User:Geometry dude/Preimage Theorem

Theorem
Let $\varphi \colon M \to N$ be a smooth map between smooth manifolds $M,N$ and let $p \in N$ be a regular value of $\varphi$. Then the preimage $\varphi^{-1}(y)$ together with the natural inclusion $\iota:\varphi^{-1}(y) \to M$ is an embedded smooth submanifold of $M$ of dimension $\dim M - \dim N$.

Proof
Centr

We need: theorem 1.5.11; Prop 1.7.3; and the fact that the set where $\varphi$ is a submersion is open.

A subset of a second countable, Hausdorff topological space is second-countable, Hausdorff.

Proposition 1.5.15

Consider the map $\tilde \varphi: \varphi^{-1}(p) \to N$ obtained by restricting the domain of $\varphi$ to $\varphi^{-1}(p)$.

$\varphi$ is smooth and hence continuous.

$\lbrace p \rbrace \subseteq N$ is closed and since the continuous preimage of closed set is closed, $\varphi^{-1}(p)$ is closed.

Why is $\varphi^{-1}(p)$ open?

Injectivity of $\iota$ is trivial. <-> inclusions are not necessarily injective! hausdorff!!

Injectivity of $\iota_*$ follows from the constant rank theorem.

Why is $\iota$ open? Invariance of domain http://mathworld.wolfram.com/DomainInvarianceTheorem.html

Also known as: regular value theorem; regular level set theorem