0708-1300/Class notes for Thursday, October 4
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Movie Time
With the word "immersion" in our minds, we watch the movie "Outside In". Also see the movie's home, a talk I once gave, and the movie itself, on google video.
Class Notes
Definition
Let [math]\displaystyle{ \theta : M^m \rightarrow N^n\! }[/math] be a smooth map between manifolds. If for each [math]\displaystyle{ p \in M\! }[/math] the differential [math]\displaystyle{ d\theta_p : T_p M \rightarrow T_{\theta(p)} N\! }[/math] is surjective, [math]\displaystyle{ \theta\! }[/math] is called a submersion.
Theorem
If [math]\displaystyle{ \theta : M^m \rightarrow N^n\! }[/math] is a smooth map between manifolds and for some [math]\displaystyle{ p \in M\! }[/math] the differential [math]\displaystyle{ d\theta_p : T_p M \rightarrow T_{\theta(p)} N\! }[/math] is surjective then there exist charts [math]\displaystyle{ \phi : U \rightarrow U' \subset \mathbb{R}^m\! }[/math] and [math]\displaystyle{ \psi : V \rightarrow V' \subset \mathbb{R}^n\! }[/math] on [math]\displaystyle{ M\! }[/math] and [math]\displaystyle{ N\! }[/math] respectively such that
- [math]\displaystyle{ \phi(p) = 0\! }[/math]
- [math]\displaystyle{ \psi\left(\theta(p)\right) = 0\! }[/math]
- The diagram
commutes, where [math]\displaystyle{ \pi : \mathbb{R}^m = \mathbb{R}^n \times \mathbb{R}^{m-n} \rightarrow \mathbb{R}^n\! }[/math] is the canonical projection.
