,2π).

Then we know that a map

X:K3

given by

X(θ,ϕ)=((2+cosθ)cosϕ,(2+cosθ)sinϕ,sinθ)

is a parametrization for almost all of M. Now, via the projection π3:3 we get the restriction

G=π3|M:M,(θ,ϕ)sinθ

G=G(θ,ϕ)=sinθ is a function whose critical sets are determined by

G(θ,ϕ)=(Gθ,Gϕ)(θ,ϕ)=(0,0),

this is if and only if θ=π2,3π2.

These two values for θ give the critical sets

X(π/2,ϕ)=(2cosϕ,2sinϕ,1)
X(3π/2,ϕ)=(2cosϕ,2sinϕ,1)

which represent two extremal circles over the torus M.

Observe that the Hessian for this function is

Hess(G)=[sinθ000]

which clearly it reveals itself as of rankHess(G)=1 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.[1]. An update at [2]