In mathematics, particularly in the field of 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.
[1][2] Definition.
be a smooth map between manifolds.
We say that a point
is a regular value of
are the tangent spaces of
be a smooth map, and let
be a regular value of
is a submanifold of
im
{\displaystyle y\in {\text{im}}(f),}
then the codimension of
is equal to the dimension of
Also, the tangent space of
is equal to
ker ( d
There is also a complex version of this theorem:[3] Theorem.
be two complex manifolds of complex dimensions
be a holomorphic map and let
im
{\displaystyle y\in {\text{im}}(g)}
rank
{\displaystyle {\text{rank}}(dg_{x})=m}
is a complex submanifold of
of complex dimension
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