Unfolding (functions)

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In mathematics, an unfolding of a smooth real-valued function ƒ on a smooth manifold, is a certain family of functions that includes ƒ.

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Definition

Let be a smooth manifold and consider a smooth mapping Let us assume that for given and we have . Let be a smooth -dimensional manifold, and consider the family of mappings (parameterised by ) given by We say that is a -parameter unfolding of if for all In other words the functions and are the same: the function is contained in, or is unfolded by, the family

Example

Let be given by An example of an unfolding of would be given by

As is the case with unfoldings, and are called variables, and and are called parameters, since they parameterise the unfolding.

Well-behaved unfoldings

In practice we require that the unfoldings have certain properties. In , is a smooth mapping from to and so belongs to the function space As we vary the parameters of the unfolding, we get different elements of the function space. Thus, the unfolding induces a function The space , where denotes the group of diffeomorphisms of etc., acts on The action is given by If lies in the orbit of under this action then there is a diffeomorphic change of coordinates in and , which takes to (and vice versa). One property that we can impose is that

where "" denotes "transverse to". This property ensures that as we vary the unfolding parameters we can predict – by knowing how the orbit foliates – how the resulting functions will vary.

Versal unfoldings

There is an idea of a versal unfolding. Every versal unfolding has the property that , but the converse is false. Let be local coordinates on , and let denote the ring of smooth functions. We define the Jacobian ideal of , denoted by , as follows:

Then a basis for a versal unfolding of is given by the quotient

.

This quotient is known as the local algebra of . The dimension of the local algebra is called the Milnor number of . 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 . A calculation shows that

This means that 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.

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