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

M

{\displaystyle M}

be a smooth manifold and consider a smooth mapping

f

:

M

→

R

.

{\displaystyle f:M\to \mathbb {R} .}

Let us assume that for given

x

0

∈

M

{\displaystyle x_{0}\in M}

and

y

0

∈

R

{\displaystyle y_{0}\in \mathbb {R} }

we have

f

(

x

0

)

=

y

0

{\displaystyle f(x_{0})=y_{0}}

. Let

N

{\displaystyle N}

be a smooth

k

{\displaystyle k}

-dimensional manifold, and consider the family of mappings (parameterised by

N

{\displaystyle N}

) given by

F

:

M

×

N

→

R

.

{\displaystyle F:M\times N\to \mathbb {R} .}

We say that

F

{\displaystyle F}

is a

k

{\displaystyle k}

-parameter unfolding of

f

{\displaystyle f}

if

F

(

x

,

0

)

=

f

(

x

)

{\displaystyle F(x,0)=f(x)}

for all

x

.

{\displaystyle x.}

In other words the functions

f

:

M

→

R

{\displaystyle f:M\to \mathbb {R} }

and

F

:

M

×

{

0

}

→

R

{\displaystyle F:M\times \{0\}\to \mathbb {R} }

are the same: the function

f

{\displaystyle f}

is contained in, or is unfolded by, the family

F

.

{\displaystyle F.}

Example

Let

f

:

R

2

→

R

{\displaystyle f:\mathbb {R} ^{2}\to \mathbb {R} }

be given by

f

(

x

,

y

)

=

x

2

+

y

5

.

{\displaystyle f(x,y)=x^{2}+y^{5}.}

An example of an unfolding of

f

{\displaystyle f}

would be

F

:

R

2

×

R

3

→

R

{\displaystyle F:\mathbb {R} ^{2}\times \mathbb {R} ^{3}\to \mathbb {R} }

given by

F

(

(

x

,

y

)

,

(

a

,

b

,

c

)

)

=

x

2

+

y

5

+

a

y

+

b

y

2

+

c

y

3

.

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

As is the case with unfoldings,

x

{\displaystyle x}

and

y

{\displaystyle y}

are called variables, and

a

,

{\displaystyle a,}

b

,

{\displaystyle b,}

and

c

{\displaystyle c}

are called parameters, since they parameterise the unfolding.

Well-behaved unfoldings

In practice we require that the unfoldings have certain properties.