Unfolding (functions)

In mathematics there exists and idea of an unfolding. Given a function, we can talk about unfolding the function, to give a family of functions.

Let $M$ and $\mathbb {R}$ be two smooth manifolds, and consider a smooth mapping $f:M\to \mathbb {R} .$ Let us assume that for given $x_{0}\in M$ and $y_{0}\in \mathbb {R}$ we have $f(x_{0})=y_{0}$. Let $N$ be a smooth manifold, and consider the family of mapping (parameterised by $N$) given by $F:M\times N\to \mathbb {R} .$ We say that $F$ is an unfolding of $f$ at $x_{0}\in M$ if $F(x,0)=f(x)$ for all $x.$ In other words the functions $f:M\to \mathbb {R}$ and $F:M\times \{0\}\to \mathbb {R}$ are the same: the function $f$ is contained in , or is unfolded by, the family $F.$

Let $f:\mathbb {R} ^{2}\to \mathbb {R}$ be given by $f(u,v)=u^{2}+v^{5}.$ An example of an unfolding of $f$ would be $F:\mathbb {R} ^{2}\times \mathbb {R} ^{3}\to \mathbb {R}$ given by

$F((x,y),(a,b,c))=x^{2}+y^{5}+ay+by^{2}+cy^{3}.$

As is the case with unfoldings, $x$ and $y$ are called variables and $a,$ $b,$ and $c$ are called parameters - since they parameterise the unfolding.

In application we require that the unfoldings have certain nice properties. \mathbb{R}otice that $f$ is a smooth mapping from $M$ to $\mathbb {R}$ and so belongs to the function space $C^{\infty }(M,\mathbb {R} ).$ As we vary the parameters of the unfolding we get different elements of the function space. Thus, the unfolding induces a function $\Phi :N\to C^{\infty }(M,\mathbb {R} ).$ The space ${\mbox{diff}}(M)\times {\mbox{diff}}(\mathbb {R} ),$ where ${\mbox{diff}}(M)$ denotes the group of diffeomorphisms of $M$ etc, acts on $C^{\infty }(M,\mathbb {R} ).$ The action is given by $(\phi ,\psi )\cdot f=\psi \circ f\circ \phi ^{-1}.$ If $g$ lies in the orbit of $f$ under this action then there is a diffeomorphic change of coordinates in $M$ and $\mathbb {R}$ which takes $g$ to $f$ (and vise versa). One nice property that we may like to impose is that

${\mbox{Im}}(\Phi )\pitchfork {\mbox{orb}}(f)$

where "$\pitchfork$" denotes "transverse to". This property ensures that as we vary the unfolding parameters we can predict - by knowing how the orbit foliate $C^{\infty }(M,\mathbb {R} )$ - how the resulting functions will vary.

There is an idea of a versal unfolding. Every versal unfolding has the property that ${\mbox{Im}}(\Phi )\pitchfork {\mbox{orb}}(f)$, but the converse is false. Let $x_{1},\ldots ,x_{n}$ be local coordinates on $M$, and let ${\mathcal {O}}(x_{1},\ldots ,x_{n})$ denote the ring of smooth functions. We define the Jacobian ideal of $f,$ denoted by $J_{f}$ as follows:

$J_{f}:=\left\langle {\frac {\partial f}{\partial x_{1}}},\ldots ,{\frac {\partial f}{\partial x_{n}}}\right\rangle .$

Then a basis for a versal unfolding of $f$ is given by quotient

${\frac {\mathcal {O}}{J_{f}}}$