In mathematics, an unfolding of a smooth real-valued function ƒ on a smooth manifold, is a certain family of functions that includes ƒ.
-dimensional manifold, and consider the family of mappings (parameterised by
is contained in, or is unfolded by, the family
are called parameters, since they parameterise the unfolding.
In practice we require that the unfoldings have certain properties.
is a smooth mapping from
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
denotes the group of diffeomorphisms of
under this action then there is a diffeomorphic change of coordinates in
" 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.
There is an idea of a versal unfolding.
Every versal unfolding has the property that
{\displaystyle \operatorname {Im} (\Phi )\pitchfork \operatorname {orb} (f)}
, but the converse is false.
be local coordinates on
denote the ring of smooth functions.
We define the Jacobian ideal of
, as follows: Then a basis for a versal unfolding 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.
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.
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.
Sometimes unfoldings are called deformations, versal unfoldings are called versal deformations, etc.