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math214:02-10 [2020/02/10 08:40]
pzhou created 		math214:02-10 [2020/02/11 20:47] (current)
pzhou [Tubular Neighborhood Theorem]
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 ====== 2020-02-10, Monday ====== ====== 2020-02-10, Monday ======
-$$\gdef\wt \widetilde, \gdef\RM\backslash$+$$\gdef\wt\widetilde \gdef\RM\backslash$$ 
 ===== Whitney Approximation Theorem. ===== ===== Whitney Approximation Theorem. =====
 **Thm** Suppose $M$ is a smooth manifold, $F: M \to \R^k$ is a continuous map. $\delta: M \to \R$ is a positive function. Then, we can find a smooth function $\wt F:  M \to \R^k$, such that $|F(x) - \wt F(x)| < \delta(x)$ for all $x \in M$. Furthermore, if $F$ is already smooth on a closed set $A$, we can choose $\wt F = F$ on $A$.  **Thm** Suppose $M$ is a smooth manifold, $F: M \to \R^k$ is a continuous map. $\delta: M \to \R$ is a positive function. Then, we can find a smooth function $\wt F:  M \to \R^k$, such that $|F(x) - \wt F(x)| < \delta(x)$ for all $x \in M$. Furthermore, if $F$ is already smooth on a closed set $A$, we can choose $\wt F = F$ on $A$. 
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   - By extension of smooht function lemma, we may find a smooth function $F_0: M \to \R$ that agrees with $F$ on $A$. Define $$ U_0 = \{x \in M | | F(x) - F_0(x)| < \delta(x) \}.$$   - By extension of smooht function lemma, we may find a smooth function $F_0: M \to \R$ that agrees with $F$ on $A$. Define $$ U_0 = \{x \in M | | F(x) - F_0(x)| < \delta(x) \}.$$
   - For each point $x \in M$, define $U_x = \{ y \in M \RM A |F(y) - F(x)| < \delta(x)/2, \z{ and } \delta(x)/2 < \delta(y)  \}$.    - For each point $x \in M$, define $U_x = \{ y \in M \RM A |F(y) - F(x)| < \delta(x)/2, \z{ and } \delta(x)/2 < \delta(y)  \}$. 
-  - The collection of open sets $\{U_x\}$ covers $M \RM A$, we choose a countable subcover+  - The collection of open sets $\{U_x\}$ covers $M \RM A$, we choose a countable subcover $\{U_{x_i}\}_{i=1}^\infty$, and set $U_i = U_{x_i}$.  
 +  - Do a partition of unity $\{\varphi_0, \varphi_i\}$ subject to $\{U_i\}_{i=0}$. Then define the function as $$ \wt F(x) = F_0(x) \varphi_0(x) + \sum_{i=1}^\infty F(x_i) \varphi_i(x) $$ Claim, this function works.  
 + 
 +===== Tubular Neighborhood Theorem ===== 
 +The big plan: we want to be able to approximate a $C^0$ map $F: N \to M$ by a $C^\infty$ map $\wt F: N \to M$, such that the $C^0$ distance of $F$ and $\wt F$ is small. In order to do this, we first embed $M$ to $\R^m$ for some big $m$, 
 +$$ \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 $M$.  
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math214/02-10.1581352803.txt.gz · Last modified: 2020/02/10 08:40 by pzhou

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