In mathematics

Mathematics

Mathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...

, particularly in differential topology

Differential topology

In mathematics, differential topology is the field dealing with differentiable functions on differentiable manifolds. It is closely related to differential geometry and together they make up the geometric theory of differentiable manifolds.- Description :...

, the preimage theorem is a theorem concerning the preimage of particular points in a manifold

Manifold

In mathematics , a manifold is a topological space that on a small enough scale resembles the Euclidean space of a specific dimension, called the dimension of the manifold....

 under the action of a smooth map.

Statement of Theorem

Definition. Let be a smooth map between manifolds. We say that a point is a regular value of f if for all the map is surjective. Here, and are the tangent space

Tangent space

In mathematics, the tangent space of a manifold facilitates the generalization of vectors from affine spaces to general manifolds, since in the latter case one cannot simply subtract two points to obtain a vector pointing from one to the other....

s of X and Y at the points x and y.
Theorem. Let be a smooth map, and let be a regular value of f. Then is a submanifold of X. Further, the codimension

Codimension

In mathematics, codimension is a basic geometric idea that applies to subspaces in vector spaces, and also to submanifolds in manifolds, and suitable subsets of algebraic varieties.The dual concept is relative dimension.-Definition:...

of this manifold in X is equal to the dimension of Y.