Unfolding (functions)

In mathematics, an unfolding of a smooth real-valued function "ƒ" on a smooth manifold, is a certain family of functions that includes "ƒ".
Definition.
Let formula_1 be a smooth manifold and consider a smooth mapping formula_2 Let us assume that for given formula_3 and formula_4 we have formula_5. Let formula_6 be a smooth formula_7-dimensional manifold, and consider the family of mappings (parameterised by formula_8) given by formula_9 We say that formula_10 is a formula_7-parameter unfolding of formula_12 if formula_13 for all formula_14 In other words the functions formula_15 and formula_16 are the same: the function formula_12 is contained in, or is unfolded by, the family formula_18
Example.
Let formula_19 be given by formula_20 An example of an unfolding of formula_12 would be formula_22 given by 
As is the case with unfoldings, formula_24 and formula_25 are called variables, and formula_26 formula_27 and formula_28 are called parameters, since they parameterise the unfolding.
Well-behaved unfoldings.
In practice we require that the unfoldings have certain properties. In formula_29, formula_12 is a smooth mapping from formula_1 to formula_29 and so belongs to the function space formula_33 As we vary the parameters of the unfolding, we get different elements of the function space. Thus, the unfolding induces a function formula_34 The space formula_35, where formula_36 denotes the group of diffeomorphisms of formula_1 etc., acts on formula_33 The action is given by formula_39 If formula_40 lies in the orbit of formula_12 under this action then there is a diffeomorphic change of coordinates in formula_1 and formula_29, which takes formula_40 to formula_12 (and vice versa). One property that we can impose is that 
where "formula_47" denotes "transverse to". This property ensures that as we vary the unfolding parameters we can predict – by knowing how the orbit foliates formula_48 – how the resulting functions will vary.
Versal unfoldings.
There is an idea of a versal unfolding. Every versal unfolding has the property that 
formula_46, but the converse is false. Let formula_50 be local coordinates on formula_1, and let formula_52 denote the ring of smooth functions. We define the Jacobian ideal of formula_12, denoted by formula_54, as follows:
Then a basis for a versal unfolding of formula_12 is given by the quotient
This quotient is known as the local algebra of formula_12. The dimension of the local algebra is called the Milnor number of formula_12. The minimum number of unfolding parameters for a versal unfolding is equal to the Milnor number; that is not to say that every unfolding with that many parameters will be versal. Consider the function formula_60. A calculation shows that
This means that formula_62 give a basis for a versal unfolding, and that 
is a versal unfolding. A versal unfolding with the minimum possible number of unfolding parameters is called a miniversal unfolding.
Bifurcations sets of unfoldings.
An important object associated to an unfolding is its bifurcation set. This set lives in the parameter space of the unfolding, and gives all parameter values for which the resulting function has degenerate singularities.
Other terminology.
Sometimes unfoldings are called deformations, versal unfoldings are called versal deformations, etc.