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math214:02-05 [2020/02/04 22:40]
127.0.0.1 external edit 		math214:02-05 [2020/02/05 23:22] (current)
pzhou
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 ** Example and Non-example of smooth embeddings** ** Example and Non-example of smooth embeddings**
  
-  - $S^1 \into \R^2$. We view $S^1 = \R / 2\pi \Z$, and the map is $\theta \mapsto (\cos(\theta), \sin(\theta))$. (Yes)+  - $S^1 \into \R^2 $. We view $S^1 = \R / 2\pi \Z$, and the map is $\theta \mapsto (\cos(\theta), \sin(\theta))$. (Yes)
   - $(0, 2\pi) \into \R^2$.  $\theta \mapsto (\cos(\theta), \sin(\theta))$. (Still yes)   - $(0, 2\pi) \into \R^2$.  $\theta \mapsto (\cos(\theta), \sin(\theta))$. (Still yes)
-  - The figure eight curve. one maps $(-\pi, \pi)$ to $\R^2$ as a figure eight, $$ \beta(t) = (\sin(2t), \sin(t) )$, we see that $\lim_{t \to \pm \pi} \beta(t) = (0,0) = \beta(0)$. Hence it is example that an injective smooth immersion is not a smooth embedding.+  - The figure eight curve. one maps $(-\pi, \pi)$ to $\R^2$ as a figure eight, $$ \beta(t) = (\sin(2t), \sin(t) )$$, we see that $\lim_{t \to \pm \pi} \beta(t) = (0,0) = \beta(0)$. Hence it is example that an injective smooth immersion is not a smooth embedding.
   - Another example is the dense curve on a two torus. Pick an irrational number $c \in \R - \Q$. Then we can consider the map $\phi: \R \to T^2 = \R^2 / \Z^2$, $x \mapsto [(x, cx)]$.     - Another example is the dense curve on a two torus. Pick an irrational number $c \in \R - \Q$. Then we can consider the map $\phi: \R \to T^2 = \R^2 / \Z^2$, $x \mapsto [(x, cx)]$.  
  
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 To show that such nice $v$ exists, we will consider all possible directions $[v] \in \R\P^{N-1}$, and use Sard theorem to say the bad directions are negligible. Let $\Delta_M \subset M \times M$ be the diagonal. And let $M \subset TM$ as the zero section. Then we have two maps To show that such nice $v$ exists, we will consider all possible directions $[v] \in \R\P^{N-1}$, and use Sard theorem to say the bad directions are negligible. Let $\Delta_M \subset M \times M$ be the diagonal. And let $M \subset TM$ as the zero section. Then we have two maps
-$$ \kappa: (M \times M) \RM \Delta_M \to \R \P^N, \quad (p,q) \mapsto [p-q] $$ +$$ \kappa: (M \times M) \RM \Delta_M \to \R \P^{N-1}, \quad (p,q) \mapsto [p-q] $$ 
-$$ \tau: TM \RM M \to  \R \P^N, \quad (p, w) \mapsto [w] $$+$$ \tau: TM \RM M \to  \R \P^{N-1}, \quad (p, w) \mapsto [w] $$
 For both map, the source manifold is $2n$-dimensional, and the target manifold is $N-1$ dimensional, by assumption $N-1 > 2n$, hence the image of $\kappa$ and $\tau$ are the singular value set, hence negligible.  For both map, the source manifold is $2n$-dimensional, and the target manifold is $N-1$ dimensional, by assumption $N-1 > 2n$, hence the image of $\kappa$ and $\tau$ are the singular value set, hence negligible. 
  
math214/02-05.1580884803.txt.gz · Last modified: 2020/02/04 22:40 by 127.0.0.1

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