The preimage theorem, also known as the regular value theorem or the submersion theorem, is a fundamental result in differential topology. It describes the structure of the preimage of a point under a smooth map between smooth manifolds, provided that the point is a regular value of the map.
Formally, let $ f: M \to N $ be a smooth map between smooth manifolds, and let $ y \in N $. A point $ y $ is called a regular value of $ f $ if for every $ x \in f^{-1}(y) $, the differential $ df_x: T_xM \to T_yN $ is surjective. The preimage theorem states that if $ y $ is a regular value of $ f $, then the preimage $ f^{-1}(y) $ is a smooth submanifold of $ M $, and its dimension is given by $ \dim M - \dim N $.
This theorem is a direct consequence of the implicit function theorem and is widely used in geometry and topology to construct submanifolds and analyze level sets of smooth functions. It also forms the basis for concepts such as transversality and plays a crucial role in proofs involving intersection theory and degree theory.
Special cases of the preimage theorem include the construction of embedded submanifolds via constraints (e.g., spheres as level sets of distance functions) and underlie important results such as Sard’s theorem, which asserts that the set of critical values has measure zero.
The theorem generalizes to various settings, including maps between Banach manifolds (in infinite-dimensional contexts), under appropriate conditions.