topology In mathematics, topology (from the Greek words , and ) is concerned with the properties of a geometric object that are preserved under continuous deformations, such as stretching, twisting, crumpling, and bending; that is, without closing ho ...

and in

calculus Calculus, originally called infinitesimal calculus or "the calculus of infinitesimals", is the mathematics, mathematical study of continuous change, in the same way that geometry is the study of shape, and algebra is the study of generalizati ...

, a round function is a

scalar function In mathematics and physics, a scalar field is a function associating a single number to every point in a space – possibly physical space. The scalar may either be a pure mathematical number (dimensionless) or a scalar physical quantity ...

M\to

, over a

manifold In mathematics, a manifold is a topological space that locally resembles Euclidean space near each point. More precisely, an n-dimensional manifold, or ''n-manifold'' for short, is a topological space with the property that each point has a ...

M

, whose critical points form one or several connected components, each homeomorphic to the

circle A circle is a shape consisting of all points in a plane that are at a given distance from a given point, the centre. Equivalently, it is the curve traced out by a point that moves in a plane so that its distance from a given point is const ...

S^1

, also called critical loops. They are special cases of Morse-Bott functions.

For instance

For example, let

M

be the

torus In geometry, a torus (plural tori, colloquially donut or doughnut) is a surface of revolution generated by revolving a circle in three-dimensional space about an axis that is coplanar with the circle. If the axis of revolution does not ...

. Let :

K=(0,2\pi)\times(0,2\pi).\,

Then we know that a map :

X\colon K\to^3\,

given by :

X(\theta,\phi)=((2+\cos\theta)\cos\phi,(2+\cos\theta)\sin\phi,\sin\theta)\,

is a parametrization for almost all of

M

. Now, via the projection

\pi_3\colon^3\to

we get the restriction :

G=\pi_3, _M\colon M\to,  (\theta,\phi) \mapsto \sin \theta \,

G=G(\theta,\phi)=\sin\theta

is a function whose critical sets are determined by :

\ G(\theta,\phi)=
\left(,\right)\!\left(\theta,\phi\right)=(0,0),\,

this is if and only if

\theta=,\

. These two values for

\theta

give the critical sets :

X(,\phi)=(2\cos\phi,2\sin\phi,1)\,

X(,\phi)=(2\cos\phi,2\sin\phi,-1)\,

which represent two extremal circles over the torus

M

. Observe that the Hessian for this function is :

(G)=
\begin 
-\sin\theta & 0 \\ 0 & 0 \end

which clearly it reveals itself as rank of

{\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

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. 1

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Differential geometry Geometric topology Types of functions