mathematics Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...

, particularly in the field of

differential topology In mathematics, differential topology is the field dealing with the topological properties and smooth properties of smooth manifolds. In this sense differential topology is distinct from the closely related field of differential geometry, which ...

, the preimage theorem is a variation of the implicit function theorem concerning the

preimage In mathematics, the image of a function is the set of all output values it may produce. More generally, evaluating a given function f at each element of a given subset A of its domain produces a set, called the "image of A under (or through) ...

of particular points in a

manifold In mathematics, a manifold is a topological space that locally resembles Euclidean space near each point. More precisely, an n-dimensional manifold, or ''n-manifold'' for short, is a topological space with the property that each point has a n ...

under the action of a

smooth map In mathematical analysis, the smoothness of a function is a property measured by the number of continuous derivatives it has over some domain, called ''differentiability class''. At the very minimum, a function could be considered smooth if it ...

Statement of Theorem

''Definition.'' Let

f : X \to Y

be a smooth map between manifolds. We say that a point

y \in Y

is a ''regular value of''

f

if for all

x \in f^(y)

the map

d f_x: T_x X \to T_y Y

surjective In mathematics, a surjective function (also known as surjection, or onto function) is a function that every element can be mapped from element so that . In other words, every element of the function's codomain is the image of one element of i ...

. Here,

T_x X

and

T_y Y

are the

tangent space In mathematics, the tangent space of a manifold generalizes to higher dimensions the notion of '' tangent planes'' to surfaces in three dimensions and ''tangent lines'' to curves in two dimensions. In the context of physics the tangent space to a ...

s of

X

and

Y

at the points

x

and

y.

''Theorem.'' Let

f: X \to Y

be a smooth map, and let

y \in Y

be a regular value of

f.

Then

f^(y)

is a submanifold of

X.

y \in \text(f),

then the

codimension In mathematics, codimension is a basic geometric idea that applies to subspaces in vector spaces, to submanifolds in manifolds, and suitable subsets of algebraic varieties. For affine and projective algebraic varieties, the codimension equals the ...

f^(y)

is equal to the dimension of

Y.

Also, the

f^(y)

x

is equal to

\ker(df_x).

There is also a complex version of this theorem:. ''Theorem.'' Let

X^n

and

Y^m

be two

complex manifolds In differential geometry and complex geometry, a complex manifold is a manifold with an atlas of charts to the open unit disc in \mathbb^n, such that the transition maps are holomorphic. The term complex manifold is variously used to mean a com ...

of complex dimensions

n > m.

Let

g : X \to Y

be a

holomorphic In mathematics, a holomorphic function is a complex-valued function of one or more complex variables that is complex differentiable in a neighbourhood of each point in a domain in complex coordinate space . The existence of a complex derivativ ...

map and let

y \in \text(g)

be such that

\text(dg_x) = m

for all

x \in g^(y).

Then

g^(y)

is a complex submanifold of

X

of complex dimension

n - m.

References

{{topology-stub Theorems in differential topology

Statement of Theorem

See also

References