Preimage theorem

Short description: On the preimage of points in a manifold under the action of a smooth map

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.

Statement of Theorem

Definition. Let [math]\displaystyle{ f : X \to Y }[/math] be a smooth map between manifolds. We say that a point [math]\displaystyle{ y \in Y }[/math] is a regular value of [math]\displaystyle{ f }[/math] if for all [math]\displaystyle{ x \in f^{-1}(y) }[/math] the map [math]\displaystyle{ d f_x: T_x X \to T_y Y }[/math] is surjective. Here, [math]\displaystyle{ T_x X }[/math] and [math]\displaystyle{ T_y Y }[/math] are the tangent spaces of [math]\displaystyle{ X }[/math] and [math]\displaystyle{ Y }[/math] at the points [math]\displaystyle{ x }[/math] and [math]\displaystyle{ y. }[/math]

Theorem. Let [math]\displaystyle{ f: X \to Y }[/math] be a smooth map, and let [math]\displaystyle{ y \in Y }[/math] be a regular value of [math]\displaystyle{ f. }[/math] Then [math]\displaystyle{ f^{-1}(y) }[/math] is a submanifold of [math]\displaystyle{ X. }[/math] If [math]\displaystyle{ y \in \text{im}(f), }[/math] then the codimension of [math]\displaystyle{ f^{-1}(y) }[/math] is equal to the dimension of [math]\displaystyle{ Y. }[/math] Also, the tangent space of [math]\displaystyle{ f^{-1}(y) }[/math] at [math]\displaystyle{ x }[/math] is equal to [math]\displaystyle{ \ker(df_x). }[/math]

There is also a complex version of this theorem:

Theorem. Let [math]\displaystyle{ X^n }[/math] and [math]\displaystyle{ Y^m }[/math] be two complex manifolds of complex dimensions [math]\displaystyle{ n \gt m. }[/math] Let [math]\displaystyle{ g : X \to Y }[/math] be a holomorphic map and let [math]\displaystyle{ y \in \text{im}(g) }[/math] be such that [math]\displaystyle{ \text{rank}(dg_x) = m }[/math] for all [math]\displaystyle{ x \in g^{-1}(y). }[/math] Then [math]\displaystyle{ g^{-1}(y) }[/math] is a complex submanifold of [math]\displaystyle{ X }[/math] of complex dimension [math]\displaystyle{ n - m. }[/math]

See also

• Fiber (mathematics) – Set of all points in a function's domain that all map to some single given point
• Level set – Subset of a function's domain on which its value is equal