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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), | + | - $S^1 \into \R^2 $. We view $S^1 = \R / 2\pi \Z$, and the map is $\theta \mapsto (\cos(\theta), |
- $(0, 2\pi) \into \R^2$. | - $(0, 2\pi) \into \R^2$. | ||
- | - 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) )$$, we see that $\lim_{t \to \pm \pi} \beta(t) = (0,0) = \beta(0)$. Hence it is example that an injective smooth immersion is not a smooth embedding. |
- Another example is the dense curve on a two torus. Pick an irrational number $c \in \R - \Q$. Then we can consider the map $\phi: \R \to T^2 = \R^2 / \Z^2$, $x \mapsto [(x, cx)]$. | - Another example is the dense curve on a two torus. Pick an irrational number $c \in \R - \Q$. Then we can consider the map $\phi: \R \to T^2 = \R^2 / \Z^2$, $x \mapsto [(x, cx)]$. | ||
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To show that such nice $v$ exists, we will consider all possible directions $[v] \in \R\P^{N-1}$, | To show that such nice $v$ exists, we will consider all possible directions $[v] \in \R\P^{N-1}$, | ||
- | $$ \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, | For both map, the source manifold is $2n$-dimensional, | ||