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--- math214:02-10 [2020/02/11 20:44]
pzhou
+++ math214:02-10 [2020/02/11 20:47] (current)
pzhou [Tubular Neighborhood Theorem]
@@ Line 12: / Line 12: @@
 ===== Tubular Neighborhood Theorem =====
-The big plan: we want to be able to approximate a $C^0$ map $F: N \to M$ by a $C^\infty$ map $\wt F: N \to M$, such that the $C^0$ distance of $F$ and $\wt F$ is small.
+The big plan: we want to be able to approximate a $C^0$ map $F: N \to M$ by a $C^\infty$ map $\wt F: N \to M$, such that the $C^0$ distance of $F$ and $\wt F$ is small. In order to do this, we first embed $M$ to $\R^m$ for some big $m$,
+$$ \iota: M \into \R^m $$
+ then we smooth the composition $\iota \circ F: N \to M$, to get $\wt G: N \to \R^m$, $C^0$-close to the original image of $\iota(M)$. Finally, we project $\wt G(N)$ back onto $M$. This smoothing-then-project-back operation gives a smooth map from $N$ to $M$.

Lecture Notes