In mathematics, particularly in differential topology, the preimage theorem is a variation of the implicit function theorem concerning the preimage of particular points in a manifold under the action of a smooth map.
Definition. Let
f
:
X
→
Y
be a smooth map between manifolds. We say that a point
y
∈
Y
is a regular value of f if for all
x
∈
f
−
1
(
y
)
the map
d
f
x
:
T
x
X
→
T
y
Y
is surjective. Here,
T
x
X
and
T
y
Y
are the tangent spaces of X and Y at the points x and y.
Theorem. Let
f
:
X
→
Y
be a smooth map, and let
y
∈
Y
be a regular value of f; then
f
−
1
(
y
)
is a submanifold of X. If
y
∈
im
(
f
)
, then the codimension of
f
−
1
(
y
)
is equal to the dimension of Y. Also, the tangent space of
f
−
1
(
y
)
at
x
is equal to
ker
(
d
f
x
)
.
