Preimage Theorem

= Theorem =

Let $$y$$ be a regular value of a smooth submersion $$f:X \to Y$$.

Then the preimage $$f^{-1}(y)$$ is a smooth submanifold of $$X$$, with $$\dim f^{-1}(y) = \dim X - \dim Y$$.

= Proof =

Let $$k,l$$ be natural numbers with $$k \geq l$$.

By the Local Submersion Theorem, there exists coordinates in some open sets of $$x,y$$ such that $$f(x_1, x_2, \ldots, x_k)=(x_1, \ldots,x_l)$$ and $$y$$ corresponds to $$(0, \ldots, 0)$$.

Let $$V$$ be that neighborhood of $$x$$.

Then $$f^{-1}(y) \cap V$$ is the set of points where $$x_1=0, \ldots, x_l=0$$.

The functions $$x_{l+1}, \ldots, x_k$$ therefore form a coordinate system on the set $$f^{-1}(y) \cap V$$, which is a relatively open subset of $$f^{-1}(y)$$.

Together these functions then form a diffeomorphism to a Euclidean space.

We also have, by the regular value properties of $$y$$, a surjection of tangent spaces from $$x$$ to $$y$$.

This ensures smoothness of the solution set $$f^{-1}(y)$$.